非局部条件下若干微分包含解的存在性
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摘要
微分包含是非线性分析理论的一个重要分支,它在微分方程、工程技术、国民经济、最优控制及控制论等领域有着广泛应用.解的存在性、解对初值的连续依赖性、解的渐近行为以及解集的拓扑结构等方面是微分包含理论研究的基本内容.关于微分包含解的存在性已经有了许多的研究成果,但其中大部分研究成果都是关注Cauchy问题或周期问题.
近年来,非局部化条件下的微分方程或包含越来越受到人们的关注.这种非局部化条件包含了许多边值条件,比如初值、周期、反周期、积分、多点平均等边值条件,因此,此条件更具有一般性,在实际应用上也更具有广泛性.就我们所知,最早研究非局部条件下发展方程的是文献[63],作者Byszewski研究了一类半线性发展方程解的存在唯一性.从此,拉开了研究非局部化条件下微分方程或包含的序幕,请参见文献[62-68].
在文献[62]中,Paicu-Vrabie讨论了下面非局部条件的发展包含u(t)+Au(t)Эf(t) f(t)∈F(t,u) u(0)=g(u),其中要求算子A生成紧连续算子半群和非局部函数g具有紧性,证明了这类发展包含连续解的存在性.到目前为止,关于非局部条件的发展方程解(适度解、强解、古典解)的存在性、唯一性、稳定性,已经有了不少深刻结果,但很少涉及Banach空间里弱解的存在性.同时我们注意到这些文献中,在研究具有非局部初始条件的发展方程或包含问题时,不少都假设非局部函数满足一定的紧性条件和算子A强连续算子半群生成元或增生算子.然而,“在非局部函数失去紧性条件和算子A不具有这种结构条件下,相应的非局部问题是否存在解”本文将对这个问题给出一个肯定的回答.本文分四个部分主要研究了几类微分方程及包含解的存在性以及解集的结构.
第一部分我们考虑一类非局部发展方程可解性问题.在扰动项为非单调的情况下,运用极大单调算子以及拟单调算子的性质,给出解存在的充分条件.这部分考察下面发展方程：
其中I=[0,丁].假设
(i)对几乎所有t∈I,A:V→V~*是单调的且半连续.
(ii)B：V→V*是半连续和弱连续,并且对于在V里弱收敛函数列un→u,有
(iii)存在非负常数C_1,C_2,C_3,C_4使得
在I上几乎处处成立.(iv)对几乎所有t∈I,g：H→H是线性连续函数且‖g(u)‖≤‖(T)‖.
定理3.1如果假设(i)-(iv)成立,则问题(0.0.1)至少有一个解.
第二部分我们在Banach空间中讨论了一个非线性发展包含的非局部问题.当非线性算子A满足一致单调条件时,借助于集值分析理论和不动点定理,得到了凸和非凸两种情况下解的存在性定理.对于非凸情形,使用单值的Leray-Schauder替换定理获得解存在的充分条件.对于凸情形,利用集值的Leray-Schauder替换定理获得同样结论.利用Tolstonogov端点连续选择定理,证明了端点解的存在性和强松驰定理,并将得到的结论应用于右端项不连续情况下偏微分方程,给出了这类偏微分方程解的存在性.
这部分首先考虑下面发展包含：
其中算子A:I×V→V~*,F:I×H→2~(V*)
(H1)算子A：I×V→V*满足：
(i)t→A(t,χ)是可测的；
(ii)对于几乎所有t∈I,A(t):V→V_*是一致单调且半连续,即存在一个常数C>0,使得对于所有χ1,χ2∈V有
(iii)对于所有χ∈V,存在一个常数C1>0和一个非负函数a(·)∈L。(I)使得
(iv)对于所有u∈V,存在C_2>0,C_0,0,b_1(·)∈L_1(t)使得或
(H2)F:I×H→P_k(V_*)是一个集值函数且满足：
(i)(t,χ)→F(t,χ)是图像可测的；
(ii)对于几乎所有t∈I,χ→F(l,χ)是下半连续;
(iii)对于所有χ∈V,存在一个常数C3>0和一个非负函数b2(·)∈Lq(I)使得
其中1≤k (H3)对几乎所有l∈I,连续函数φ：H→H满足
定理4.1如果假设(H1)-(H3)成立,则问题(0.0.2)至少有一个解.下面我们研究凸的情况,此时关于F的假定如下：
(H4)F:I×H→P_(kc)(V_*)是一个集值函数且满足：(i)(t,x)→F(,x)是图像可测的;
(ⅱ)对于几乎所有t∈I,x→F(t,x)是闭图像；并且(H2)(ⅲ)成立.
定理4.2如果假设(H1),(H3)和(H4)成立,则问题(0.0.2)至少有一个解且解集在C(I,H)里弱紧的.
其次,考虑下面端点问题
这里extF(t,x)表示F(t,x)的端点集.这种情况下,利用端点选择定理及Schauder不动点定理建立问题(0.0.3)解的存在性,此时关于F的假定如下：
(H5)F:I×H→P_(wkc)(H)是一个集值函数且满足：
(ⅰ)(t,x)→F(t,x)是图像可测的；
(ⅱ)对于几乎所有t∈I,x→F(t,x)是h-连续；并且(H2)(ⅲ)成立.
定理4.3若假设(H1),(H3)和(H5)成立,则问题(0.0.3)存在解.
下面研究问题(0.0.3)的松弛定理,我们需要加强对F的假定条件.
(H6)F关于x满足：存在函数β(t)∈L+∞(I)使得对于任意x1,x2∈Lp(I,H),则都有
并且(H5)的所有假定都成立.
定理4.4若假设(H1),(H3)和(H6)成立,则Se=S这里Se(?)C(I,H).
为了说明结果的应用,我们举一个例子.
设I=[0,b],Ω(?)RN是一个边界(?)Ω光滑的有界区域.考虑如下积分边值问题：其中假设
(A)函数f(t,x,u)关于u不连续,且假定
为了研究问题(4.4.1)的可解性,我们需要对六(l,χ,u)(i=1,2)进行假定：(H6)(i)fi(f,x,u)(i=1,2)是Nemitsky可测(对于所有u：I×Ω→R是可测的,u→fi(f,x,u)(i-1,2)是可测的);
(ii)存在a2(t)∈Lq(t)+,C>0,使得对于几乎所有
其中1≤k 定理4.5如果假设(A),(H6)成立,则问题(0.0.4)的解存在,即存在解u∈Lp(I,W1,p(Ω)),且(?)∈Lq(I,W-1,q(Ω)).
第三部分我们研究了非线性项在非单调的情况下一类发展方程解存在性.在非线性项A满足拟单调的情况下,运用通论方法、极大单调算子与拟单调算子的性质以及Banach空间不动点定理,证明了发展包含解的存在性以及稠密性.接着运用连续选择定理,讨论了其端点解的存在性和松弛定理,并将这一结果应用到一类最优控制问题中.
首先,我们考虑下面非单调情况下发展包含：其中设I=[0,丁],算子A：I×V→V*是拟单调算子,F：I×H→2v*是满足相应条件的集值函数,φ是从H→H上的几乎处处连续函数.假定
(II1)算子A：I×V→V*满足：
(i)t→A(t,χ)是可测的；
(ii)对于几乎所有t∈I,A(t):V→V*是拟单调且半连续;
(iii)对于所有χ∈V,存在一个常数C1>0和一个非负函数a(·)∈Lq(I)使得
(iv)对于所有u∈V,存在C2>O,Co>0,b1(·)∈L1(t)使得
(H2) F:I×H→P_(fc)(V~*)是一个集值函数且满足：
(i)对于每一个χ∈H,t→F(t,χ)是可测的；
(ii)对于几乎所有l∈I,χ→F(l,χ)在其图像H×Vw*圪是闭的(这里Vw*表示空间V*的弱拓扑);
(iii)对于所有χ∈H,存在一个常数C3>0和一个非负函数b2(·)∈Lq(I)使得
其中1≤k (H3)对几乎所有l∈I,φ：Ⅱ→Ⅱ是线性连续函数且‖(u)‖≤‖u(T)‖.
(H4) F:I×H→P_(wkc)(H)是一个集值函数且满足：
(i)对于每一个χ∈Ht→F(t,χ)是可测的；
(ii)对于几乎所有l∈I,χ→F(l,χ)是h连续的;
(iii)对于所有χ∈V,存在一个常数C3>0和一个非负函数b2(·)∈Lq(I)使得
其中1≤k (H5)F：I×H→Pukc(H)是一个集值函数且满足：
(iv)存在一个可积函数L：I→R+,使得对于每一个χ1,χ2∈Lp(I,H),都有
并且(H4)成立.
定理5.1如果假设(H1)-(H3)成立,则问题(0.0.5)的解集S是非空的,且在Wpq里弱紧的,在C(I,H)里是紧的.
定理5.2如果假设(H1),(H3)和(H4)成立,则问题(0.0.5)的端点解集Sc≠(?).
定理5.3如果假设(H1),(H3)和(H5)成立,则Se=S这里Se(?)C(I,H).
为了定理的应用,举一个最优控制的例子.
设I=[0,b],Z(?)RN是一个边界光滑的有界区域.考虑如下的最优化控制问题：
使得其中在相应的假设条件下,运用前面的定理,证明了上述问题是最优可控的.
第四部分我们在Sobolev空间框架下讨论了一类偏微分包含的边值问题.当右端项分别满足一定条件时,借助于集值分析理论和不动点定理,获得了凸和非凸两种情况下边值问题解的存在性定理.对于非凸情形,使用单值的Schauder不动点定理获得解存在的充分条件.对于凸情形,利用集值的Kakutani不动点定理获得同样结论.利用Tolstonogov端点连续选择定理,证明了端点解的存在性和端点解的稠密性(强松驰定理).
这部分,考虑下面的边值问题：
假设
(Fl)算子H:Ω×R×R~N→P_k(R)是集值函数且满足：
(i)(x,u,s)→H(x,u,s)是图像可测;
(ii)对于几乎所有z∈Ω,(u,s)→H(x,u,s)是下半连续的;
(iii)对于几乎所有χ∈Ω,存在一个非负函数b(χ)∈Lp((Ω),使得其中当0≤α,β<1时,当α=β=1时,这里C>0满足
(F2)H:Ω×R×R~N→P_(kc)(R)是一个集值函数且满足：
(i)(x,u,s)→H(x,u,s)是图像可测;
(ii)对于几乎所有x∈Ω,(u,s)→H(x,u,s)是闭图像；并且(F1)(iii)成立.
(F3)H:Ω×R×R~N→P_(kc)(R)是一个集值函数且满足：
(i)(x,u,s)→H(x,u,s)是图像可测;
(ii)对于几乎所有x∈Ω,(u,s)→H(x,u,s)是h-连续；且(F1)(iii)成立,p>N.
定理6.1如果假设(F1)成立,则问题(0.0.6)存在解u∈W01,p(Ω).
定理6.2如果假设(F2)成立,则问题(0.0.6)存在解u∈W01,p(Ω)且解集在W01,p(Ω)里是弱紧的.
定理6.3如果假设(F3)成立,则问题(0.0.6)存在解u∈W1,p(Ω).
(F4)H:Ω×R×R~N→P_(kc)(R)是一个集值函数且满足：
(iv)对任意χ∈Ω,存在一个可积函数p：Ω→R1使得且(F3)成立.
定理6.4如果假设(F4)成立且p(χ)≤γ<λ(λ是-△在Ω上Dirichlet边界条件的第一特征值),则Se=S这里Se(?)C(Ω).
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→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 (H3)For all almost t∈I,there exists a continuous function φ:H→H such that
Theorem4.1Assumed(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 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 where
The 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 that
where1≤k 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 (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 have
where C3>0, b2(·)∈Lq(I) and1≤k (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 that
Theorem6.4Assumed (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).
近年来,非局部化条件下的微分方程或包含越来越受到人们的关注.这种非局部化条件包含了许多边值条件,比如初值、周期、反周期、积分、多点平均等边值条件,因此,此条件更具有一般性,在实际应用上也更具有广泛性.就我们所知,最早研究非局部条件下发展方程的是文献[63],作者Byszewski研究了一类半线性发展方程解的存在唯一性.从此,拉开了研究非局部化条件下微分方程或包含的序幕,请参见文献[62-68].
在文献[62]中,Paicu-Vrabie讨论了下面非局部条件的发展包含u(t)+Au(t)Эf(t) f(t)∈F(t,u) u(0)=g(u),其中要求算子A生成紧连续算子半群和非局部函数g具有紧性,证明了这类发展包含连续解的存在性.到目前为止,关于非局部条件的发展方程解(适度解、强解、古典解)的存在性、唯一性、稳定性,已经有了不少深刻结果,但很少涉及Banach空间里弱解的存在性.同时我们注意到这些文献中,在研究具有非局部初始条件的发展方程或包含问题时,不少都假设非局部函数满足一定的紧性条件和算子A强连续算子半群生成元或增生算子.然而,“在非局部函数失去紧性条件和算子A不具有这种结构条件下,相应的非局部问题是否存在解”本文将对这个问题给出一个肯定的回答.本文分四个部分主要研究了几类微分方程及包含解的存在性以及解集的结构.
第一部分我们考虑一类非局部发展方程可解性问题.在扰动项为非单调的情况下,运用极大单调算子以及拟单调算子的性质,给出解存在的充分条件.这部分考察下面发展方程：
其中I=[0,丁].假设
(i)对几乎所有t∈I,A:V→V~*是单调的且半连续.
(ii)B：V→V*是半连续和弱连续,并且对于在V里弱收敛函数列un→u,有
(iii)存在非负常数C_1,C_2,C_3,C_4使得
在I上几乎处处成立.(iv)对几乎所有t∈I,g：H→H是线性连续函数且‖g(u)‖≤‖(T)‖.
定理3.1如果假设(i)-(iv)成立,则问题(0.0.1)至少有一个解.
第二部分我们在Banach空间中讨论了一个非线性发展包含的非局部问题.当非线性算子A满足一致单调条件时,借助于集值分析理论和不动点定理,得到了凸和非凸两种情况下解的存在性定理.对于非凸情形,使用单值的Leray-Schauder替换定理获得解存在的充分条件.对于凸情形,利用集值的Leray-Schauder替换定理获得同样结论.利用Tolstonogov端点连续选择定理,证明了端点解的存在性和强松驰定理,并将得到的结论应用于右端项不连续情况下偏微分方程,给出了这类偏微分方程解的存在性.
这部分首先考虑下面发展包含：
其中算子A:I×V→V~*,F:I×H→2~(V*)
(H1)算子A：I×V→V*满足：
(i)t→A(t,χ)是可测的；
(ii)对于几乎所有t∈I,A(t):V→V_*是一致单调且半连续,即存在一个常数C>0,使得对于所有χ1,χ2∈V有
(iii)对于所有χ∈V,存在一个常数C1>0和一个非负函数a(·)∈L。(I)使得
(iv)对于所有u∈V,存在C_2>0,C_0,0,b_1(·)∈L_1(t)使得或
(H2)F:I×H→P_k(V_*)是一个集值函数且满足：
(i)(t,χ)→F(t,χ)是图像可测的；
(ii)对于几乎所有t∈I,χ→F(l,χ)是下半连续;
(iii)对于所有χ∈V,存在一个常数C3>0和一个非负函数b2(·)∈Lq(I)使得
其中1≤k
定理4.1如果假设(H1)-(H3)成立,则问题(0.0.2)至少有一个解.下面我们研究凸的情况,此时关于F的假定如下：
(H4)F:I×H→P_(kc)(V_*)是一个集值函数且满足：(i)(t,x)→F(,x)是图像可测的;
(ⅱ)对于几乎所有t∈I,x→F(t,x)是闭图像；并且(H2)(ⅲ)成立.
定理4.2如果假设(H1),(H3)和(H4)成立,则问题(0.0.2)至少有一个解且解集在C(I,H)里弱紧的.
其次,考虑下面端点问题
这里extF(t,x)表示F(t,x)的端点集.这种情况下,利用端点选择定理及Schauder不动点定理建立问题(0.0.3)解的存在性,此时关于F的假定如下：
(H5)F:I×H→P_(wkc)(H)是一个集值函数且满足：
(ⅰ)(t,x)→F(t,x)是图像可测的；
(ⅱ)对于几乎所有t∈I,x→F(t,x)是h-连续；并且(H2)(ⅲ)成立.
定理4.3若假设(H1),(H3)和(H5)成立,则问题(0.0.3)存在解.
下面研究问题(0.0.3)的松弛定理,我们需要加强对F的假定条件.
(H6)F关于x满足：存在函数β(t)∈L+∞(I)使得对于任意x1,x2∈Lp(I,H),则都有
并且(H5)的所有假定都成立.
定理4.4若假设(H1),(H3)和(H6)成立,则Se=S这里Se(?)C(I,H).
为了说明结果的应用,我们举一个例子.
设I=[0,b],Ω(?)RN是一个边界(?)Ω光滑的有界区域.考虑如下积分边值问题：其中假设
(A)函数f(t,x,u)关于u不连续,且假定
为了研究问题(4.4.1)的可解性,我们需要对六(l,χ,u)(i=1,2)进行假定：(H6)(i)fi(f,x,u)(i=1,2)是Nemitsky可测(对于所有u：I×Ω→R是可测的,u→fi(f,x,u)(i-1,2)是可测的);
(ii)存在a2(t)∈Lq(t)+,C>0,使得对于几乎所有
其中1≤k
第三部分我们研究了非线性项在非单调的情况下一类发展方程解存在性.在非线性项A满足拟单调的情况下,运用通论方法、极大单调算子与拟单调算子的性质以及Banach空间不动点定理,证明了发展包含解的存在性以及稠密性.接着运用连续选择定理,讨论了其端点解的存在性和松弛定理,并将这一结果应用到一类最优控制问题中.
首先,我们考虑下面非单调情况下发展包含：其中设I=[0,丁],算子A：I×V→V*是拟单调算子,F：I×H→2v*是满足相应条件的集值函数,φ是从H→H上的几乎处处连续函数.假定
(II1)算子A：I×V→V*满足：
(i)t→A(t,χ)是可测的；
(ii)对于几乎所有t∈I,A(t):V→V*是拟单调且半连续;
(iii)对于所有χ∈V,存在一个常数C1>0和一个非负函数a(·)∈Lq(I)使得
(iv)对于所有u∈V,存在C2>O,Co>0,b1(·)∈L1(t)使得
(H2) F:I×H→P_(fc)(V~*)是一个集值函数且满足：
(i)对于每一个χ∈H,t→F(t,χ)是可测的；
(ii)对于几乎所有l∈I,χ→F(l,χ)在其图像H×Vw*圪是闭的(这里Vw*表示空间V*的弱拓扑);
(iii)对于所有χ∈H,存在一个常数C3>0和一个非负函数b2(·)∈Lq(I)使得
其中1≤k
(H4) F:I×H→P_(wkc)(H)是一个集值函数且满足：
(i)对于每一个χ∈Ht→F(t,χ)是可测的；
(ii)对于几乎所有l∈I,χ→F(l,χ)是h连续的;
(iii)对于所有χ∈V,存在一个常数C3>0和一个非负函数b2(·)∈Lq(I)使得
其中1≤k
(iv)存在一个可积函数L：I→R+,使得对于每一个χ1,χ2∈Lp(I,H),都有
并且(H4)成立.
定理5.1如果假设(H1)-(H3)成立,则问题(0.0.5)的解集S是非空的,且在Wpq里弱紧的,在C(I,H)里是紧的.
定理5.2如果假设(H1),(H3)和(H4)成立,则问题(0.0.5)的端点解集Sc≠(?).
定理5.3如果假设(H1),(H3)和(H5)成立,则Se=S这里Se(?)C(I,H).
为了定理的应用,举一个最优控制的例子.
设I=[0,b],Z(?)RN是一个边界光滑的有界区域.考虑如下的最优化控制问题：
使得其中在相应的假设条件下,运用前面的定理,证明了上述问题是最优可控的.
第四部分我们在Sobolev空间框架下讨论了一类偏微分包含的边值问题.当右端项分别满足一定条件时,借助于集值分析理论和不动点定理,获得了凸和非凸两种情况下边值问题解的存在性定理.对于非凸情形,使用单值的Schauder不动点定理获得解存在的充分条件.对于凸情形,利用集值的Kakutani不动点定理获得同样结论.利用Tolstonogov端点连续选择定理,证明了端点解的存在性和端点解的稠密性(强松驰定理).
这部分,考虑下面的边值问题：
假设
(Fl)算子H:Ω×R×R~N→P_k(R)是集值函数且满足：
(i)(x,u,s)→H(x,u,s)是图像可测;
(ii)对于几乎所有z∈Ω,(u,s)→H(x,u,s)是下半连续的;
(iii)对于几乎所有χ∈Ω,存在一个非负函数b(χ)∈Lp((Ω),使得其中当0≤α,β<1时,当α=β=1时,这里C>0满足
(F2)H:Ω×R×R~N→P_(kc)(R)是一个集值函数且满足：
(i)(x,u,s)→H(x,u,s)是图像可测;
(ii)对于几乎所有x∈Ω,(u,s)→H(x,u,s)是闭图像；并且(F1)(iii)成立.
(F3)H:Ω×R×R~N→P_(kc)(R)是一个集值函数且满足：
(i)(x,u,s)→H(x,u,s)是图像可测;
(ii)对于几乎所有x∈Ω,(u,s)→H(x,u,s)是h-连续；且(F1)(iii)成立,p>N.
定理6.1如果假设(F1)成立,则问题(0.0.6)存在解u∈W01,p(Ω).
定理6.2如果假设(F2)成立,则问题(0.0.6)存在解u∈W01,p(Ω)且解集在W01,p(Ω)里是弱紧的.
定理6.3如果假设(F3)成立,则问题(0.0.6)存在解u∈W1,p(Ω).
(F4)H:Ω×R×R~N→P_(kc)(R)是一个集值函数且满足：
(iv)对任意χ∈Ω,存在一个可积函数p：Ω→R1使得且(F3)成立.
定理6.4如果假设(F4)成立且p(χ)≤γ<λ(λ是-△在Ω上Dirichlet边界条件的第一特征值),则Se=S这里Se(?)C(Ω).
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→
(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
Theorem4.1Assumed(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
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 where
The 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 that
where1≤k
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
(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 have
where C3>0, b2(·)∈Lq(I) and1≤k
(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 that
Theorem6.4Assumed (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).
引文
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[3] J. P. Aubin, I. Ekeland, Applied Nonlinear Analysis [M]. John Wiley Sons Inc. NewYork,(1984).
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[12] H. Brezis, Op′erateurs Maximaux Monotones et Semigroupes de Contractions dans lesEspaces de Hilbert [M]. North Holland: Amsterdam,(1973).
[13] H. Attouch, A. Damlamian, On Multivalued Evolution Equations in Hilbert Spaces [J].Israel Journal of Mathematics,12(1972),373-390.
[14] T. Wazewski, Syst′emes de Commande et′equations Au Contingent [J], Bulletin of thePolish Academy of Sciences,9(1961),152-155.
[15] A. Cellina, M. Marchi, Nonconvex Perturbations of Maximal Monotone DiferentialInclusions [J]. Israel Journal of Mathematics,46(1983),1-11.
[16] D. Wagner, Survey of Measurable Selection Theorems [J]. SIAM Journal on Controland Optimization,15(1977)859-903.
[17] A. Bressan, On Diferential Relation with Lower Continuous Right-Hand Side, an Ex-istence Theorem [J]. Journal of Diferential Equations,37(1980)89-97.
[18] B. Cornet, Existence of Slow Solutions for a Class of Diferential Inclusions [J]. Journalof Mathematical Analysis and Applications,96(1983)130-147.
[19] J. P. Aubin, A. Cellina, Diferential Inclusions [M]. Berlin: Springer-Verlag,(1984).
[20] A. A. Tolstonogov, Diferential Inclusions in Banach Spaces [M]. Netherland: KluwerAcademic,(2000).
[21] A. Bressan, G. Colombo, Existensions and Selections of Maps with DecomposableValues [J]. Studia Math.90(1988)69-86.
[22] A. A. Tolstonogov, Extreme Continuous Selectors of Multivalued Maps and Their Ap-plications [J]. J. Diferential Equations.122(1995),161-180.
[23] A. A. Tolstonogov, D. A. Tolsonogov, Continuous Extreme Selectors of Multifunctionswith Decomposable Values. I: Existence Theorems, II: Relaxation Theorems [J]. Set-Valued Anal.4(1996),173-203,237-269.
[24] A. Plis, On Trajectories of Orientor Fields [J]. Bull Acad. Polon. Sci.13(1965),565-569.
[25] J. Davy, Properties of the Solution Set of a Generalized Diferential Equation [J]. Bull.Australian Math. Soc.6(1972),379-398.
[26] A. F. Filippov, Classical Solutions of Diferential Equations with Multi-valued Right-Hand Side [J]. SIAM J. Control. Optim.6(1967),609-621.
[27] H. Kaczynski, C. Olech, Existence of Solutions of Orientor Fields with Non-ConvexRight-Hand Side [J]. Ann. Polon. Math.29(1974),61-66.
[28] H. Antosiewicz, A. Cellina, Continuous Selections and Diferential Relations [J]. J. Dif.Equ.19(1975),386-398.
[29] A. Bressan, On Diferential Relations with Lower Semicontinuous Right-Hand Side. AnExistence Theorem [J]. J. Dif. Equ.37(1980),89-97.
[30] J. S. Lojasiewicz, The Existence of Solutions of for Lower Semicontinuous OrientorFields [J]. Bull. Acad. Polon. Sci.28(1980),483-487.
[31] R. A. Adams, Sobolev spaces [M]. Academic Press, New York,(1975).
[32] F. S. De Blasi, Existence and Stability of Solutions for Autonomous Multivalued Dif-ferential Equation in Banach Space [J]. Atti. Accad. Naz. Lincei, Ser. VIII.60(1976),767-774.
[33] C. Bardaro, P. Pucci, Some Contributions to the Theory of Multivalued DiferentialEquations [J]. Atti. Sem. Math. Fis. Modena.32(1983),175-202.
[34] F. S. De Blasi, J. Myjak, The Generic Property of Existence of Solutions for a Class ofMultivalued Diferential Equations in Hilbert Spaces [J]. Funkc. Ekvacioj.21(1978),271-278.
[35] F. S. De Blasi, G.. Pianigiani, Non-Convexvalued Diferential Inclusions in BanachSpace [J]. J. Math. Anal. Appl.157(1991),469-494.
[36] A. A. Tolstonogov, I. A. Finogenko, On Solutions of the Diferential Inclusion withLower Semicontinuous Right-Hand Side in Banach Space [J]. Sb. Mathem.125(1984),199-230(in Russian).
[37] H. Brezis, Op′erateurs Maximaux Monotones et Semigroupes de Contractions dans lesEspaces de Hilbert [M], North Holland: Amsterdam,(1973).
[38] H. Attouch, A. Damlamian, On Multivalued Evolution Equations in Hilbert Spaces[J].Israel Journal of Mathematics,12(1972),373-390.
[39] N. S. Papageorgiou, N. Yannakakis, Existence of Extremal Solutions for NonlinearEvolution Inclusions [J]. Archivum Mathematicum,37(2001),9-23.
[40] S. Guillaume, Subdiferential Evolution Inclusion in Nonconvex Analysis [J]. Positivity,4(2000),357-395.
[41] A. Cellina, M. Marchi, Nonconvex Perturbations of Maximal Monotone DiferentialInclusions [J]. Israel Journal of Mathematics,46(1983)1-11.
[42] D. Kravvaritis, N. S. Papageorgiou, Multivalued Perturbations of Subdiferential TypeEvolution Equations in Hilbert Spaces [J]. Journal of Diferential Equations,76(1988),238-255.
[43] N. S. Papageorgiou, F. Papalini, On the Structure of the Solution Set of EvolutionInclusions with Time-dependent Subdiferentials [J]. Rendiconti del Seminario Matem-atico della Universita di Padova,65(1997),163-187.
[44] T. Cardinali, F. Papalini, Existence Theorems for Nonlinear Evolutions Inclusions [J].Annali di Matematica Puraed Applicata,173(1997),1-11.
[45] R. Cascaval, I. I. Vrabie, Existence of periodic solutions for a class of nonlinear evolutionequations [J]. Rev. Mat. Univ. Complut. Madrid.7(1994),325-338.
[46] N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbertspaces [J]. Proc. Amer. Math. Soc.120(1994),185-192.
[47] N. Hirano, N. Shioji, Invariant sets for nonlinear evolution equations, Cauchy problemsand periodic problems [J]. Abstr. Appl. Anal.3(2004),183-203.
[48] A. Paicu, Periodic solutions for a class of nonlinear evolution equations in Banachspaces [J]. An.stiint.“Al.I.Cuza”, Iasi, Ser. Nouaˇ. Mat. LV (2009),107-118.
[49] I. I. Vrabie, Periodic soutions for nonlinear evolution equations in a Banach space [J].Proc. Amer. Math. Soc.109(3)(1990),653-661.
[50] S. Aizicovici, N.S. Papageorgiou, V. Staicu, Periodic solutions of nonlinear evolutioninclusions in Banach spaces [J].J. Nonlinear Convex. Anal.7(2006),163-177.
[51] C. Castaing, D.P. Monteiro-Marques, Periodic solutions of evolution problems associ-ated with a moving convex set [J]. C.R. Acad. Sci. Paris, Se′rie A.321(1995),531-536.
[52] V. Lakshmikantham, N.S. Papageorgiou, Periodic solutions of nonlinear evolution in-clusions [J]. Comput. Appl. Math.52(1994),277-286.
[53] N. S. Papageorgiou, Periodic trajectories for evolution inclusions associated with time-dependent subdiferentials [J]. Ann. Univ. Sci. Budapest.37(1994),139-155.
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