Topological Structure of the Solution Sets for a Nonlinear Delay Evolution

Abstract

We consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact interval. The nonlinearity, having convex and closed values, is upper hemicontinuous with respect to the solution variable. A basic question on whether there exists a solution set carrying $R_{\delta }$-structure remains unsolved when the operator families generated by the principal part lack compactness. One of our main goals is to settle this question in the affirmative. Moreover, we prove that the solution map, having compact values, is an $R_{\delta }$-map, which maps any connected set into a connected set. It is then exploited to deal with the existence in the large for a nonlocal problem. Finally, several examples are worked out in detail, illustrating the applicability of our general results.

Multi-valued perturbations on stochastic evolution equations driven by fractional Brownian motions

“…By the assumption imposed on F, it can be derived that Sel F (u) = ∅ (see, e.g. [34,44]) and Sel The first main result of the present paper is contained in the following theorem.…”

“…The notion of R δ -structure can also be found in, e.g. [4,44]. We remark that the contractibility of a set implies that its homology is exactly the same as the homology of a one-point space, and it has a sufficiently simple structure.…”

Multi-valued random dynamics of partly dissipative reaction–diffusion system with discontinuous nonlinearity on RN

“…by noticing (3.6) and lemma 2.4, with the help of theorem 5.1 in [50]. At the same time, taking account into (3.6), (3.9) and (3.13), it can be derived that for every…”

“…This together with (2.2) yields that f ∈ F(t, u(t)) for a.e. t ∈ J, with the help of theorem 5.1 in [50].…”

“…These can be found in, e.g. the monographs [5,14,24] and the papers [6,50], and are provided in the present paper for easy reference. Moreover, we introduce a general theorem on the existence of pullback attractor for multi-valued random dynamical system, which will be used for the system under consideration.…”