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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.

Analyse critique de : Lambert (Franklin) - Berends (Frits), Vous avez dit : sabbat de sorcières ? La singulière histoire des premiers Conseils Solvay / préface de Thibault Damour. – Les Ulis : edp sciences, 2019. – 322 p. – (Sciences & histoire). – 1 vol. broché de 16,5 × 24 cm. – 34,00 €. – isbn 978-2-7598-2371-0.

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. Universidade de Vigo/CISUG Agencia Estatal de Investigación | Ref. MTM2017-85054-C2-1-P

We present a general formulation of the averaging method in the setting of a semilinear equation L x = e N ( x , e ) , being L a linear Fredholm mapping of index zero. Our general approach provides new results even in the classical periodic framework. Among the applications we obtained there are: a partial answer to an open problem related to the Liebau phenomenon, the multiplicity of periodic solutions for a planar system with delay and the existence of solution for a nonlocal chemical reactor.

Analyse critique de : De Bock (Dirk) - Vanpaemel (Geert), Rods, Sets and Arrows. – Cham : Springer Nature Switzerland, 2019. – xxii, 293 p. – 1 vol. relié de 18 × 26 cm. – 105,99 €. – isbn 978-3-030-20598-0.

Les contributions de Paul Mansion aux mathématiques sont illustrées par son action pour l’emploi de la théorie des limites en géométrie élémentaire, par une description de son ouvrage Résumé d’analyse infinitésimale utilisé pour l’enseignement de cette branche à l’Université de Gand, par la discussion de son point de vue sur les infiniment petits et par l’étude de l’influence de son confrère De Tilly sur ses contributions à la géométrie non euclidienne. On discute enfin le rôle de Mansion comme animateur des mathématiques en Belgique. * * * The mathematical contributions of Paul Mansion to mathematics are illustrated by his action toward using the theory of limits in elementary geometry, by describing his textbook Résumé d’analyse infinitésimale used in his teaching of this discipline at the University of Gand, by discussing his viewpoint about the infinitesimals and by studying the influence of his fellow-member De Tilly on his contributions to non-Euclidian geometry. Finally, the role of Mansion in promoting mathematics in Belgium is emphasized.

This paper is devoted to study the existence of solutions and their regularity in the p(t)–Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois–Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.

We study the existence of homoclinic solutions for a class of Lagrangian systems $\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

The generalized ordinary differential equations (shortly GODEs), introduced by J. Kurzweil in 1957, encompass other types of equations. The first main result of this paper extends to GODEs some classical conditions on the existence of a periodic solution of a nonautonomous ODE. By means of the correspondence between impulse differential equations (shortly IDEs) and GODEs, we translate the result to IDEs. Instead of the classical hypotheses that the functions on the righthand side of an IDE are piecewise continuous, it is enough to require that they are integrable in the sense of Lebesgue, allowing such functions to have many discontinuities. Our second main result provides conditions for the existence of a bifurcation point with respect to the trivial solution of a periodic boundary value problem for a GODE depending upon a parameter, and, again, we apply such result to IDEs. The machinery employed to obtain the main results are the topological degree theory, tools from the theory of compact operators and an Arzela-Ascoli-type theorem for regulated functions.

For a smooth mapping \(f : \overline D \subset {\mathbb R}^n \to {\mathbb R}^n\) with 0∉f(∂D), D open, bounded with smooth boundary ∂D, the Kronecker index of f on ∂D is defined as the integral over ∂D of some (n − 1)-differential form depending upon f and its partial derivatives. For n = 2, it is the Cauchy index or winding number of f on ∂D, the algebraic number of turns of f around the closed curve ∂D. The Gauss linking number of two curves is a special case of the Kronecker index in \(\mathbb R^3\). Various expressions of the Kronecker index are given, as well as applications to the Ginzburg-Landau theory of liquid crystals. Writing the Kronecker index as an integral over \(\overline D\) leads in a natural way to the more general concept of the Brouwer degree for a continuous mappings \(f : \overline D \to {\mathbb R}^n\) without smoothness assumption upon ∂D. Its fundamental properties are established, including an axiomatic characterization and its localization to a neighborhood of an isolated zero, the Brouwer index. Applications are given to generalized implicit function theorems. An easy extension is made to mappings \(f : \overline D \subset X \to Y\) between oriented normed spaces X and Y of the same finite dimension, and some results on its computation are given, including reductions to situations of smaller dimension. Borsuk’s fruitful topological concept of retract leads to the obtention of the Brouwer fixed point theorem for continuous self-mappings of various classes of sets, and to the definition of an index of fixed points for continuous mappings defined in retracts of \({\mathbb R}^n\) like convex sets, wedges or cones in \({\mathbb R}^n\). The role of dimension is illustrated by the hairy ball theorem and the Brouwer degree is applied to the linking of curves and its use in various minimax theorems of critical point theory.

In many situations, the Brouwer degree can be combined with a suitable limit process to obtain existence results for the zeros and fixed points of mappings in some infinite-dimensional topological vector spaces. In this way, extensions of the Kakutani fixed point theorem are obtained for set-valued self-mappings of compact subsets in locally convex topological vector spaces (Ky Fan fixed point theorem), with their important special cases for single-valued mappings in locally convex topological vector spaces (Tychonov fixed point theorem) and normed spaces (Schauder fixed point theorem). Another useful and versatile consequence is an intersection property of sets, the Knaster-Kuratowski-Mazurkiewicz theorem and its multi-form generalizations. One of its applications is the Hartmann-Stampacchia theorem for variational inequalities, generalizing some conclusions of the minimization of functionals on convex sets. An useful extension for various problems in mechanics is given by the hemivariational inequalities of Panagiotopoulos. The study of equations in reflexive Banach spaces, and in particular the surjectivity of weakly or strongly coercive monotone mappings, is another useful consequence of the Brouwer degree, with applications to some quasilinear Dirichlet problems. Non-monotone mappings can be considered as well by a similar approach, including the study of stationary Navier-Stokes equations. An short introduction to the fruitful degree theory very similar to Brouwer’s one, developed by Leray and Schauder for the compact perturbations of the identity in normed spaces, is also given.

For a system of first order ordinary differential equations, the Poincare method reduces the search of its periodic solutions to the search of their initial conditions, a finite-dimensional problem. This approach is applied to bounded perturbations of linear problems, and when the Cauchy problem is not uniquely solvable, a variant due to Stampacchia is introduced. A theorem of Krasnosel’skii-Perov estimates the Brouwer degree of the Poincare operator in terms of the degree of the right-hand member of the differential system. It is applied to compute the Brouwer degree of some gradient mappings under various conditions. For the study of periodic solutions of first order differential systems, the method of guiding functions, introduced by Krasnosel’skii and its school, is the analogous of the Lyapunov method in stability. It reduces the existence of a periodic solution to the search of a suitable real function, whose gradient has a behavior akin to the one of the right-hand member of the differential system. Various situations are considered, depending upon the continuability or the non-continuability over the period of the solutions of the Cauchy problem. The method of guiding functions is also extended to a more general class of equations important in mechanics, the evolution complementarity systems, whose periodic solutions are searched in a suitable closed convex cone.

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, μ) depending upon a parameter μ, including the detection of the bifurcation points through linearization of F (theorem of Krasnosel’skii) and the global behavior of the bifurcation branches (theorem of Rabinowitz). Finally, the extension of the Brouwer fixed point theorem to set-valued mappings with convex values (theorem of Kakutani) is considered and applied to the existence of equilibria in game theory and economics following von Neumann and Nash.

The history of the Brouwer fixed point theorem, closely linked to the history of the Brouwer degree, is particularly intricated and is a case story showing the ‘nonlinear’ character of the evolution of mathematics. After describing and commenting the fundamental contribution of Brouwer, it is shown how it has been in one way or another anticipated by Poincare (with his n-dimensional intermediate value theorem), Hadamard (with his first published version of the theorem) and Bohl (with his no-retraction theorem for a n-cube). Subsequent new proofs of the fixed point theorem, by Alexander, by Birkhoff and Kellogg, and by Knaster, Kuratowski, Mazurliewicz, and its infinite-dimensional extensions, by Birkhoff and Kellogg, by Schauder and by Tychonov, are then described. The role of the Brouwer fixed point theorem in the theory of games and economics, in the hands of von Neumann, Kakutani, and Nash, and the numerical approximation of the fixed points by Scarf, and by Kellogg, Li and Yorke are then analyzed. A rediscovery of Poincare’s internmediate value theorem has led, not only to the proof of its equivalence to Brouwer fixed point theorem by Miranda, but also to a long and vivacious quarrel between Cinquini and Scorza-Dragoni for deciding if the Brouwer fixed point theorem is or is not a ‘topological theorem’. The chapter ends with the independent formulation of the Brouwer fixed point theorem in terms of variational inequalities, due to Harmann-Stampacchia and Karamardian, and to the endless quest for an ‘elementary proof’ of a theorem whose applications are exceptionally diverse and numerous.

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.

As the computation of the Brouwer degree is especially developed for two-dimensional mappings, the chapter collects a number of results in this direction. It starts with the method of lower and upper solutions for periodic solutions of second order differential equations, developed from the method of Stampacchia. It is followed by the results of Ortega about the use of the Brouwer index in the study of the stability of periodic solutions of second order equations of Duffing type, with convex or with periodic nonlinearities. Positive solutions of some perturbations of positively homogeneous Hamiltonian systems are then considered, in the line of Fabry and Fonda, and have for special case the perturbed asymmetric piecewise-linear second order equations considered by Fucik and Dancer. Special techniques for the computation of the Brouwer degree (or of the Kronecker index) are then developed for planar multilinear mappings, including the use of Sturm sequences when the mappings are not factorized. The case of mappings defined by holomorphic functions is also considered, and applied to questions of stability and control for linear ordinary differential and difference equations, associated to the Routh–Hurwitz and Schur–Cohn stability criteria for linear differential and difference systems.

Jean-Nicolas Noël (Dombrot-le-Sec, Vosges, 1783 - Liège, 1867) a consacré sa carrière à l’enseignement et à la didactique des mathématiques et de la physique, du niveau primaire jusqu’à l’université, respectivement en France, au Grand-Duché de Luxembourg et en Belgique. Ses manuels de mathématiques et de physique ont connu de nombreuses éditions. Appelé par l’Université de Liège en 1835, il a pris part activement, vers 1850, à une querelle sur l’enseignement des fondements du calcul différentiel et intégral et sur son utilisation en géométrie dans l’enseignement secondaire. Les partisans traditionnels de la méthode des limites (infinifuges) s’opposaient aux défenseurs de l’emploi des infiniment petits et des infiniment grands (infinicoles), dont Noël a été un ardent partisan. Ses contributions sont analysées à la lumière des points de vue actuels dans ce débat. * * * Jean-Nicolas Noël (Dombrot-le-Sec, Vosges, 1783 - Liège, 1867) has devoted his career to the teaching and didactics of mathematics and physics, from primary school to university, respectively in France, Luxemburg and Belgium. His mathematical and physical textbooks have seen many editions. Called by the University of Liège in 1835, he has actively taken part around 1850 to a dispute about how to teach the foundations of the calculus and to use it in geometry at the level of high schools. The traditional defenders of the method of limits (infinifuges) are opposed to the ones defending the use of infinitely small and infinitely large quantities (infinicoles), strongly supported by Noël. His contributions are analyzed in the light of the points of view of the present day in this debate.

This paper is concerned with the second order differential equation with not necessarily linear nonlocal boundary condition. The existence of solutions is obtained using the properties of the Leray–Schauder degree. The results generalize and improve some known results with linear nonlocal boundary conditions.

In this paper, we discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. It is well known that these equations can be realized as local degenerate elliptic problems in a half-cylinder of $$\mathbb {R}^{N+1}_{+}$$ together with a nonlinear Neumann boundary condition, through the extension technique in periodic setting. Exploiting this fact, and by combining the Moser iteration scheme in the nonlocal framework with an abstract multiplicity result valid for differentiable functionals due to Ricceri, we show that the problem under consideration admits at least three periodic solutions with the property that their Sobolev norms are bounded by a suitable constant. Finally, we provide a concrete estimate of the range of these parameters by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain.

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 ℂn to ℂn are obtained using Brouwer degree.

We prove the multiplicity of periodic solutions for an equation in a separable Hilbert space $H$, with $T$-periodic dependence in time, of the type $$ ddot x+{cal A}x+ abla_xV(t,x)=e(t),. $$ Here, ${cal A}$ is a semi-negative definite bounded selfadjoint operator, with nontrivial null-space ${cal N}({cal A})$, the function $V(t,x)$ is bounded above, periodic in $x$ along a basis of ${cal N}({cal A})$, with $ abla_xV$ having its image in a compact set, and $e(t)$ has mean value in ${cal N}({cal A})^perp$. Our results generalize several well-known theorems in the finite-dimensional setting, as well as a recent existence result by Boscaggin, Fonda and Garrione.

We consider the system of difference equations Δ ( Δ ⁢ u n - 1 1 - | Δ ⁢ u n - 1 | 2 ) = ∇ V n ( u n ) + h n , u n = u n + T ( n ∈ ℤ ) , $\Delta\bigg{(}\frac{\Delta u_{n-1}}{\sqrt{1-|\Delta u_{n-1}|^{2}}}\bigg{)}=% \nabla V_{n}(u_{n})+h_{n},\quad u_{n}=u_{n+T}\quad(n\in\mathbb{Z}),$ with Δ ⁢ u n = u n + 1 - u n ∈ ℝ N ${\Delta u_{n}=u_{n+1}-u_{n}\in{\mathbb{R}}^{N}}$ , V n = V n ⁢ ( x ) ∈ C 2 ⁢ ( ℝ N , ℝ ) ${V_{n}=V_{n}(x)\in C^{2}({\mathbb{R}}^{N},\mathbb{R})}$ , V n + T = V n ${V_{n+T}=V_{n}}$ , h n + T = h n ${h_{n+T}=h_{n}}$ for all n ∈ ℤ ${n\in\mathbb{Z}}$ and some positive integer T, V n ⁢ ( x ) ${V_{n}(x)}$ is ω i ${\omega_{i}}$ -periodic ( ω i > 0 ${\omega_{i}>0}$ ) with respect to each x i ${x_{i}}$ ( i = 1 , … , N ${i=1,\ldots,N}$ ) and ∑ j = 1 T h j = 0 ${\sum_{j=1}^{T}h_{j}=0}$ . Applying a modification argument to the corresponding problem with a left-hand member of p-Laplacian type, and using Morse theory, we prove that if all its solutions are non-degenerate, then the difference system above has at least 2 N ${2^{N}}$ geometrically distinct T-periodic solutions.

We identify a family of Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes such that the radially symmetric prescribed curvature problem with Neumann boundary condition is solvable on a ball of small radius. Such family contains some examples of interest in Cosmology.

Using Brouwer degree and convex analysis, we obtain geometric conditions for the existence of solutions of multipoint boundary value problems for n-dimensional difference systems of the formΔu(k)=f(k,u(k))(k=0,1,…,T-1),u(T)=∑k=0T-1g(k)u(k),when T is a positive integer, the vector fields f(k, u) point outside a bounded convex set of Rn on its boundary, and the diagonal matrix function g(k) satisfies suitable conditions, which cover the periodic boundary conditions.

The paper computes the Brouwer degree of some classes of homogeneous polynomials defined on quaternions and applies the results, together with a continuation theorem of coincidence degree theory, to the existence and multiplicity of periodic solutions of a class of systems of quaternionic valued ordinary differential equations. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

An analysis of the mathematical contributions of Juliusz Schauder through the writings of contemporary mathematicians, and in particular Leray, Hadamard, Banach, Randolph, Federer, Lorentz, Ladyzhenskaya, Ural'tseva, Orlicz, G\aa rding and Pietsch.