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.