In this work, we formulate the definition of periodicity for functions defined on isolated time scales. The introduced definition is consistent with the known formulations in the discrete and quantum calculus settings. Using the... more
In this work, we formulate the definition of periodicity for functions defined on isolated time scales. The introduced definition is consistent with the known formulations in the discrete and quantum calculus settings. Using the definition of periodicity, we discuss the existence and uniqueness of periodic solutions to a family of linear dynamic equations on isolated time scales. Examples in quantum calculus and for mixed isolated time scales are presented.
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In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time... more
In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton–Holt equation. The first main theorem provides conditions for the existence of a unique $ \omega $ -periodic solution that is globally asymptotically stable, which addresses the first Cushing–Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing–Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton–Holt model in the discrete and quantum cases, and it complements existing conclusions on periodic time scales. This work can furthermore guide other applications of the recently introduced periodicity on isolated time scales.
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The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which... more
The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which essentially improve a number of related ones from the literature. A couple of examples illustrate the value of the results obtained.
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In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time... more
In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton–Holt equation. The first main theorem provides conditions for the existence of a unique $ \omega $ -periodic solution that is globally asymptotically stable, which addresses the first Cushing–Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing–Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton–Holt model in the discrete and...
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In this paper, we introduce the epidemic model following the hypothesis of the disease flow Susceptible → Infected → Susceptible, short SIS, on time scales. After a brief introduction of time scales, we present dynamic systems... more
In this paper, we introduce the epidemic model following the hypothesis of the disease flow Susceptible → Infected → Susceptible, short SIS, on time scales. After a brief introduction of time scales, we present dynamic systems representing the SIS-model on time scales and derive its solution sets. We are discussing the stability of the steady states before investigating a modified SIS-model including a birth and death rate. Throughout, examples are used to illustrate the results. 2010 Mathematics Subject Classification: 34N05, 92D25
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In this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or... more
In this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.
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In this paper, we develop a new technique to deduce oscillation of a second-order noncanonical advanced differential equation by using established criteria for second-order canonical advanced differential equations. We illustrate our... more
In this paper, we develop a new technique to deduce oscillation of a second-order noncanonical advanced differential equation by using established criteria for second-order canonical advanced differential equations. We illustrate our results by presenting two examples.
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We present several new sufficient conditions for Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear dynamic equations for functions defined on a time scale with values in a Banach space.
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The mathematics of time scales has recently received much attention and holds great promise in a number of areas. In this paper we propose a new area of mathematics, namely the theory of stochastic dynamic equations, which unifies the... more
The mathematics of time scales has recently received much attention and holds great promise in a number of areas. In this paper we propose a new area of mathematics, namely the theory of stochastic dynamic equations, which unifies the theories of stochastic differential and difference equations. We give an example involving stochastic dynamic equations, namely an equation modeling a stock price. AMS Subject Classification: 60J65, 26E70, 60G05, 65C30, 39A50.
This paper is a continuation of the recent study by Bohner et al [9] on oscillation properties of nonlinear third order functional differential equation under the assumption that the second order differential equation is nonoscillatory.... more
This paper is a continuation of the recent study by Bohner et al [9] on oscillation properties of nonlinear third order functional differential equation under the assumption that the second order differential equation is nonoscillatory. We consider both the delayed and advanced case of the studied equation. The presented results correct and extend earlier ones. Several illustrative examples are included.
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ABSTRACT. In this paper, we present some new improvements of dynamic Opial-type inequalities of first and higher order on time scales. We employ the new inequalities to prove several results related to the spacing between consecutive... more
ABSTRACT. In this paper, we present some new improvements of dynamic Opial-type inequalities of first and higher order on time scales. We employ the new inequalities to prove several results related to the spacing between consecutive zeros of a solution and/or a zero of its derivative of a second-order dynamic equation with a damping term. The main results are proved by making use of a recently introduced new technique for Opial dynamic inequalities, the time scales integration by parts formula, the time scales chain rule, the time scales Taylor formula, and classical as well as time scales versions of Hölder’s inequality.
ABSTRACT. In this paper, we present some new generalizations of dynamic Opial-type inequalities of higher order on time scales. The results contain as special cases many of the results currently given in literature. As an application, we... more
ABSTRACT. In this paper, we present some new generalizations of dynamic Opial-type inequalities of higher order on time scales. The results contain as special cases many of the results currently given in literature. As an application, we apply these inequalities together with a Hardy-type inequality on time scales to establish some lower bounds of the distance between zeros of a solution and/or its derivatives for a fourth-order dynamic equation. AMS (MOS) Subject Classification. 34A40, 34N05, 39A10, 39A13, 26D10, 26D15.
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In this paper, we first prove a new dynamic inequality based on an application of the time scales version of a Hardy-type inequality. Second, by employing the obtained inequality, we prove several Gehring-type inequalities on time scales.... more
In this paper, we first prove a new dynamic inequality based on an application of the time scales version of a Hardy-type inequality. Second, by employing the obtained inequality, we prove several Gehring-type inequalities on time scales. As an application of our Gehring-type inequalities, we present some interpolation and higher integrability theorems on time scales. The results as special cases, when the time scale is equal to the set of all real numbers, contain some known results, and when the time scale is equal to the set of all integers, the results are essentially new.
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A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous well-known classical special functions are demonstrated.
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This paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations.... more
This paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.
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We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to... more
We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to Gleissner's approach. If the coefficients are constant, both solution methods yield the same result. After a brief introduction to time scales, we formulate the SIR (susceptible-infected-removed) model in the general time domain and derive its solution. In the discrete case, this provides the solution to a new discrete epidemic system, which exhibits the same behavior as the continuous model. The last part is dedicated to the analysis of the limiting behavior of susceptible, infected, and removed, which contains biological relevance.
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Abstract. In this paper, we study the oscillatory behavior of a class of third-order nonlinear delay differential equations (a(t)(b(t)y′(t))′) ′ + q(t)yγ (τ (t)) = 0. Some new oscillation criteria are presented by transforming this... more
Abstract. In this paper, we study the oscillatory behavior of a class of third-order nonlinear delay differential equations (a(t)(b(t)y′(t))′) ′ + q(t)yγ (τ (t)) = 0. Some new oscillation criteria are presented by transforming this equation to the first-order delayed and advanced differential equations. Employing suitable com-parison theorems we establish new results on oscillation of the studied equation. Assumptions in our theorems are less restrictive, these criteria improve those in the recent paper [Appl. Math. Comput., 202 (2008), 102-112] and related con-tributions to the subject. Examples are provided to illustrate new results. 1.
This study focuses on nonlocal boundary value problems for elliptic ordinary and par-tial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and... more
This study focuses on nonlocal boundary value problems for elliptic ordinary and par-tial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain pa-rameter. Several conditions are obtained that guarantee the maximal regularity and Fred-holmness, estimates for the resolvent, and the completeness of the root elements of dif-ferential operators generated by the corresponding boundary value problems in Banach-valued weighted Lp spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differ-ential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.
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We consider a nonoscillatory second-order linear dynamic equation on a time scale to-gether with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also... more
We consider a nonoscillatory second-order linear dynamic equation on a time scale to-gether with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equa-tion. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench. Copyright © 2007 M. Bohner and S. Stević. This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited. 1.
We give new Lyapunov-type inequalities for linear Hamiltonian systems on arbitrary time scales, which improve recently published results and hence all the related ones in the literature. As an application, we obtain new diconjugacy... more
We give new Lyapunov-type inequalities for linear Hamiltonian systems on arbitrary time scales, which improve recently published results and hence all the related ones in the literature. As an application, we obtain new diconjugacy criteria for linear Hamiltonian systems.
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We investigate the oscillation and boundedness of first and second order dy-namic equations with mixed nonlinearities. Our results extend and improve known results for oscillation of first and second order dynamic equations that have been... more
We investigate the oscillation and boundedness of first and second order dy-namic equations with mixed nonlinearities. Our results extend and improve known results for oscillation of first and second order dynamic equations that have been established by Agarwal and Bohner. Some examples are given to illustrate the main results.
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In this paper, sufficient criteria for the existence of multiple positive periodic solutions of a certain nonlinear dynamic system with feedback control are established. This is done by the Avery-Henderson fixed point theorem and the... more
In this paper, sufficient criteria for the existence of multiple positive periodic solutions of a certain nonlinear dynamic system with feedback control are established. This is done by the Avery-Henderson fixed point theorem and the LeggettWilliams fixed point theorem. By using the method of coincidence degree, sufficient conditions are derived ensuring the existence of at least one periodic solution of a more general nonlinear dynamic system with feedback control on time scales.
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The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point... more
The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases. Copyright q 2008 Elvan Akın-Bohner et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
This paper is concerned with the oscillation of certain second-order neutral dynamic equations on a time scale. Four new oscillation criteria are presented that supplement those results given in Arun K. Tripathy (Some oscillation results... more
This paper is concerned with the oscillation of certain second-order neutral dynamic equations on a time scale. Four new oscillation criteria are presented that supplement those results given in Arun K. Tripathy (Some oscillation results for second order nonlinear dynamic equations of neutral type, Nonlinear Anal. 71,[1727][1728][1729][1730][1731][1732][1733][1734][1735] 2009).
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Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions... more
Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index
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In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.
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In this paper we establish some new oscillation criteria for a second order nonlinear delay differential equation of Emden‐Fowler type with damping term. These results extend and improve some of the well-known results in the nondelay... more
In this paper we establish some new oscillation criteria for a second order nonlinear delay differential equation of Emden‐Fowler type with damping term. These results extend and improve some of the well-known results in the nondelay case. Our results in the delay case are new and can be applied to new classes of equations which are not covered by the known criteria for oscillation. Some selected examples are provided. AMS subject classification: 34C10, 34K11, 34K40.
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This paper is concerned with oscillation of a certain class of second-order differential equations with a sublinear neutral term. Two oscillation criteria and two illustrative examples are included. In particular, the results obtained... more
This paper is concerned with oscillation of a certain class of second-order differential equations with a sublinear neutral term. Two oscillation criteria and two illustrative examples are included. In particular, the results obtained improve those reported in the literature.
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In this paper, by applying Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness, we present some existence of weak solutions for a class of functional implicit fractional differential equations of... more
In this paper, by applying Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness, we present some existence of weak solutions for a class of functional implicit fractional differential equations of Hilfer–Hadamard type. AMS Subject Classifications: 26A33.
We shall establish some new criteria for the oscillation of solutions of the fourth-order difference equation 2 a(k) 2 x(k) + q(k)f (x (g(k))) = 0 with the property that x(k)=k 2 ! 0 as k!1.
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On a Hilbert space H, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scaleinvariant self-adjoint extension in H. We prove that there is a one-to-one... more
On a Hilbert space H, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scaleinvariant self-adjoint extension in H. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.
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We study surfaces parametrized by time scale parameters, obtain an integral fomula for computing the area of time scale surfaces, introduce delta integrals over time scale surfaces, and give sucient conditions that ensure existence of... more
We study surfaces parametrized by time scale parameters, obtain an integral fomula for computing the area of time scale surfaces, introduce delta integrals over time scale surfaces, and give sucient conditions that ensure existence of these integrals. AMS (MOS) Subject Classification. 26B15, 28A75, 34N05, 39A12.
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In this article, we introduce the concepts of Bochner and Bohr almost periodic functions in quantum calculus and show that both concepts are equivalent. Also, we present a correspondence between almost periodic functions defined in... more
In this article, we introduce the concepts of Bochner and Bohr almost periodic functions in quantum calculus and show that both concepts are equivalent. Also, we present a correspondence between almost periodic functions defined in quantum calculus and N0, proving several important properties for this class of functions. We investigate the existence of almost periodic solutions of linear and nonlinear q-difference equations. Finally, we provide some examples of almost periodic functions in quantum calculus.
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In this paper, using the recently introduced concept of periodic functions in quantum calculus, we study the existence of positive periodic solutions of a certain higher-order functional q-difference equation. Just as for the well-known... more
In this paper, using the recently introduced concept of periodic functions in quantum calculus, we study the existence of positive periodic solutions of a certain higher-order functional q-difference equation. Just as for the well-known continuous and discrete versions, we use a fixed point theorem in a cone in order to establish the existence of a positive periodic solution. This paper is dedicated to Professor George A. Anastassiou on the occasion of his 60th birthday
A new definition of a multi-valued logarithm on time scales is introduced for delta-differentiable functions that never vanish. This new logarithm arises naturally from the definition of the cylinder transformation that is also at the... more
A new definition of a multi-valued logarithm on time scales is introduced for delta-differentiable functions that never vanish. This new logarithm arises naturally from the definition of the cylinder transformation that is also at the heart of the definition of exponential functions on time scales. This definition will lead to a logarithm function on arbitrary time scales with familiar and useful properties that previous definitions in the literature lacked.
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Differential and integral calculus on time scales allows to develop a theory of dynamic equations in order to unify and extend the usual differential equations and difference equations. For single variable differential and integral... more
Differential and integral calculus on time scales allows to develop a theory of dynamic equations in order to unify and extend the usual differential equations and difference equations. For single variable differential and integral calculus on time scales, we refer the reader to the textbooks [4, 5] and the references given therein. Multivariable calculus on time scales was developed by the authors [2, 3]. In [3], we presented the process of Riemann multiple delta (nabla and mixed types) integration on time scales. In the present paper, we introduce the definitions of Lebesgue multi-dimensional delta (nabla and mixed types) measures and integrals on time scales. A comparison of the Lebesgue multiple delta integral with the Riemann multiple delta integral is given. Beside this introductory section, this paper consists of two sections. In Section 2, following [3], we give the Darboux definition of the Riemann multiple delta integral and present some needed facts connected to it. The m...
In this work, we investigate the existence of multiple solutions for a class of nonhomogeneous nonlocal systems via variational methods and critical point theory. We give a new criteria for guaranteeing that the nonhomogeneous nonlocal... more
In this work, we investigate the existence of multiple solutions for a class of nonhomogeneous nonlocal systems via variational methods and critical point theory. We give a new criteria for guaranteeing that the nonhomogeneous nonlocal systems with a perturbed term have at least three solutions in an appropriate Orlicz-Sobolev space. By presenting two examples we illustrate the results. AMS (MOS) Subject Classification. 35J60, 35J70, 46E35, 58E05, 68T40, 76A02. This Paper is Dedicated to Professor Ravi P. Agarwal on the Occasion of His 70th Birthday
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In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type... more
In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type and, more specifically, of Pach- patte type.
We are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic -Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev... more
We are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic -Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at least three weak solutions for the problem in a special case.
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In this paper, we propose a new tool for modeling and analysis in finance, introducing an impulsive discrete stochastic neural network (NN) fractional-order model. The main advantages of the proposed approach are: (i) Using NNs which can... more
In this paper, we propose a new tool for modeling and analysis in finance, introducing an impulsive discrete stochastic neural network (NN) fractional-order model. The main advantages of the proposed approach are: (i) Using NNs which can be trained without the restriction of a model to derive parameters and discover relationships, driven and shaped solely by the nature of the data; (ii) using fractional-order differences, whose nonlocal property makes the fractional calculus a suitable tool for modeling actual financial systems; (iii) using impulsive perturbations, which give an opportunity to control the dynamic behavior of the model; (iv) including a stochastic term, which allows to study the effect of noise disturbances generally existing in financial assets; (v) taking into account the existence of time delayed influences. The modeling approach proposed in this paper can be applied to investigate macroeconomic systems.
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ABSTRACT. We present a survey on recent results connected to linear Hamiltonian difference systems. In order to obtain unified results on continuous and discrete Hamiltonian systems we also consider a approach via so-called time scales.
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This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping terms. Several new oscillation criteria are established for such a class of differential equations under... more
This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping terms. Several new oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results.
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We discuss the use of inequalities to obtain the solution of certain variational problems on time scales.
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Two new existence results are presented for time scale boundary value problems on infinite intervals. The first is based on a growth argument and the second on an upper and lower solution idea.
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We generalize several standard properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Some of these properties were justified earlier under certain restrictions on the graininess of the time scale. In... more
We generalize several standard properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Some of these properties were justified earlier under certain restrictions on the graininess of the time scale. In this work, we have no restrictions on the graininess.
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We prove the Gr¨ uss inequality on time scales and thus unify corresponding continuous and discrete versions from the literature. We also apply our results to the quantum calculus case.
... The Beverton–Holt dynamic equation. ... In Section 3 we will provide some necessary time scales essentials. In the last two sections we will investigate the first and second Cushing–Henson conjectures on time scales, respectively. 2.... more
... The Beverton–Holt dynamic equation. ... In Section 3 we will provide some necessary time scales essentials. In the last two sections we will investigate the first and second Cushing–Henson conjectures on time scales, respectively. 2. The Beverton–Holt equation. ...
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ABSTRACT We establish some new criteria for oscillation and asymptotic behavior of solutions of even-order half-linear advanced differential equations. We study the case of canonical and the case of noncanonical equations subject to... more
ABSTRACT We establish some new criteria for oscillation and asymptotic behavior of solutions of even-order half-linear advanced differential equations. We study the case of canonical and the case of noncanonical equations subject to various conditions.
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676 RP. Agarwai. M. Bohner I Nonlinear Analysis 33 (1998) 675-692 Result 2 (Discrete version of JacobVs condition). Let Rk and Pk be symmetric и x n-matrices with real entries such that Rk is invertible for each keZ. Let M,N GZ with M... more
676 RP. Agarwai. M. Bohner I Nonlinear Analysis 33 (1998) 675-692 Result 2 (Discrete version of JacobVs condition). Let Rk and Pk be symmetric и x n-matrices with real entries such that Rk is invertible for each keZ. Let M,N GZ with M <N - 1. Then the discrete quadratic ...
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Dedicated to Monika and Albert and Tina, Carla, David, and Carrie with ... Martin Bohner Allan Peterson DYNAMIC EQUATIONS ON TIME SCALES An Introduction with Applications BlRKHAUSER Boston Basel Berlin This Om ... Martin Bohner Allan... more
Dedicated to Monika and Albert and Tina, Carla, David, and Carrie with ... Martin Bohner Allan Peterson DYNAMIC EQUATIONS ON TIME SCALES An Introduction with Applications BlRKHAUSER Boston Basel Berlin This Om ... Martin Bohner Allan Peterson Department of ...
... During the two-week-long 1997 Rocky Mountain Mathematics Consortium Summer Conference at the University of Wyoming, this book was presented by Calvin and Allan, who were the principal speakers for the conference. ... He also enjoys a... more
... During the two-week-long 1997 Rocky Mountain Mathematics Consortium Summer Conference at the University of Wyoming, this book was presented by Calvin and Allan, who were the principal speakers for the conference. ... He also enjoys a number of sports and hobbies. ...
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ABSTRACT The objective of this note is to present new Hille and Nehari type asymptotic criteria for a class of third-order delay dynamic equations on a time scale. Assumptions in our theorems are less restrictive, whereas the proofs are... more
ABSTRACT The objective of this note is to present new Hille and Nehari type asymptotic criteria for a class of third-order delay dynamic equations on a time scale. Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those reported in the literature. The results obtained extend and improve some previous results.
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676 RP. Agarwai. M. Bohner I Nonlinear Analysis 33 (1998) 675-692 Result 2 (Discrete version of JacobVs condition). Let Rk and Pk be symmetric и x n-matrices with real entries such that Rk is invertible for each keZ. Let M,N GZ with M... more
676 RP. Agarwai. M. Bohner I Nonlinear Analysis 33 (1998) 675-692 Result 2 (Discrete version of JacobVs condition). Let Rk and Pk be symmetric и x n-matrices with real entries such that Rk is invertible for each keZ. Let M,N GZ with M <N - 1. Then the discrete quadratic ...
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This work is concerned with the oscillation of a second-order neutral retarded dynamic equation on time scales. Some new oscillation criteria are presented that improve and complement those results reported in the literature.
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We consider 2n◊2n symplectic dierence systems together with associated discrete qua- dratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair... more
We consider 2n◊2n symplectic dierence systems together with associated discrete qua- dratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system dier by
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... A tape format for transferral of image data and source programs. L. Se´vigny a , C.Hedegaard b , J.-P. Gambotto c , M. Bohner d , S. Grinaker e , DE Lloyd f , LE Garn g and JA Knecht * , h. ... Available online 28 June 2006. Abstract.... more
... A tape format for transferral of image data and source programs. L. Se´vigny a , C.Hedegaard b , J.-P. Gambotto c , M. Bohner d , S. Grinaker e , DE Lloyd f , LE Garn g and JA Knecht * , h. ... Available online 28 June 2006. Abstract. ...
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Impulsive differential equations have been an object of intensive investigation during recent years, due to the wide possibilities for their application in various fields of science and technology. This monograph deals with periodic... more
Impulsive differential equations have been an object of intensive investigation during recent years, due to the wide possibilities for their application in various fields of science and technology. This monograph deals with periodic solutions of impulsive differential ...
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In this paper, we propose a study concerning a ratio-dependent model suggested by BD Aggarwala which describes the evolution of AIDS in the Canadian society in the case of extinction. On the basis of statistical data on HIV/AIDS published... more
In this paper, we propose a study concerning a ratio-dependent model suggested by BD Aggarwala which describes the evolution of AIDS in the Canadian society in the case of extinction. On the basis of statistical data on HIV/AIDS published by the authorities of ...
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The mathematics of time scales has recently received much attention and holds great promise in a number of areas. In this paper we propose a new area of mathematics, namely the theory of stochastic dynamic equations, which unifies the... more
The mathematics of time scales has recently received much attention and holds great promise in a number of areas. In this paper we propose a new area of mathematics, namely the theory of stochastic dynamic equations, which unifies the theories of stochastic differential and difference equations. We give an example involving stochastic dynamic equations, namely an equation modeling a stock
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ABSTRACT Using the discrete fractional sum operator, we establish some inequalities of Chebyshev type.
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By means of Riccati transformation techniques we establish some oscillation criteria for the second order EmdenñFowler dynamic equation on a time scale. Such equations contain the classical EmdenñFowler equa- tion as well as their... more
By means of Riccati transformation techniques we establish some oscillation criteria for the second order EmdenñFowler dynamic equation on a time scale. Such equations contain the classical EmdenñFowler equa- tion as well as their discrete counterparts. The classical oscillation results of Atkinson (in the superlinear case) and Belohorec (in the sublinear case) are extended in this paper to EmdenñFowler dynamic
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In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type... more
In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type and, more specifically, of Pach- patte type.
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Page 1. Series in Mathematical Analysis and Applications Edited by Ravi R Agarwal and Donal O'Regan VOLUME 5 OSCILLATION THEORY FOR SECOND ORDER DYNAMIC EQUATIONS Ravi R Agarwal, Said R. Grace and ...
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... A Boundary Value Problem and Green's Function 267 8.9. Monotone Methods 271 8.10. Open Problems 272 Chapter 9. Boundary Value Problems on Infinite... more
... A Boundary Value Problem and Green's Function 267 8.9. Monotone Methods 271 8.10. Open Problems 272 Chapter 9. Boundary Value Problems on Infinite Intervals 275 by Ravi Agarwal, Martin Bohner, and Donal O'Regan 9.1. Introduction 275 9.2. ...
We consider several dynamic equations and present methods on how to solve these equations. Among them are linear equations of higher order, Euler-Cauchy equations of higher order, logistic equations (or Verhulst equations), Bernoulli... more
We consider several dynamic equations and present methods on how to solve these equations. Among them are linear equations of higher order, Euler-Cauchy equations of higher order, logistic equations (or Verhulst equations), Bernoulli equations, Riccati equations, and Clairaut equations. In order to solve Bernoulli dynamic equations, we define an important product on the set of positively regressive functions and give
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Page 1. n yam ■^ 1^ 1 r 1 ImT Ravi P. Agarwal Martin Bohner Said R. Grace Donal D'Regan Page 2. Discrete Oscillation Theory Page 3. Page 4. Discrete Oscillation Theory Ravi P. Agarwal Martin Bohner Said R. Grace Donal O'Regan... more
Page 1. n yam ■^ 1^ 1 r 1 ImT Ravi P. Agarwal Martin Bohner Said R. Grace Donal D'Regan Page 2. Discrete Oscillation Theory Page 3. Page 4. Discrete Oscillation Theory Ravi P. Agarwal Martin Bohner Said R. Grace Donal O'Regan Page 5. ...
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We introduce the Laplace transform for an arbitrary time scale. Two particular choices of time scales, namely the reals and the integers, yield the concepts of the classical Laplace transform and of the classical Z-transform. Other... more
We introduce the Laplace transform for an arbitrary time scale. Two particular choices of time scales, namely the reals and the integers, yield the concepts of the classical Laplace transform and of the classical Z-transform. Other choices of time scales yield new concepts of our Laplace transform, which can be applied to find solutions of higher order linear dynamic equations with constant coeffficients. We present several useful properties of our Laplace transform and offer formulas for the Laplace transforms of many elementary functions, among them results for the convolution of two functions on a time scale, which is introduced in this paper as well.
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In this paper we study improper integrals on time scales. We also give some mean value theorems for integrals on time scales, which are used in the proof of an analogue of the classical Dirichlet{Abel test for improper integrals. AMS... more
In this paper we study improper integrals on time scales. We also give some mean value theorems for integrals on time scales, which are used in the proof of an analogue of the classical Dirichlet{Abel test for improper integrals. AMS (MOS) Subject Classication. 39A10.
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We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established,... more
We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established, among them the Euler condition, the Legendre condition, the strengthened Legendre condition, and the Jacobi condition. AMS (MOS) Subject Classication. 39A10.
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... to a .function A # 0. Then, .for any domain D C @ such that the hozdary nf D does not contci~ G zerc qf A, i: is possible to ,find a number M= M(D) E N szrch that for N > M each of the func-tf*J2s !i(N) 1--., : J -... more
... to a .function A # 0. Then, .for any domain D C @ such that the hozdary nf D does not contci~ G zerc qf A, i: is possible to ,find a number M= M(D) E N szrch that for N > M each of the func-tf*J2s !i(N) 1--., : J - trn.r rrwtcrc D ilir wnlr rumbcr uj'zrros as A has mside I). Since the ...
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... Finally, we wish to express our thanks to the staff of Marcel Dekker, Inc., in particular Maria Allegra and Elizabeth Draper, for their cooperation during the preparation of this book for publication. Ravi Agarwal Martin Bohner... more
... Finally, we wish to express our thanks to the staff of Marcel Dekker, Inc., in particular Maria Allegra and Elizabeth Draper, for their cooperation during the preparation of this book for publication. Ravi Agarwal Martin Bohner Wan-Tong Li Page 18. Page 19. ...
Page 1. A SURVEY OF EXPONENTIAL FUNCTIONS ON TIME SCALES MARTIN BOHNER ANDALLAN PETERSON1 Abstract. ... Anal. Appl., 2000. To appear. [3] E. Akın, L. Erbe, B. Kaymakçalan and A. Peterson. Oscillation Results for a Dynamic Equation on a... more
Page 1. A SURVEY OF EXPONENTIAL FUNCTIONS ON TIME SCALES MARTIN BOHNER ANDALLAN PETERSON1 Abstract. ... Anal. Appl., 2000. To appear. [3] E. Akın, L. Erbe, B. Kaymakçalan and A. Peterson. Oscillation Results for a Dynamic Equation on a Time Scale. ...
... A Boundary Value Problem and Green's Function 267 8.9. Monotone Methods 271 8.10. Open Problems 272 Chapter 9. Boundary Value Problems on Infinite... more
... A Boundary Value Problem and Green's Function 267 8.9. Monotone Methods 271 8.10. Open Problems 272 Chapter 9. Boundary Value Problems on Infinite Intervals 275 by Ravi Agarwal, Martin Bohner, and Donal O'Regan 9.1. Introduction 275 9.2. ...
We present an oscillation criterion for rst order delay dynamic equa- tions on time scales, which contains well-known criteria for delay dierential equa- tions and delay dierence equations as special cases. We illustrate our results by... more
We present an oscillation criterion for rst order delay dynamic equa- tions on time scales, which contains well-known criteria for delay dierential equa- tions and delay dierence equations as special cases. We illustrate our results by applying them to various kinds of time scales.
The aim of this paper is to present the relationship between the classical lineariza- tion and the optimal derivative of a nonlinear ordinary differential equation. An application is presented using the quadratic error.
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A necessary and sucient condition for the nonnegativity of the discrete qua- dratic functional corresponding to a symplectic dierence system is proved using the diagonalization method. AMS (MOS) Subject Classication. 39 A 10.
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Page 1. Proceeding! of the Edinburgh Mathematical Society (1999) 42, 349-374 © POSITIVE SOLUTIONS AND EIGENVALUES OF CONJUGATE BOUNDARY VALUE PROBLEMS by RAVI P. AGARWAL, MARTIN BOHNER* and PATRICIA JY WONG (Received 12th June 1997) ...
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INTRODUCTION: Current research is focusing on injectable osteoconductive materials. Injectable CaP cements offer a minimal-invasive use, but lack osteogenic properties. In order to create a combined osteoconductive / osteogenic bone... more
INTRODUCTION: Current research is focusing on injectable osteoconductive materials. Injectable CaP cements offer a minimal-invasive use, but lack osteogenic properties. In order to create a combined osteoconductive / osteogenic bone substitute we used in this study a synthetic, injectable and resorbable Brushite / ß-tricalcium- phosphate (β-TCP) scaffold (chronOS Inject) and impregnated it with a transglutaminase (plasmatransglutaminase - F XIII). We evaluated the activity of the osteogenic protein and the biomechanics of the mixture. METHODS: Activation study: In order to evaluate the reaction of the osteogenic protein F XIII to the fluid phase of chronOS Inject (sodium hyaluronate), different pH solutions (pH 4 - 7), sodium hyaluronate and chronOS Inject were mixed with F XIII and the protein activity and F XIII release was detected with ELISA. Biomechanical study: The injectability of the chronOS Inject / F XIII mixture was assessed measuring the force required to inject the mixt...
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Abstract The talk will focus on optimal control of systems of ordinary differential equations modeling biological systems. Here the differential equations model the dynamics of system response with some component of the system being under... more
Abstract The talk will focus on optimal control of systems of ordinary differential equations modeling biological systems. Here the differential equations model the dynamics of system response with some component of the system being under direct control. The optimal control problem consists on a criterion (the objective) that is to be maximized or minimized under constraints. The mathematical theory then provides methods for the best application of the controls along the time (dynamic optimization). Some recent applications to dengue and ...
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ABSTRACT
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ABSTRACT For a certain q-difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self-adjoint extensions, i.e., the so-called Friedrichs and Kreĭn extensions. We show that for the... more
ABSTRACT For a certain q-difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self-adjoint extensions, i.e., the so-called Friedrichs and Kreĭn extensions. We show that for the interval of parameters under consideration, the Friedrichs extension and the Kreĭn extension are distinct and give values of the parameter in the von Neumann formulas that correspond to those extensions and describe their resolvent operators. A crucial rôle in our investigation plays the fact that both the Friedrichs and the Kreĭn extensions are scale invariant.
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... PHILOS TYPE CRITERIA FOR SECOND–ORDER HALF–LINEAR DYNAMIC EQUATIONS SAID R. GRACE, RAVI P. AGARWAL, MARTIN BOHNER AND DONAL O'REGAN Abstract. ... Mathematical Inequalities & Applications www.ele-math.com... more
... PHILOS TYPE CRITERIA FOR SECOND–ORDER HALF–LINEAR DYNAMIC EQUATIONS SAID R. GRACE, RAVI P. AGARWAL, MARTIN BOHNER AND DONAL O'REGAN Abstract. ... Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com
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1. Let y = e rt , so that y = r e rt and y = r 2 e rt. Direct substitution into the differential equation yields (r 2 + 3r − 4)e rt = 0. Canceling the exponential, the characteristic equation is r 2 + 3r − 4 = 0. The roots of the equation... more
1. Let y = e rt , so that y = r e rt and y = r 2 e rt. Direct substitution into the differential equation yields (r 2 + 3r − 4)e rt = 0. Canceling the exponential, the characteristic equation is r 2 + 3r − 4 = 0. The roots of the equation are r = −4 , 1. Hence the general solution is y = c 1 e t + c 2 e −4t. 2. Let y = e rt. Substitution of the assumed solution results in the characteristic equation r 2 + 5r + 6 = 0. The roots of the equation are r = −3 , −2. Hence the general solution is y = c 1 e −2t + c 2 e −3t. 4. Substitution of the assumed solution y = e rt results in the characteristic equation 3r 2 − 4r + 1 = 0. The roots of the equation are r = 1/3 , 1. Hence the general solution is y = c 1 e t/3 + c 2 e t. 6. The characteristic equation is 9r 2 − 16 = 0 , with roots r = ±4/3. Therefore the general solution is y = c 1 e −4t/3 + c 2 e 4t/3. 8. The characteristic equation is r 2 − 4r − 4 = 0 , with roots r = 2 ± 2 √ 2. Hence the general solution is y = c 1 e (2−2 √ 2)t + c 2 e (2+2 √ 2)t. 9. Substitution of the assumed solution y = e rt results in the characteristic equation r 2 + 2r − 3 = 0. The roots of the equation are r = −3 , 1. Hence the general solution is y = c 1 e −3t + c 2 e t. Its derivative is y = −3c 1 e −3t + c 2 e t. Based on the
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For a number q bigger than 1, we consider a q-difference version of a second-order singular differential operator which depends on a real parameter. We give three exact parameter intervals in which the operator is semibounded from above,... more
For a number q bigger than 1, we consider a q-difference version of a second-order singular differential operator which depends on a real parameter. We give three exact parameter intervals in which the operator is semibounded from above, not semibounded, and semibounded from below, respectively. We also provide two exact parameter sets in which the operator is symmetric and self-adjoint,
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Page 1. Kneser's theorem in q-calculus This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2005 J. Phys. A: Math. Gen. 38 6729 (http://iopscience.iop.org/0305-4470/38/30/008)... more
Page 1. Kneser's theorem in q-calculus This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2005 J. Phys. A: Math. Gen. 38 6729 (http://iopscience.iop.org/0305-4470/38/30/008) Download ...
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Page 1. A SURVEY OF EXPONENTIAL FUNCTIONS ON TIME SCALES MARTIN BOHNER ANDALLAN PETERSON1 Abstract. ... Anal. Appl., 2000. To appear. [3] E. Akın, L. Erbe, B. Kaymakçalan and A. Peterson. Oscillation Results for a Dynamic Equation on a... more
Page 1. A SURVEY OF EXPONENTIAL FUNCTIONS ON TIME SCALES MARTIN BOHNER ANDALLAN PETERSON1 Abstract. ... Anal. Appl., 2000. To appear. [3] E. Akın, L. Erbe, B. Kaymakçalan and A. Peterson. Oscillation Results for a Dynamic Equation on a Time Scale. ...
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Page 17. Half-Linear Dynamic Equations RP Agarwal1, M. Bohner1, and P. Rehak2 1 Florida Institute of Technology, Department of Mathematical Sciences, Melbourne, FL 32901, USA 2 Mathematical Institute, Academy of Sciences ...
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... can be chosen to be the zero-matrix also. (ii)(H) actually consists of two equations: While Ax= AEx+ Bu is the equation of motion, Au= CExATu is called the Euler Equation. (iii) Note that the invertibility assumption on/Ak for ...
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Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the particular concepts of the h-Laplace and q-Laplace transforms. The convolution and inversion problems for these transforms are considered... more
Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the particular concepts of the h-Laplace and q-Laplace transforms. The convolution and inversion problems for these transforms are considered in some detail.
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ABSTRACT The objective of this note is to present new Hille and Nehari type asymptotic criteria for a class of third-order delay dynamic equations on a time scale. Assumptions in our theorems are less restrictive, whereas the proofs are... more
ABSTRACT The objective of this note is to present new Hille and Nehari type asymptotic criteria for a class of third-order delay dynamic equations on a time scale. Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those reported in the literature. The results obtained extend and improve some previous results.
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... During the two-week-long 1997 Rocky Mountain Mathematics Consortium Summer Conference at the University of Wyoming, this book was presented by Calvin and Allan, who were the principal speakers for the conference. ... He also enjoys a... more
... During the two-week-long 1997 Rocky Mountain Mathematics Consortium Summer Conference at the University of Wyoming, this book was presented by Calvin and Allan, who were the principal speakers for the conference. ... He also enjoys a number of sports and hobbies. ...
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ABSTRACT We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and... more
ABSTRACT We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and similarly conditions ensuring a bounded solution from which all other solutions diverge. We also consider periodic time scales and corresponding linear dynamic equations with periodic coefficients and prove similar statements about periodic solutions to which all other solutions converge or from which all other solutions diverge.
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... to a .function A # 0. Then, .for any domain D C @ such that the hozdary nf D does not contci~ G zerc qf A, i: is possible to ,find a number M= M(D) E N szrch that for N > M each of the func-tf*J2s !i(N) 1--., : J -... more
... to a .function A # 0. Then, .for any domain D C @ such that the hozdary nf D does not contci~ G zerc qf A, i: is possible to ,find a number M= M(D) E N szrch that for N > M each of the func-tf*J2s !i(N) 1--., : J - trn.r rrwtcrc D ilir wnlr rumbcr uj'zrros as A has mside I). Since the ...
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We derive a Reid roundabout Theorem for Sturm-liouville Difference Equations of higher order by applying general results on linear Hamiltonian Difference Systems. This theorem gives conditions that are equivalent to the positive... more
We derive a Reid roundabout Theorem for Sturm-liouville Difference Equations of higher order by applying general results on linear Hamiltonian Difference Systems. This theorem gives conditions that are equivalent to the positive definiteness of a certain discrete quardratic functional; among them disconjugacy of the related Sturm-Liouville Difference Equation and solvability of a certain discrete Riccati Matrix Equation.
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... C. Concept of the Highly-Parallel PSO Certain factors - like the neighborhood structure and the synchronization-have influence on the hardware imple-mentation. ... its own computation component so that the particles could determine... more
... C. Concept of the Highly-Parallel PSO Certain factors - like the neighborhood structure and the synchronization-have influence on the hardware imple-mentation. ... its own computation component so that the particles could determine their new positions parallel to each ...
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Two new existence results are presented for time scale boundary value problems on infinite intervals. The first is based on a growth argument and the second on an upper and lower solution idea.
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The Beverton-Holt model is a classical population model which has been considered in the literature for the discrete-time case. Its continuous-time analogue is the well-known logistic model. In this paper, we consider a quantum calculus... more
The Beverton-Holt model is a classical population model which has been considered in the literature for the discrete-time case. Its continuous-time analogue is the well-known logistic model. In this paper, we consider a quantum calculus analogue of the Beverton-Holt equation. We use a recently introduced concept of periodic functions in quantum calculus in order to study the existence of periodic solutions of the Beverton-Holt q-difference equation. Moreover, we present proofs of quantum calculus versions of two so-called Cushing-Henson conjectures.
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We generalize several standard properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Some of these properties were justified earlier under certain restrictions on the graininess of the time scale. In... more
We generalize several standard properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Some of these properties were justified earlier under certain restrictions on the graininess of the time scale. In this work, we have no restrictions on the graininess.
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ABSTRACT In this work, we generalize several properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Among them are translation theorems, transforms of periodic functions, integration of transforms,... more
ABSTRACT In this work, we generalize several properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Among them are translation theorems, transforms of periodic functions, integration of transforms, transforms of derivatives and integrals, and asymptotic values.
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We study a generalization of the brain-state-in-a-box (BSB) model for a class of nonlinear discrete dynamical systems where we allow the states of the system to lie in an arbitrary convex body. The states of the classical BSB model are... more
We study a generalization of the brain-state-in-a-box (BSB) model for a class of nonlinear discrete dynamical systems where we allow the states of the system to lie in an arbitrary convex body. The states of the classical BSB model are restricted to lie in a hypercube. Characterizations of equilibrium points of the system are given using the support function of a convex body. Also, sufficient conditions for a point to be a stable equilibrium point are investigated. Finally, we study the system in polytopes. The results in this special case are more precise and have simpler forms than the corresponding results for general convex bodies. The general results give one approach of allowing pixels in image reconstruction to assume more than two values.
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Page 1. Funkcialaj Ekvacioj, 53 (2010) 381394 Linear Integral Inequalities Involving Maxima of the Unknown Scalar Functions By Snezhana G. Hristova and Kremena V. Stefanova (Plovdiv University, Bulgaria) Abstract. This ...
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Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the Laplace transform on isolated time scales, prove several properties of the Laplace transform in this case, and establish a formula for... more
Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the Laplace transform on isolated time scales, prove several properties of the Laplace transform in this case, and establish a formula for the inverse Laplace transform. The concept of convolution is considered in more detail by proving the convolution theorem and a discrete analogue of the classical theorem of Titchmarsh for the usual continuous convolution.
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Research Interests: Applied Mathematics, Ordinary Differential Equations, Nonlinear dynamics, Nonlinear, Oscillations, and 10 morePARTIAL DIFFERENTIAL EQUATION, Stability of Functional Equation, Second Order, Finite Difference Method, Numerical Analysis and Computational Mathematics, Boundary Value Problems, Boundary Value Problem, Finite Difference Methods, Oscillation, and Ordinary Differential Equation
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We prove the Gr¨ uss inequality on time scales and thus unify corresponding continuous and discrete versions from the literature. We also apply our results to the quantum calculus case.
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ABSTRACT Some new oscillation results are presented that improve those reported in [C. Zhang, T. Li, B. Sun, and E. Thandapani, On the oscillation of higher-order half-linear delay differential equations, Appl. Math. Lett. 24 (2011)... more
ABSTRACT Some new oscillation results are presented that improve those reported in [C. Zhang, T. Li, B. Sun, and E. Thandapani, On the oscillation of higher-order half-linear delay differential equations, Appl. Math. Lett. 24 (2011) 1618–1621].
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ABSTRACT Consider the second order nonlinear neutral delay differential equationwhere τ and σ are nonnegative constants, and α is a quotient of positive odd integers. Some new oscillatory criteria for Eq. (E) are established. Several... more
ABSTRACT Consider the second order nonlinear neutral delay differential equationwhere τ and σ are nonnegative constants, and α is a quotient of positive odd integers. Some new oscillatory criteria for Eq. (E) are established. Several examples which dwell upon the importance of our results are also illustrated.
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... The Beverton–Holt dynamic equation. ... In Section 3 we will provide some necessary time scales essentials. In the last two sections we will investigate the first and second Cushing–Henson conjectures on time scales, respectively. 2.... more
... The Beverton–Holt dynamic equation. ... In Section 3 we will provide some necessary time scales essentials. In the last two sections we will investigate the first and second Cushing–Henson conjectures on time scales, respectively. 2. The Beverton–Holt equation. ...
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ABSTRACT We establish some new criteria for the oscillation of second-order Emden–Fowler neutral delay differential equations. We study the case of superlinear and the case of sublinear equations subject to various conditions. The results... more
ABSTRACT We establish some new criteria for the oscillation of second-order Emden–Fowler neutral delay differential equations. We study the case of superlinear and the case of sublinear equations subject to various conditions. The results obtained show that the presence of a neutral term in a differential equation can cause or destroy oscillatory properties. Several examples are provided to illustrate the relevance of new theorems. © 2013, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.
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Dedicated to Monika and Albert and Tina, Carla, David, and Carrie with ... Martin Bohner Allan Peterson DYNAMIC EQUATIONS ON TIME SCALES An Introduction with Applications BlRKHAUSER Boston Basel Berlin This Om ... Martin Bohner Allan... more
Dedicated to Monika and Albert and Tina, Carla, David, and Carrie with ... Martin Bohner Allan Peterson DYNAMIC EQUATIONS ON TIME SCALES An Introduction with Applications BlRKHAUSER Boston Basel Berlin This Om ... Martin Bohner Allan Peterson Department of ...