Structure of solution sets to the nonlocal problems

Cheng et al. Boundary Value Problems (2016) 2016:26 DOI 10.1186/s13661-016-0529-3 RESEARCH Open Access Structure of solution sets to the nonlocal problems Yi Cheng1* , Ben Niu1 and Cuiying Li2 * Correspondence: Department of Mathematics, Bohai University, Jinzhou, 121012, P.R. China Full list of author information is available at the end of the article 1 Abstract This paper deals with the structural properties of the solution set for a class of nonlinear evolution inclusions with nonlocal conditions. For the nonlocal problems with a convex-valued right-hand side it is proved that the solution set is compact Rδ ; it is the intersection of a decreasing sequence of nonempty compact absolute retracts. Then for the cases of a nonconvex-valued perturbation term it is proved that the solution set is path connected. Finally some examples of nonlinear parabolic problems are given. MSC: 34B15; 34B16; 37J40 Keywords: evolution inclusions; nonlocal conditions; compact Rδ ; path connected 1 Introduction In this paper, we study the structural properties of the solution set for a class of nonlinear evolution inclusions initiated in [] with nonlocal conditions. The nonlocal Cauchy problems of evolution inclusions were investigated by Aizicovici and Lee [], Aizicovici and McKibben [], Aizicovici and Staicu [], García-Falset [], García-Falset and Reich [], and Paicu and Vrabie [], and by the references therein. However, as far as we know, not much work has been done for the topological structure of the solution set studied in this paper. In the past the topological structure of the solution set of differential inclusions in RN has been investigated by Himmelberg and Van Vleck [] and DeBlasi and Myjak []. Himmelberg and Van Vleck considered the topological structure of the solution set to the following differential inclusions:   ẋ(t) ∈ F t, x(t) , x() = , and they showed that the solution set was an Rδ set (see page ). For the Cauchy problems the topological structure of the solution set of evolution inclusions was examined primarily by Papageorgiou and Shahzad [], Andres and Pavlǎcková [], Chen et al. [] and Papageorgiou and Yannakakis [] in a Banach space. For the optimization of this subject, we refer the reader to []. The corresponding work for multivalued evolution systems is limited to the Cauchy or periodic problems. The paper is largely motivated by the work of [] in which the existence of solutions for first-order nonlinear evolution inclusions were proved. In [], they only showed that there exists at least one solution for the cases of a © 2016 Cheng et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Cheng et al. Boundary Value Problems (2016) 2016:26 Page 2 of 17 convex and of a nonconvex-valued perturbation term to the nonlocal problems, also prove that, under appropriate hypotheses, the extremal solution set is dense in the solution set of a system with a convexified right-hand side. However, it has to be noted that the topological structure of the solution set for this type of nonlinear evolution inclusions is an interesting problem which we intend to study in this paper. In this paper we first of all prove that the solution set of nonlinear time-dependent evolution inclusions with a convex-valued right-hand side is compact Rδ ; it is the intersection of a decreasing sequence of nonempty compact absolute retracts in C(I, H). Second, we go further and show that the solution set is path-connected in C(I, H) for the case of a nonconvex-valued orientor field. Finally, some examples are also given to illustrate the effectiveness of our results. In particular, control systems given in this paper with a prior feedback, and systems with discontinuities, have a built-in multivalued character which is modeled appropriately by evolution inclusions. 2 Preliminaries In this section we recall some basic definitions and facts from multivalued analysis, which will be needed later in this work. For further details we refer to the books of Hu and Papageorgiou [, ] and Zeidler []. Let X be a separable Banach space. The following notations are needed: Pf (X ) = {A ⊂ X : A is a nonempty; closed subset of X }, Pfc (X ) = {A ⊂ X : A is a nonempty; closed and convex subset of X }. Definition . A multifunction H:  → Pf (X ) is called ‘measurable’, if for all y ∈ X , R+ valued function x → d(y, H(x)) = inf{y – v, v ∈ H(x)} is measurable. For given A, B ∈ Pf (X ), let   dH (A, B) = max sup d(a, B), sup d(b, A) . a∈A b∈B The function dH : Pf (X ) × Pf (X ) → R+ is a metric on Pf (X ) and is called the ‘Hausdorff metric’. Definition . Let Z be a separable Banach space, a multifunction H : X → Z \∅ is called ‘h-upper semicontinuous’ (h-usc) if, for every x ∈ X , the function x → dH (H(x ), H(x)) is continuous. Let Z be a complete metric space, also H : X → Pf (Z) is called ‘h-continuous’ (resp. ‘h-Lipschitz’) if it is continuous (resp. Lipschitz) as a function from Z into Pf (Z, dH ). For details refer to []. Let (V , H, V ∗ ) be an evolution triple where the embedding V → H → V ∗ is compact. Let ·, · ∗ denote the pairing of an element x ∈ V ∗ and an element y ∈ V . Let ·· be the inner product on H, then ·, · ∗ = ·· , if x, y ∈ H. The dual space of Lp (I, V ) is Lq (I, V ∗ ) where  < q ≤ p < ∞, p + q = , and I is an interval in R. The norm in Banach space Lp (I, V ) will be denoted by  · Lp (I,V ) . Due to the reflexivity of V , both Lp (I, V ) and Lq (I, V ∗ ) are reflexive Banach spaces (see Zeidler [], p.). Cheng et al. Boundary Value Problems (2016) 2016:26 Page 3 of 17 We define a Banach space Wpq (I) = {x : x ∈ Lp (I, V ), ẋ ∈ Lq (I, V ∗ )} furnished with the norm xW = xLp (I,V ) + ẋLq (I,V ∗ ) . The pairing between Lp (I, V ) and Lq (I, V ∗ ) is denoted by ·, · ∗∗ . p p By SG we denote the set of all Lp (I, H)-selectors of a multifunction G, i.e. SG = {f ∈ p Lp (I, H) : f (x) ∈ G(x) a.e. for x ∈ I}. We say that the set SG is decomposable if χA f + χAc f ∈ p p p SG where (f , f , A) ∈ SG × SG × . We recall some of the topological concepts which will be used to characterize the solution set of the evolution inclusion. Definition . A subset A ⊂ X is called ‘path connected’, if for every x, y ⊂ A, there exists a path h : [, ] → A which joins x to y. For A ⊂ X nonempty, we claim that A is a retract of X , if there exists a continuous map f : X → A such that f |A = identity. It is clear to see that a retract A ⊂ X is closed. Definition . A closed subset A of X is called an absolute retract, if for any closed subset C in every metric space Y , every continuous map f : C → A can (...truncated)