TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

TABLE OF CONTENTS

Title and Author(s)		Page
	Positive solutions for second order four-point boundary value problems at resonance Liu Yang and Chunfang Shen ABSTRACT. Using Leggett-Williams norm-type theorem due to D. O'Regan and M. Zima, we establish the existence of positive solution for a class of second-order four point boundary value problem under different resonance conditions. An example is given to illustrate the main results.	1
	Impulsive problems for fractional evolution equations and optimal controls in infinite dimensional spaces JinRong Wang, Yong Zhou and Wei Wei ABSTRACT. In this paper, a class of impulsive fractional evolution equations and optimal controls in infinite dimensional spaces is considered. A suitable concept of a $PC$-mild solution is introduced and a suitable operator mapping is also constructed. By using a $PC$-type Ascoli-Arzela theorem, the compactness of the operator mapping is proven. Applying a generalized Gronwall inequality and Leray-Schauder fixed point theorem, the existence and uniqueness of the $PC$-mild solutions is obtained. Existence of optimal pairs for system governed by impulsive fractional evolution equations is also presented. Finally, an example illustrates the applicability of our results.	17
	Nonexpansive mappings on Hilbert's metric spaces Bas Lemmens ABSTRACT. This paper deals with the iterative behavior of nonexpansive mappings on Hilbert's metric spaces $(X,d_X)$. We show that if $(X,d_X)$ is strictly convex and does not contain a hyperbolic plane, then for each nonexpansive mapping, with a fixed point in $X$, all orbits converge to periodic orbits. In addition, we prove that if $X$ is an open $2$-simplex, then the optimal upper bound for the periods of periodic points of nonexpansive mappings on $(X,d_X)$ is $6$. The results have applications in the analysis of nonlinear mappings on cones, and extend work by Nussbaum and others.	45
	Lower and upper solutions to semilinear boundary value problems: an abstract approach Alessandro Fonda and Rodica Toader ABSTRACT. We provide an abstract setting for the theory of lower and upper solutions to some semilinear boundary value problems. In doing so, we need to introduce an abstract formulation of the Strong Maximum Principle. We thus obtain a general version of some existence results, both in the case where the lower and upper solutions are well-ordered, and in the case where they are not so. Applications are given, e.g. to boundary value problems associated to parabolic equations, as well as to elliptic equations.	59
	Positive solutions for generalized nonlinear logistic equations of superdiffusive type Antonio Iannizzotto and Nikolaos S. Papageorgiou ABSTRACT. We consider a generalized version of the $p$-logistic equation. Using variational methods based on the critical point theory and truncation techniques, we prove a bifurcation-type theorem for the equation. So, we show that there is a critical value $\lambda_*>0$ of the parameter $\lambda>0$ such that the following holds: if $\lambda>\lambda_*$, then the problem has two positive solutions; if $\lambda=\lambda_*$, then there is a positive solution; and finally, if $0<\lambda<\lambda_*$, then there are no positive solutions.	95
	Bifurcation of Fredholm maps I; the index bundle and bifurcation Jacobo Pejsachowicz ABSTRACT. We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $\beta(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $\beta(f)$ is derived from the index bundle of the linearization of the family along the trivial branch by means of the generalized $J$-homomorphism. Using the Agranovich reduction and a cohomological form of the Atiyah-Singer family index theorem, due to Fedosov, we compute the bifurcation index of a multiparameter family of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch. In this way we obtain criteria for bifurcation of solutions of nonlinear elliptic equations which cannot be achieved using the classical Lyapunov-Schmidt method.	115
	Bifurcation in the case of twofold degeneration Yakov Dymarskii and Dmytrii Nepiypa ABSTRACT. This paper studies typical (in a sense) bifurcations in the case of twofold degeneration linearized operator. We use an original approach of quasi-linear representation.	169
	On homotopy Conley index for multivalued flows in Hilbert spaces Zdzislaw Dzedzej and Grzegorz Gabor ABSTRACT. An approximation approach is applied to obtain a homotopy version of the Conley type index in Hilbert spaces considered by the first author and W. Kryszewski. The definition given in the paper is more elementary and, as a by-product, gives a natural connection between indices of Kunze and Mrozek in a finite-dimensional case. Some geometric properties of the index from a paper of the second author are discussed in an infinite dimensional situation. Some additional properties for gradient differential inclusions are also presented.	187