| Differential inclusion is an important branch of nonlinear analysis, which has ex-tensive practical applications in many fields such as differential equation, engineering, economics, optimal control and optimization theory. The existence of solutions, con-tinuation of solutions, dependence on initial conditions and parameters and topological properties of the set of solutions are basic contents of differential inclusions. The main content of this paper is to study the existence and topological properties of solutions for several types of differential inclusions. There are many results about the existence of solutions of differential inclusions, however, many of the results were concerned about the Cauchy or periodic problems.In recent years, differential equations or inclusions with nonlocal condition has paid more and more people’s attention. Since such nonlocal conditions contains a variety of boundary conditions, such as initial, periodic, reverse periodic, integral and multi-point average boundary value conditions, therefore, it has the generality and the universality in the practical application. As far as we know,[63] was the first article on the nonlocal condition, the author proved the existence and uniqueness of solution for a class of semilinear evolution equations. Next, the prelude of research of differential equations or inclusions with nonlocal conditions was opened, see [63-68].In [63], Paicu-Vrabie discussed the existence of solutions to the following semilinear evolution differential inclusion u(t)+Au(t)Эf(t) f(t)∈F(t,u) u(0)=g(u),(0.0.7) where A generates the compactness and equicontinuity of the semigroup and g has compactness, proved the existence of continuous solutions. So far, with respect to the existence, uniquness and stable of (moderate, strong, and classical) solutions of evolution equations with nonlocal conditions, and there are already many profound results in [62-69], but rarely with the existence of weak solutions. At the same time we note that many of these documents all assume that nonlocal function meets certain conditions of compactness and A is a strongly continuous semigroups of operator or accretive operator in studying the evolution equations or inclusions with nonlocal con-ditions. However, one may ask that whether there are the similar results without the assumption on the compactness or equicontinuity of the semigroup. This article will give a positive answer to this question. This article divided into four parts studys the existence of solutions of several types of differential equations and inclusions as well as the structure of the solution set.In the first part, we consider the solvability problem of a class of evolution e-quations with nonlocal conditions. Under the non-monotone of disturbance, sufficient conditions for the existence of solutions are given by using the nature of maximal monotone operators and quasi monotone.In this part, we consider the following evolution equation where I=[0,T]. Assume that(i) For all almost t E I, A:V'V*is monotone and demicontinuous.(ii) B:V'V*is both continuous and weakly continuous. Furthermore, for any sequence{un} in V with un-u in V, we have(iii) There exist positive constants C1,C2,C3and c4such that(iv) For almost all t∈I, g:H'H is a linear continuous function which satisfies‖g(u)‖≤‖u(T)‖.Theorem3.1Assumed (i)-(iv) are satisfied, the equation (0.0.8) has at least one solution.In the second part, we discuss a class of nonlinear evolution inclusions with local condition in Banach spaces. When A satisfies uniformly monotone condition, using techniques from multivalued analysis and fixed point theory, we establish the existence theorems for convex and nonconvex cases. In the nonconvex case, we obtain the suf-ficient conditions for the existence of solutions by using single-valued Leray-Schauder alternative theorem. In the convex case, the desirable result has been obtained by using set-valued Leray-Schauder alternative theorem. On the basis of Tolstonogov ex-tremal continuous selection theorem, we prove the existence of extremal solutions and the density of extremal solutions (the strong relaxation theorem). Moreover we apply the results obtained to a class of partial differential equations with a discontinuous right-hand side, the sufficient conditions of existence for solutions are given.In this part, we firstly consider the following evolution inclusions where A:I×V'V*, F:I×H'2V* We need the following hypotheses on the data problem(0.0.9).(H1)A:I×V'V*is an operator such that(i)f'A(t,x)is measurable;(ii)for each t∈I,the operator A(t,·):V'V*is uniformly monotone and hemicontinuous.that is,there exists a constant C2>0such that for all x1,x2∈V,and the map s'<A(t,x+sz),y>is continuous on[0,1]for all x,y,z∈V(iii)There exists a constant C1>0and a nonnegative function α(·)∈Lq(I)such that‖A(t,x)‖v*≤α(t)+C1‖x‖for all x∈V,a.e. on I.(iv)There exists C2>0,Co>0,b1(·)∈L1(t)such that or(H2)F:I×H'Pk(V*)is a multifunction such that(i)(t,x)'F(t,x)is graph measurable；(ii)for almost all t∈I,x'F(t,x)is lower semicontinuous(LSC)；(iii)there exists an nonnegative function b2(·)∈Lq(I)and a constant C3>0such that where1≤k<p.(H3)For all almost t∈I,there exists a continuous function φ:H'H such thatTheorem4.1Assumed(H1)一(H3)are satisfied,the equation(0.0.9)has at least one solution. Next, we consider the convex case, the assumption on F is the following:(H4) F:I×H'Pkc(V*)is a multifunction such that(i)(t,x)'F(t,x) is graph measurable;(ii) for almost all t∈I,x'F(t,x) has a closed graph; and (H2)(iii) hold.Theorem4.2Assumed (H1),(H3) and (H4) are satisfied, the equation (0.0.9) has at least one solution, moreover the solution set is weakly compact in C(I,H).Furthermore, we also consider the extremal problem of following evolution inclu-sion where extF(t,x) denotes the extremal point set of F(t,x). We need the following hypotheses:(H5) F:I×H'Pwkc(H) is a multifunction such that(i)(t,x)'F(t,x) is graph measurable;(ii) for almost all t∈I,x'F(t,x) is h-continuous; and (H2)(iii) hold.Theorem4.3Assumed (H1),(H3) and (H5) are satisfied, the equation (0.0.10) has at least one solution.For the relation theorem of problem (0.0.10), we need the following hypotheses.(H6) for each t∈I, there is an integrable function β(t)∈L∞(t) such that and (H5) hold.Theorem4.4Assumed (H1),(H3) and (H6) are satisfied, and β(t)≤θ<C, then Se=S, where the closure is taken in C(I, H).As an application of the previous results, we introduce an example.Let I=[0,b],Ω be a bounded domain in RN with smooth boundary(?)Ω. We consider the following nonlinear evolution equation with the interval boundary value condition whereThe hypotheses on the data of this problem are the following:(A) Since f(t,x,u) is not continuous, we introduce the functions f1(t,x,u) and f2(t,x,u) defined by (H6)(i) fi(t,x,u)(i=1,2) are Nemitsky-measurable,(i.e.,u:I×Ω'R for all measurable,u'fi(t,x,u)(i=1,2) is measurable.(ii) there exists a2(t)∈Lq(t)+,C>0, such thatwhere1≤k<p.Theorem4.5If the hypothesis (A),(H6) holds, then problem (0.0.11) has a nonempty set of solutions u∈LP(I,W1,p(Ω)) such that (?)∈Lg(I,W-1.q)).In the three part, we consider the existence of solution for a class of nonlinear evolution inclusions without monotone condition. When A satisfies pseudomonotone condition, using homotopy method, techniques from the maximal monotone and the pseudomonotone, and Banach fixed point theory, we establish the existence theorems and the density of solutions for evolution inclusions. On the basis of extremal contin-uous selection theorem, we prove the existence of extremal solutions and the strong relaxation theorem. Moreover we apply the results obtained to a nonlinear hyperbolic optimal control problem. We firstly consider the following evolution inclusions without monotone condition where,=[0,T],A:i×V'V*is a pseudomonotone operator,F:I×H'2V*is a multifunction φ:H'H is a linear continuous map for almost all t∈I.The precise hypotheses on the data of problem(0.0.12)are the following:(H1)A:I×V'V*is an operator such that:(i)t'A(t,x)is measurable;(ii)For almost all t∈I,A(t):V'V*is demicontinuous,pseudomonotone;(iii) For almost all t∈I,and all x∈V,we have with C1>0,α(·)∈Lq(I);(iv)There exists C2>0,Co>0,b1(·)∈L1(t)such that(H2)F:I×H'Pfc,(V*)is a multifunctiOn such that:(i) For every x∈H,t'F(t,x)is measurable;(ii)For almost all t∈I,'F(t,x)is sequentially closed in H×(?)(here by (?) we denote the Banach space V*furnished with the weak topology)；(iii) For almost all t∈I,all x∈H,we have where C3>0,b2(·)∈Lq(I)and1≤＝k<p.(H3)For all almost t∈I,φ:H'H is a linear continuous fuuctiion which satisfies‖g(u)‖≤‖u(T)‖.(H4)F:I×H'Pwkc(H)is a multifunction such that:(i)For every x∈H t'F(t,x)is measurable； (ii) For almost allt∈I,x'F(t,x) is h-continuous;(iii) For almost all t∈I, all x∈H,we havewhere C3>0, b2(·)∈Lq(I) and1≤k<p.(H5) F:I×H'Pwkc(H) is a multifunction such that:(iv) for each t∈I, there is an integrable function L:I'R_, such that and (H4) hold.Theorem5.1Assumed (H1)-(H3) are satisfied, then the solution set S of problem (0.0.12) is nonempty, weakly compact in Wpq and compact in C(I,H).Theorem5.2Assumed (H1),(H3) and (H4) are satisfied, the extremal solution set of equation (0.0.12) Se≠0.Theorem5.3Assumed (H1),(H3) and (H5) are satisfied, then Se=S, where the closure is taken in C(I,H).In this section as an application of the abstract theory, we study a nonlinear hyperbolic optimal control problem (distributed parameter system).Let I=[0,b], Z be a bounded domain in RN with smooth boundary. We consider the following optimal control problem with a state-dependent control constraint set: s.t. where Under the appropriate assumptions, we prove the optimal control of the above-mentioned problem by using the previous theorems.In the four part, we discuss the boundary value problems for a class of partial differential inclusions in Sobolev space. When the right-hand side satisfies some condi-tions, using techniques from multivalued analysis and fixed point theory, we establish the existence theorems for convex and nonconvex cases. In the nonconvex case, we obtain the sufficient conditions for the existence of solutions by using single-valued Leray-Schauder alternative theorem. In the convex case, the desirable results has been obtained by using set-valued Leray-Schauder alternative theorem. On the basis of ex-tremal continuous selection theorem, we prove the existence of extremal solutions and the density of extremal solutions (the strong relaxation theorem).In this part, we consider the following boundary value problem: We need the following hypotheses:(F1) H:Ω×R×RN'Pk(R)is a multifunction such that(ⅰ)(x,u,s)'H(x,u,s) is graph measurable;(ⅱ) for almost all x∈Ω,(u,s)'H(x,u,s) is LSC;(ⅲ) for a.e. x∈Ω, where b(x)∈LP(Ω), and either(1)0≤α,β<1,c1(x)∈Lp/(1-α),C2(x)∈Lp/(1-β)(Ω), or(2) α=β=1, Cmax{‖C1‖∞,‖c2‖∞}<1where C>0satisfies‖u‖1,p≤C‖-△u‖p. for every u∈W1,p(Ω).(F2) H:Ω×R×RN'Pkc(R) is a multifunction such that(ⅰ)(x,u,s)'H(x,u,s) is graph measurable; (ⅱ) for almost all x∈Ω,(u,s)'H(x,u,s) has a closed graph; and (F1)(ⅲ) holds.(F3) H:Ω×R×RN'Pkc(R) is a multifunction such that(ⅰ)(x,u,s)'H(x,u,s) is graph measurable;(ⅱ) for almost all x∈Ω,(u,s)'H(x,u,s) is h-continuous; and (F1)(ⅲ) holds, where p> N.Theorem6.1Assumed (F1) is satisfied, the equation (0.0.13) has at least one solution u∈W01，p(Ω).Theorem6.2Assumed (F2) is satisfied, the equation (0.0.13) has at least one solution, moreover the solution set is weakly compact in W01，p(Ω).Theorem6.3Assumed (F3) is satisfied, the extremal solution set of equation (0.0.13) Se≠(?).(F4) H:Ω×R×RN'Pkc(R) is s multifunction satisfying hypothesis (F3), and (ⅳ) for each x∈Ω, there is an integrable function ρ:Ω'R+such thatTheorem6.4Assumed (F4) is satisfied, and ρ(x)≤γ<λ(λ denotes the first eigen-value of negative Laplacian with Dirichlet boundary conditions), then Se=S, where the closure is taken in C(Ω). |
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