Jean Mawhin

Coincidence degree and nonlinear differential equations

Alternative problems : An historical perspective.- Coincidence degree for perturbations of Fredholm mappings.- A generalized continuation theorem and existence theorems for Lx = Nx.- Two-point boundary value problems : Nonlinearities without special structure.- Approximation of solutions - The projection method.- Quasibounded perturbations of Fredholm mappings.- Boundary-value problems for some semilinear elliptic partial differential equations.- Periodic solutions of ordinary differential equations with quasibounded nonlinearities and of functional differential equations.- Coincidence index, multiplicity and bifurcation theory.- Coincidence degree for k-set contractive perturbations of linear Fredholm mappings.- Nonlinear perturbations of fredholm mappings of nonzero index.

Topological Degree Methods in Nonlinear Boundary Value Problems

Introduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.

Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces

and falls in the scope of fixed point theory for the Hammerstein operators, i.e., operators which can be written KN with K : Z-+ X linear. References to works devoted to this theory can be found in the extensive bibliography given in the survey papers of Dolph and Minty [l] and of Ehrmann [2]. If X and Z are Banach spaces and L-lN is completely continuous on the closure cl Sz of some open bounded set D C X, a very powerful device for proving the existence of fixed points of L-IN is Leray-Schauder degree theory for mappings of the form I T with I the identity and T : cl i2 -+ X completely continuous. Since the publication of the basic and now classical paper of Leray and Schauder [3], their theory has been extended in several ways, A number of them conserve the form of the mapping but allow X to be a more general topological vector space (Rothe [4], Leray [5], Nagumo [6], Browder [7], Altman [8, 93, Klee [lo]), th eir approach remaining in the spirit

Multiple Solutions of the Periodic Boundary-value Problem for Some Forced Pendulum-type Equations

If f(x) = c > 0 and e is a constant t?, then integrating (0.1) over (0.272) after multiplication of both members of the equation by x’ shows that the only possible solutions are constant, and hence satisfy a sin x = C. Consequently, if we exclude the trivial translation by a multiple of 275 (0.1) will have no solution if ]dJ > a, one solution if lP] = a and two solutions if lc!l < a. In 0th er t erms, the intersection of the range of the operator d2/dt2 + c(d/dt) f u sin(.) acting on Zn-periodic functions of class g* with the subspace of constant functions in the space C([O, 2711) of real continuous functions on [0,27r] is the closed interval [--a, a], whose interior points are images of two distinct solutions and boundary points of one. We try to obtain similar results for the case where e is no more constant and

Coincidence degree and periodic solutions of neutral equations

This paper is devoted to the problem of existence of periodic solutions for some nonautonomous neutral functional differential equations. It is essentially an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations recently obtained by one of the authors [2] and of a generalized Leray-Schauder theory developed by the second one [3, 41. Although their proofs are surprisingly simple, the obtained results are nontrivial extensions to the neutral case of a number of recent existence theorems for periodic solutions of functional differential equations. In particular, Section 3 generalizes some existence criteria due to one of the authors [5] and a corresponding recent extension by J. Cronin [6], the example following Theorem 4.1 improves a condition for existence given by Lopes [ 141 for the equation of a transmission line problem, and Theorem 5.1 generalizes a result due to R. E. Fennel1 [7]. Lastly, criteria analogous to Theorem 5.2 for the retarded case can be found in [8]. For partly related results concerning periodic solutions of neutral functional differential equations, see [9].

A Continuation Approach To Superlinear Periodic Boundary-value-problems

This paper deals with the problem of the existence of T-periodic solutions for the first order differential system x’ = I;( t, x), (1.1) where F: [0, T] x R” + R” is a Caratheodory function. In what follows, we prove some results for the solvability of the periodic BVP in the case when the dimension ,of the space is even. Such a limitation is motivated by our interest in applications to the second order equation d’+ g(t, u, u’)=O, (1.2) which takes the form of (1.1) when it is written as the equivalent system u’ = v 0’ = -g(t, u, v). (1.3)

Periodic solutions of some vector retarded functional differential equations

has no nontrivial T-periodic solution (see the books of Krasnosel’skii [6], Reissig, Sansone, and Conti [14], and Roseau [15]). This result has been extended to the case of the vector retarded functional differential equation by Fennel1 [2]. When the requirement upon the linear part is not satisfied, supplementary conditions upon f are needed to ensure the existence of T-periodic solutions, as follows immediately from the Fredholm alternative [4] for the special case off independent of x. Such auxiliary conditions appear in papers of Lazer [7] (for a second order scalar differential equation), Ezeilo [l], Sedsiwy [16], Villari [19], and the author [8] (f or a third order scalar differential equation), Sedsiwy [17] and Reissig [ll, 121 (for the n-th order scalar case) and the same authors [18, 131 f or the corresponding vector case. Also, Fennel1 [2] has extended Lazer’s result and method to the retarded case. Those results are proved by various methods (Brouwer, Schauder, or Leray-Schauder fixed point theorems) which make the arguments rather tedious. On the other hand, all the equations considered in those papers are

Critical points of convex perturbations of some indefinite quadratic forms and semi-linear boundary value problems at resonance

Abstract Critical points of convex perturbations of indefinite quadratic forms are obtained from the dual least action principle. The main result leads to necessary and sufficient conditions for the existence of a critical point when the corresponding Euler equation is scalar or when the perturbation is strictly convex. Applications are given to periodic solution of hamiltonian systems and to systems of semi-linear beam equations. A global version of the averaging method is given.

On Periodic-solutions of Forced Pendulum-like Equations

The existence and multiplicity of T-periodic solutions of the forced pen- dulum equation y”(t) + q’(t) + A sin J(t) = e(t) (1) have been considered recently by a number of authors and we refer to [2,3] for the history and a survey of this problem. The main techniques used up to now have been upper and lower solutions, topological degree and variational methods. A precise and explicit characterizarion of the set of e E

Leray-Schauder degree: a half century of extensions and applications

The Leray-Schauder degree is defined for mappings of the form