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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.
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.
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.
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.
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.
We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an 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.
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.