HAL (Le Centre pour la Communication Scientifique Directe), Feb 20, 2023
In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The ... more In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The first one asserts that the Jacobian has a divergence structure. The second one, that some vector fields involving the cofactors of the Jacobian are divergence free. We illustrate the fundamental impact of these properties on research, from the times of Jacobi to our days.
An existence condition for a zero of holomorphic functions in a disk is stated and proved in a ve... more An existence condition for a zero of holomorphic functions in a disk is stated and proved in a very simple way using the mean value property. It contains as special cases Bolzano's theorem and Brouwer fixed point theorem in a disk for holomorphic functions, the fundamental theorem of algebra and an asymptotic condition for the existence of zeros of transcendental entire functions. An elementary proof of the used mean value property is given. For more information see https://ejde.math.txstate.edu/special/01/m2/abstr.html
After giving a new proof of the existence of a stable limit cycle for the relativistic Van der Po... more After giving a new proof of the existence of a stable limit cycle for the relativistic Van der Pol equation d dtẋ √ 1 −ẋ 2 + k(x 2 − 1)ẋ + x = 0, we find sufficient conditions upon f and g in order that the relativistic Liénard equation d dtẋ √ 1 −ẋ 2 + f (x)ẋ + g(x) = 0 has a stable limit cycle. The approach and the existence conditions are distinct from those recently obtained by Pérez-González, Torregrosa and Torres [8].
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz £asopis pro pes to van i mate mat iky, rod. 111 (1986), Praha
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign c... more The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions. A more complicated proof, using Cauchy's argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from C n to C n are obtained using Brouwer degree.
Proceedings of the American Mathematical Society, 2017
The solvability of the resonant nonlocal boundary value problem x = f (t, x, x), x(0) = 0, x (1) ... more The solvability of the resonant nonlocal boundary value problem x = f (t, x, x), x(0) = 0, x (1) = 1 0 x (s)dg(s), with f : [0, 1] × R n × R n → R n continuous, g = diag(g 1 ,. .. , g n), g i : [0, 1] → R of bounded variation, 1 0 dg i (s) = 1 (i = 1,. .. , n), is studied using the Leray-Schauder continuation theorem. The a priori estimates follow from the existence of an open bounded convex subset C ⊂ R n , such that, for each t ∈ [0, 1] and x ∈ C, the vector fields f (t, x, •) satisfy suitable geometrical conditions on ∂C. The special cases where C is a ball or a parallelotope are considered.
Mawhin Jean. Les héritiers de Pierre-François Verhuist : une population dynamique. In: Bulletin d... more Mawhin Jean. Les héritiers de Pierre-François Verhuist : une population dynamique. In: Bulletin de la Classe des sciences, tome 13, n°7-12, 2002. pp. 349-378
The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equ... more The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent.
Progress in Nonlinear Differential Equations and Their Applications, 2020
The estimation of the Brouwer degree for some classes of mappings in \({\mathbb R}^n\) starts wit... more The estimation of the Brouwer degree for some classes of mappings in \({\mathbb R}^n\) starts with the homogeneous polynomial mappings. The orientation-preserving mappings are limits of mappings having everywhere a positive Jacobian and contain as special cases the monotone mappings, the holomorphic mappings and the quaternionic monomials. The symmetry of a mapping provides useful information about its degree. The case of odd mappings considered by Borsuk is especially interesting, with applications to the discussion of elliptic differential operators, to the Lusternik-Schnirelmann covering theorem of the unit sphere in \({\mathbb R}^n\), the measure of non-compactness of the unit sphere in an infinite-dimensional Banach space and the Krasnosel’skii genus of symmetrical sets, important in critical point theory. For S1-equivariant mappings, a reduction formula for the computation of their Brouwer degree is obtained, which is useful for the study of closed orbits of Hamiltonian systems. The results on the degree of odd mappings give direct proofs of the Brouwer theorems on the invariance of domain and of dimension, as well as the Banach-Mazur homeomorphism theorem for locally one-to-one proper mappings in \({\mathbb R}^n\). The Leray product formula for the computation of the Brouwer degree of the composition of two mappings in \({\mathbb R}^n\) is applied to the Jordan-Brouwer separation theorem about the equality of the number of connected components of the complements of two disjoint compact sets of \({\mathbb R}^n\) and to the computation of the Brouwer degree of one-to-one mappings. An application to a class of mappings with orientation-preserving character occurring in the theory of elasticity is also given.
Mawhin Jean. La tentative belge d'édition des œuvres complètes d'Euler vue par Henri Bosm... more Mawhin Jean. La tentative belge d'édition des œuvres complètes d'Euler vue par Henri Bosmans. In: Bulletin de la Classe des sciences, tome 21, 2010. Le Père Henri Bosmans SJ (1852-1928) historien des mathématiques. pp. 105-133
Using some recent extensions of upper and lower solutions techniques and continuation theorems to... more Using some recent extensions of upper and lower solutions techniques and continuation theorems to the periodic solutions of quasilinear equations of p-Laplacian type, we prove the existence of positive periodic solutions of equations of the form (|xʹ|p-2xʹ)ʹ + f(x)xʹ + g(x) = h(t) with p > 1, f arbitrary and g singular at 0. This extends results of Lazer and Solimini for the undamped ordinary differential case.
Progress in Nonlinear Differential Equations and Their Applications, 2020
After extending the homotopy invariance of the Brouwer degree to situations where not only the ma... more After extending the homotopy invariance of the Brouwer degree to situations where not only the mapping but also its set of definition may depend upon a parameter, the Leray–Schauder continuation theorem is stated and proved, giving conditions under which a subcontinuum of the set of solutions \({\mathcal S} = F^{-1}(\{0\})\) connects the slice \({\mathcal D}_a\) to the slice \({\mathcal D}_b\) of the open bounded set \({\mathcal D} \subset {\mathbb R}^n \times [a,b]\). Applications are given to the method of a priori bounds for proving the existence of a zero of a mapping, and to the Leray–Schauder alternative. Important special cases are the perturbed linear mappings, the surjectivity of coercive and monotone mappings, boundary conditions on the mapping leading to the existence of a zero in the set, further fixed point theorems and degree conditions implying the uniqueness of the zero or the fixed point. Other applications are given to the bifurcation of the zeros of mappings F(x, ...
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
We survey recent results on the multiplicity of T-periodic solutions of differential systems of t... more We survey recent results on the multiplicity of T-periodic solutions of differential systems of the form " u p 1 − |u | 2 " + uF (t, u) = e(t) when F (t, u) is ω i-periodic with respect to u i (i = 1,. .. , N). Several techniques of critical point theory are used. 1. Motivation: the forced pendulum and corresponding systems The periodic problem for the forced pendulum equation u + a sin u = e(t), u(0) = u(T), u (0) = u (T) (1.1) has been for almost one century a source of inspiration for ordinary differential equations and nonlinear functional analysis, and a cornerstone for most nonlinear techniques (see e.g. [16, 18]). In particular its solutions are the critical points of the Lagrangian action functional L(u) := T 0 u 2 2 + a cos u + eu dt in the Sobolev space H 1 T = {u ∈ H 1 ([0, T ]) : u(0) = u(T)}. In 1922, Hamel [12] proved that for each e ∈ C([0, T ]) such that e := T −1 T 0 e(t) dt = 0, there exists at least one solution of (1.1) minimizing L(u) over T-periodic C...
A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem I... more A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem It is proved that the existence of such a couple (σ, τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.
We essentially use Lemma 2.1 which is proved in . Accordingly, the set Ω must be "a bounded domai... more We essentially use Lemma 2.1 which is proved in . Accordingly, the set Ω must be "a bounded domain in ℝ N (N ≥ ) with boundary ∂Ω of class C ", instead of "an open bounded set in ℝ N with boundary ∂Ω of class C ". This change does not affect the validity of the results stated in the paper. But, it affects the proof of Lemma 3.1. In this view, the only necessary modification is the following.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 20, 2023
In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The ... more In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The first one asserts that the Jacobian has a divergence structure. The second one, that some vector fields involving the cofactors of the Jacobian are divergence free. We illustrate the fundamental impact of these properties on research, from the times of Jacobi to our days.
An existence condition for a zero of holomorphic functions in a disk is stated and proved in a ve... more An existence condition for a zero of holomorphic functions in a disk is stated and proved in a very simple way using the mean value property. It contains as special cases Bolzano's theorem and Brouwer fixed point theorem in a disk for holomorphic functions, the fundamental theorem of algebra and an asymptotic condition for the existence of zeros of transcendental entire functions. An elementary proof of the used mean value property is given. For more information see https://ejde.math.txstate.edu/special/01/m2/abstr.html
After giving a new proof of the existence of a stable limit cycle for the relativistic Van der Po... more After giving a new proof of the existence of a stable limit cycle for the relativistic Van der Pol equation d dtẋ √ 1 −ẋ 2 + k(x 2 − 1)ẋ + x = 0, we find sufficient conditions upon f and g in order that the relativistic Liénard equation d dtẋ √ 1 −ẋ 2 + f (x)ẋ + g(x) = 0 has a stable limit cycle. The approach and the existence conditions are distinct from those recently obtained by Pérez-González, Torregrosa and Torres [8].
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz £asopis pro pes to van i mate mat iky, rod. 111 (1986), Praha
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign c... more The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions. A more complicated proof, using Cauchy's argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from C n to C n are obtained using Brouwer degree.
Proceedings of the American Mathematical Society, 2017
The solvability of the resonant nonlocal boundary value problem x = f (t, x, x), x(0) = 0, x (1) ... more The solvability of the resonant nonlocal boundary value problem x = f (t, x, x), x(0) = 0, x (1) = 1 0 x (s)dg(s), with f : [0, 1] × R n × R n → R n continuous, g = diag(g 1 ,. .. , g n), g i : [0, 1] → R of bounded variation, 1 0 dg i (s) = 1 (i = 1,. .. , n), is studied using the Leray-Schauder continuation theorem. The a priori estimates follow from the existence of an open bounded convex subset C ⊂ R n , such that, for each t ∈ [0, 1] and x ∈ C, the vector fields f (t, x, •) satisfy suitable geometrical conditions on ∂C. The special cases where C is a ball or a parallelotope are considered.
Mawhin Jean. Les héritiers de Pierre-François Verhuist : une population dynamique. In: Bulletin d... more Mawhin Jean. Les héritiers de Pierre-François Verhuist : une population dynamique. In: Bulletin de la Classe des sciences, tome 13, n°7-12, 2002. pp. 349-378
The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equ... more The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent.
Progress in Nonlinear Differential Equations and Their Applications, 2020
The estimation of the Brouwer degree for some classes of mappings in \({\mathbb R}^n\) starts wit... more The estimation of the Brouwer degree for some classes of mappings in \({\mathbb R}^n\) starts with the homogeneous polynomial mappings. The orientation-preserving mappings are limits of mappings having everywhere a positive Jacobian and contain as special cases the monotone mappings, the holomorphic mappings and the quaternionic monomials. The symmetry of a mapping provides useful information about its degree. The case of odd mappings considered by Borsuk is especially interesting, with applications to the discussion of elliptic differential operators, to the Lusternik-Schnirelmann covering theorem of the unit sphere in \({\mathbb R}^n\), the measure of non-compactness of the unit sphere in an infinite-dimensional Banach space and the Krasnosel’skii genus of symmetrical sets, important in critical point theory. For S1-equivariant mappings, a reduction formula for the computation of their Brouwer degree is obtained, which is useful for the study of closed orbits of Hamiltonian systems. The results on the degree of odd mappings give direct proofs of the Brouwer theorems on the invariance of domain and of dimension, as well as the Banach-Mazur homeomorphism theorem for locally one-to-one proper mappings in \({\mathbb R}^n\). The Leray product formula for the computation of the Brouwer degree of the composition of two mappings in \({\mathbb R}^n\) is applied to the Jordan-Brouwer separation theorem about the equality of the number of connected components of the complements of two disjoint compact sets of \({\mathbb R}^n\) and to the computation of the Brouwer degree of one-to-one mappings. An application to a class of mappings with orientation-preserving character occurring in the theory of elasticity is also given.
Mawhin Jean. La tentative belge d'édition des œuvres complètes d'Euler vue par Henri Bosm... more Mawhin Jean. La tentative belge d'édition des œuvres complètes d'Euler vue par Henri Bosmans. In: Bulletin de la Classe des sciences, tome 21, 2010. Le Père Henri Bosmans SJ (1852-1928) historien des mathématiques. pp. 105-133
Using some recent extensions of upper and lower solutions techniques and continuation theorems to... more Using some recent extensions of upper and lower solutions techniques and continuation theorems to the periodic solutions of quasilinear equations of p-Laplacian type, we prove the existence of positive periodic solutions of equations of the form (|xʹ|p-2xʹ)ʹ + f(x)xʹ + g(x) = h(t) with p > 1, f arbitrary and g singular at 0. This extends results of Lazer and Solimini for the undamped ordinary differential case.
Progress in Nonlinear Differential Equations and Their Applications, 2020
After extending the homotopy invariance of the Brouwer degree to situations where not only the ma... more After extending the homotopy invariance of the Brouwer degree to situations where not only the mapping but also its set of definition may depend upon a parameter, the Leray–Schauder continuation theorem is stated and proved, giving conditions under which a subcontinuum of the set of solutions \({\mathcal S} = F^{-1}(\{0\})\) connects the slice \({\mathcal D}_a\) to the slice \({\mathcal D}_b\) of the open bounded set \({\mathcal D} \subset {\mathbb R}^n \times [a,b]\). Applications are given to the method of a priori bounds for proving the existence of a zero of a mapping, and to the Leray–Schauder alternative. Important special cases are the perturbed linear mappings, the surjectivity of coercive and monotone mappings, boundary conditions on the mapping leading to the existence of a zero in the set, further fixed point theorems and degree conditions implying the uniqueness of the zero or the fixed point. Other applications are given to the bifurcation of the zeros of mappings F(x, ...
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
We survey recent results on the multiplicity of T-periodic solutions of differential systems of t... more We survey recent results on the multiplicity of T-periodic solutions of differential systems of the form " u p 1 − |u | 2 " + uF (t, u) = e(t) when F (t, u) is ω i-periodic with respect to u i (i = 1,. .. , N). Several techniques of critical point theory are used. 1. Motivation: the forced pendulum and corresponding systems The periodic problem for the forced pendulum equation u + a sin u = e(t), u(0) = u(T), u (0) = u (T) (1.1) has been for almost one century a source of inspiration for ordinary differential equations and nonlinear functional analysis, and a cornerstone for most nonlinear techniques (see e.g. [16, 18]). In particular its solutions are the critical points of the Lagrangian action functional L(u) := T 0 u 2 2 + a cos u + eu dt in the Sobolev space H 1 T = {u ∈ H 1 ([0, T ]) : u(0) = u(T)}. In 1922, Hamel [12] proved that for each e ∈ C([0, T ]) such that e := T −1 T 0 e(t) dt = 0, there exists at least one solution of (1.1) minimizing L(u) over T-periodic C...
A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem I... more A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem It is proved that the existence of such a couple (σ, τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.
We essentially use Lemma 2.1 which is proved in . Accordingly, the set Ω must be "a bounded domai... more We essentially use Lemma 2.1 which is proved in . Accordingly, the set Ω must be "a bounded domain in ℝ N (N ≥ ) with boundary ∂Ω of class C ", instead of "an open bounded set in ℝ N with boundary ∂Ω of class C ". This change does not affect the validity of the results stated in the paper. But, it affects the proof of Lemma 3.1. In this view, the only necessary modification is the following.
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