Structure of the solution set for a partial differential inclusion

Cheng et al. Advances in Difference Equations (2015) 2015:380 DOI 10.1186/s13662-015-0723-0 RESEARCH Open Access Structure of the solution set for a partial differential inclusion Yi Cheng1* , Ravi P Agarwal2,3 , Afif Ben Amar4 and Donal O’Regan5 * Correspondence: Department of Mathematics, Bohai University, Jinzhou, 121013, P.R. China Full list of author information is available at the end of the article 1 Abstract In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compact Rδ if the perturbation term of the related partial differential inclusion is convex, and its solution set is path-connected if the perturbation term is nonconvex. MSC: 34B15; 34B16; 37J40 Keywords: biharmonic problem; differential inclusion; set-valued mapping; path-connected; compact Rδ 1 Introduction In this paper, we examine the following biharmonic problem of the partial differential inclusion: ⎧  ⎪ ⎨  u ∈ H(x, u, ∇u, u) u =  on ∂, ⎪ ⎩ ∂u =  on ∂. ∂n a.e. in , (.) Here  is a bounded domain in RN with a smooth boundary ∂, and H : ×R×RN ×R → R \ {∅} is a set-valued map. Biharmonic equations with Dirichlet boundary conditions were studied by Lions-Magenes [, ], Mozolevski-Süli [, ], Amrouche-Fontes [], and Amrouche-Raudin [, ]. Boundary value problems involving partial differential equations with discontinuous nonlinearities which may be reduced to boundary value problems for partial differential inclusions were studied by Carl-Heikkilä [, ] and Chang [, ] (we refer the reader also to the work of Marano [, ]). In [], Xue-Cheng studied periodic problems for a nonlinear evolution inclusion, defined on an evolution triple of spaces, driven by a monotone operator, and with a perturbation term which is multivalued. They established existence theorems for periodic solutions, extremal periodic solutions and a strong relaxation theorem in Banach spaces, which are similar to those in Xue-Yu [] in infinite dimensional spaces. In [], Cheng-Cong-Xue considered the following boundary © 2015 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. Advances in Difference Equations (2015) 2015:380 Page 2 of 18 value problem:  u ∈ G(x, u, ∇u) u =  on ∂, a.e. in , (.) and they established the existence of solutions for inclusions with convex- and nonconvexvalued perturbations, extremal solutions, and a strong relaxation theorem in a strong solution sense. The multivalued term in problem (.) that we consider contains not only the gradient but also a Laplacian item, and we obtain results in a weak solution sense. The Lipschitz condition (H(F) (iv) in []) with respect to the second variable u of the multifunction G is essential to get a strong relaxation theorem in Cheng-Cong-Xue []. However, we only need a one-sided Lipschitz condition to get this result. Furthermore, the topological structure of the solution set is discussed (this is not considered in [] and []). Inspired by Cheng-Cong-Xue [], in this paper we prove existence theorems for both ‘convex’ and ‘nonconvex’ cases by using techniques from multivalued analysis and fixed point theory. For related works on this subject, we refer the reader to [, –] and the references therein. Based on the Baire category method, De Blasi and Pianigiani in [] gave an existence result for the following problem:  ∇u ∈ ext F(x, u) a.e. x ∈ , u(x) = ϕ(x), x ∈ ∂. (.) In this paper, we will also consider the differential inclusion in which H(x, u, ∇u, u) will be replaced by its extreme point set ext H(x, u, ∇u, u). We show that the resulting problem always has a solution (‘extremal solutions’) and the solution set is dense in the solution set of the convexified version of the problem (‘strong relaxation theorem’). Also we address the structural properties of the solution sets for this type of biharmonic inclusion problem. In [], Himmelberg and Van Vleck studied the topological structure of the solution set in RN for the ordinary differential inclusions:   ẋ(t) ∈ F t, x(t) , x() = , and proved that the solution set is an Rδ -set. For Cauchy problems the topological structure of the solution set of evolution inclusions was examined by Bothe [], AndresPavlackova [], Gabor-Grudzka [], and Chen-Wang-Zhou [] in a Banach space, Bakowska-Gabor [], and O’Regan [] in Fréchet spaces. We also refer the reader to the works of Papageorgiou-Shahzad [] for the first-order evolution inclusion and Papageorgiou-Yannakakis [] for the second-order evolution inclusion where the structure of solution sets was discussed. Following their lead, in this paper, we obtain the Rδ -structure of the solution set for a biharmonic differential inclusion based on the space variable x ∈ . We prove that the solution set of the biharmonic inclusion problem in the convex-valued case is compact Rδ in C(), and the solution set is path-connected in the case of a nonconvex-valued orientor field. The plan of our paper is as follows. In Section , we collect some preliminary results which will be used in this work. In Section , we present some basic assumptions and Cheng et al. Advances in Difference Equations (2015) 2015:380 Page 3 of 18 existence theorems for the both convex and nonconvex multivalued terms. Here, our results are based on the Leray-Schauder alternative. In Section , a relaxation theorem is established. Finally the properties of the solution set is given in Section . 2 Preliminaries In this section, we introduce some basic definitions and facts which are essential tools in the later sections; see Hu-Papageorgiou [] for details. Let RN (N ≥ ) be the N -dimensional real Euclidean space. Throughout this paper the symbol  denotes a nonempty, bounded, open set of RN , with a smooth boundary ∂. Moreover, from now on, ‘measurable’ simply means Lebesgue measurable. Given two nonnegative constants k, p ≥ , we denote by W k,p () the space of all real-valued functions defined on  whose weak partial derivatives up to the order k lie in Lp (), ∂u equipped with W k,p () the usual norm  · k,p . If u ∈ W ,p (), we set u = ni= ∂x , i ∂u N ∇u = grad u = ( ∂xi )i= . For any real number p > , we denote by q the dual exponent of p (and throughout the paper we assume p > ). Let V be a Hausdorff topological space and a multifunction F :  (...truncated)