Banach空间中具多值扰动微分包含解的存在性及其渐近性质
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摘要
Banach空间上的微分包含理论是非线性分析中非常活跃的一个分支.从七十年代开始,美国、罗马尼亚和日本等国的著名数学家(如V.Barbu、J.P.Aubin、T.Kato、N.H.Pavel等)就开始从事这方面的研究工作(见[2, 9, 13, 71]).近几十年来,这一领域的研究对近代物理和工程技术中出现的非线性问题和控制论的研究有着重要的理论意义和应用价值.由于Volterra方程(见[2])、偏微分方程(见[9, 13, 71])、控制论和最优化中研究的许多问题都可以转化为微分包含问题,因此在一定的条件下研究微分包含解(包括强解、弱解、温和解和积分解)的存在性以及渐近性态问题就显得非常重要.本文就是在Banach空间中讨论了具多值扰动微分包含解的存在性及其渐近行为,共分四章:
本文第一章主要考虑以下半线性非局部微分包含解的存在性这里F是一上半连续多值映射, g : C([0,T];E)→E是一给定的连续映射,线性算子A(可能无界)是一紧半群的无穷小生成元.
本章中我们主要利用多值不动点定理和紧性方法给出上述非局部微分包含解的存在性定理(见Theorem 1.2).证明的关键在于我们设法构造了一个新的特殊的集值映射,然后利用集值分析和非紧性测度理论证明了该集值映射是一个在给定圆盘上具闭凸值的上半连续的紧算子,正是由于该算子的良好性质便于我们构造了连续函数空间里一个相对紧的解序列,从而我们能够得到上述主要结论.如果在F和g上附加的是渐近条件或强有界条件,我们同样能够得到定理1.2中的结论(见Corollary 1.6和Remark 1.7).这些结果推广了文献[5, 64]中的相应结论至非局部多值情形.由于我们不再需要多值扰动F的Lipschitz型条件,因此这些结论即使对单值情形也是新的.在这一章的最后,我们还给出了这些结果在偏微分方程中的应用.
第二章我们继续致力于研究上述多值微分包含问题,其中A是强连续有界线性算子族{S(t) : t∈[0,T]}的生成元, F是一个upper?Carathe′odory多值映射和g是某给定的算子.
本章中我们主要利用不动点技巧、非紧测度性质、集值分析以及微分包含理论的相关已知结果,讨论了一般Banach空间中半线性微分包含适度解的存在性(见Lemma 2.9和Theorem 2.7).行文中,引理2.9给出的不等式对于整个定理2.7的证明起着至关重要的作用.在定理2.7中,我们既没有对Banach空间附加任何条件,也没有假设半群的紧性,因此我们的结果推广了文[22, 28, 30, 88, 89,91]中的主要定理.
第三章在实Banach空间中考虑如下发展型微分包含解的存在性这里线性无界算子族{A(t)}t∈[0,d]生成一强连续发展系统U(t, s), F仍是一多值映射.
在这章中,我们首先证明了当g是全连续算子时上述发展包含适度解的存在性(见Theorem 3.5).在定理3.5中,对于包含的线性部分我们只假设其生成强连续的发展系统,既不需其紧性,甚至也不需其等度连续性.主要是在其证明中,设法构造了一个新的非紧测度,正是该正则测度便于我们寻找连续函数空间中的非空紧凸子集,从而大大降低了对发展系统的要求.因此该定理又从本质上进一步改善了第二章中给出的结果.其次讨论了当g是Lipschitz连续算子时该发展包含适度解的存在性(见Theorem 3.11).在定理3.11的证明中,我们充分利用了对非紧性测度的估计和叠加算子的性质,从而在不需要空间可分性和发展系统紧性的情形下得到了上述主要定理.因此我们的结果推广了这方面的许多工作(如文献[7, 22, 28, 30, 41, 47, 88, 91]).最后,我们应用定理3.5给出的结果讨论了半线性偏微分方程的一个例子.
第四章主要处理下列非线性非局部多值问题积分解的存在性及其渐近性态:其中耗散算子,生成压缩半群S(t), F是相应于其第二变量的弱上半连续多值映射, X~*是一致凸的Banach空间.
4.1节中首先回忆了Banach空间的一些几何性质,接着介绍了一些基本概念,并给出了非自治耗散系统积分解的存在唯一性和Be′nilan不等式.在4.2节中,我们讨论了半群S(t)是等度连续和g是全连续情形下,上述非线性微分包含积分解的存在性(见Theorem 4.15). 4.3节得到了g是Lipschitz单值算子和多值映射F是关于Hausdor?距离的Lipschitz型情形下积分解的存在性(见Theorem4.17).在最后的4.4节中,首先讨论了殆非扩张曲线的渐近性质,找寻殆非扩张曲线与我们所研究的耗散系统积分解之间的内在联系,并利用这些内在联系研究积分解u(t)在t趋于无穷时的渐近行为(见Theorem 4.23和Theorem 4.26).我们的结果改进了文[22, 58, 79, 80, 86, 87, 91, 92]中的许多已知结果.
The theory of differential inclusions in Banach spaces is a very active branchin that of nonlinear analysis. Since the 1970s, many famous mathematicians (such asV.Barbu, J.P.Aubin, T.Kato, N.H.Pavel, etc.) in US, Romania and Japan and so onhave started to be engaged in a series of studies(see [2, 9, 13, 71]) in this subject. Inrecent dozens of years, the research work in the field is extremely crucial to nonlinearproblems and control theory in modern physics and engineering technology. Volterraequations(see [2]), partial differential equations(see [9, 13, 71]) and many problems incontrol theory and optimization theory can be rewritten in the abstract form as differ-ential inclusions. Therefore, the study of the existence of solutions (including strongsolutions, weak solutions, mild solutions and integral solutions) and asymptotic prop-erties for differential inclusions is very important. In this thesis, we mainly discuss theexistence and properties of solutions for nonlocal differential inclusions with multival-ued perturbtation in Banach spaces . It consists of four chapters.
In Chapter 1, we discuss the existence of mild solutions for the following nonlocalinitial valued problem :
where F is an upper semicontinuous multifunction, g : C([0, T]; E)→E is a givenEffvalued function and the linear operator (usually unbounded) A : D(A) (?) E→E isthe densely defined generator of a compact semigroup S (t) for t > 0 in a real Banachspace E.
we prove a new result on the existence of mild solutions for the nonlocal multi-valued differential inclusion in a Banach space by using the technique of multi-valuedfixed point theorem and compactness methods (see Theorem 1.2). In our proof, we tryto construct a new and special set-valued mapping. Secondly, by careful analysis, themultivalued mapping is an upper semicontinuous and compact operator with closedand convex values on a given disc. Finally, we make full use of the properties of the set-valued mapping to manage to construct a relatively compact solution sequence inC([0, T]; E). Thus, we can obtain the main result. Moreover, under the asymptoticconditions and strong boundedness conditions on F and g, we also obtain the sameresults, respectively (see Corollary 1.6 and Remark 1.7). Our results extend the mainones in [5, 64] to the nonlocal multi-valued case. Since the Lipschitz type conditionon perturbation F is not required, our results are new even in the case that the pertur-bation F is single-valued. Finally, Using the established results, we take two examplesin order to point out the effectiveness of the abstract results proved in the former.
Chapter 2 is devoted to continuing to study the class of nonlocal multivaluedproblem when F is an upper-Carathe′odory multifunction , g : C([0, T]; E)→E is agiven operator and A : D(A) ff E→E is the densely defined infinitesimal generatorof a strongly continuous semigroup of bounded linear operators {S(t) : t∈[0, T]} ingeneral Banach spaces.
In this chapter, we deal with the existence of mild solutions for the semilineardifferential inclusions in general Banach spaces by using multi-valued fixed point the-orem on upper semicontinuity, the measure of noncompactness and some known re-sults about the theory of semilinear differential inclusions and multivalued analysis(see Lemma 2.9 and Theorem 2.7). Throughout this chapter, the new inequality inLemma 2.9 is very key to the proof of Theorem 2.7. In Theorem 2.7, we haven’t anyhypothesis for the Banach space E. Moreover, we don’t also assume the semigroupS(t) is compact. Therefore our results extend those in [22, 28, 30, 88, 89, 91] to thecase that the nonlocal multi-valued differential inclusions.
The purpose of Chapter 3 is to present the existence of mild solutions for the fol-lowing nonlocal evolution differential inclusion with the upper semicontinuous non-linearity:in a real Banach space. Here the family of linear unbounded operators {A(t)}t∈[0,d]generates a strongly continuous evolution system U(t, s) and F is a multifunction.
In this chapter, we first prove the existence result of the above evolution inclusionunder the assumption the nonlocal condition g is completely continuous (see Theorem 3.5). In Theorem 3.5, we only suppose that the linear part of the inclusion generates astrongly continuous evolution system U(t, s), which is not assumed to be equicontin-uous or compact. In proof, we introduce a new measure of noncompactness, which isvery crucial to construct a nonempty, compact and convex subset in C([0, d]; X). Thisallows us to improve considerably the results in the second chapter by replacing theequicontinuity of C0ffsemigroup by a strictly weaker condition. Subsequently, we dealwith the same problem concerning the case that g is Lipschitz continuous (see Theo-rem 3.11). It is worth mentioning that we try to make full use of the estimations to theHausdorff measure of noncompactness and the properties of the superposition operatorin our proof of Theorem 3.11. Thus, we may derive the existence of mild solutionswithout the assumption of separability on Banach space X and that of compactnesson the associated evolution system U(t, s). Therefore our results improve and extendsome known results in this field (see, for example, [7, 22, 28, 30, 41, 47, 88, 91]) andthe references therein). Finally, in the last section we discuss an example of semilinearpartial differential equations.
Chapter 4 is concerned with this existence and asymptotic properties of integralsolutions for the following nonlinear nonlocal initial value problem:in a real Banach space when A : D(A) ff X→X is a nonlinear mffdissipative operatorwhich generates a contraction semigroup S (t) and F is a weakly upper semicontinuousmultifunction with respect to its second variable in a real Banach space with uniformlyconvex dual X~*.
In this chapter, we prove the existence of integral solutions of the above Cauchyproblem . Moreover, we discuss the asymptotic properties of integral solutions. InSection 4.1, we first recall some facts about geometric properties of Banach spaces.In the sequel, we introduce some basic definitions and give the existence and unique-ness result and Be′nilan inequalities about the integral solutions of the nonautonomousdissipative system in Banach spaces. In Section 4.2, we discuss the case that g iscompletely continuous and S (t) is equicontinuous (see Theorem 4.15). The existenceresult for the above problem is stated when g is Lipschitz and F is Lipschitz-type map- ping about the Hausdorff metric in Section 4.3 (see Theorem 4.17). Finally, we firstdiscuss the asymptotic properties of almost nonexpansive curves. In the sequel, we tryto look for the relations between almost nonexpansive curves and integral solutions ofgiven dissipative system. Then, by the relations between almost nonexpansive curvesand integral solutions, we study the asymptotic behavior of integral solutions in Sec-tion 4.4 (see Theorem 4.23 and Theorem 4.26). These results extend and improvesome known results in [22, 58, 79, 80, 86, 87, 91, 92].
本文第一章主要考虑以下半线性非局部微分包含解的存在性这里F是一上半连续多值映射, g : C([0,T];E)→E是一给定的连续映射,线性算子A(可能无界)是一紧半群的无穷小生成元.
本章中我们主要利用多值不动点定理和紧性方法给出上述非局部微分包含解的存在性定理(见Theorem 1.2).证明的关键在于我们设法构造了一个新的特殊的集值映射,然后利用集值分析和非紧性测度理论证明了该集值映射是一个在给定圆盘上具闭凸值的上半连续的紧算子,正是由于该算子的良好性质便于我们构造了连续函数空间里一个相对紧的解序列,从而我们能够得到上述主要结论.如果在F和g上附加的是渐近条件或强有界条件,我们同样能够得到定理1.2中的结论(见Corollary 1.6和Remark 1.7).这些结果推广了文献[5, 64]中的相应结论至非局部多值情形.由于我们不再需要多值扰动F的Lipschitz型条件,因此这些结论即使对单值情形也是新的.在这一章的最后,我们还给出了这些结果在偏微分方程中的应用.
第二章我们继续致力于研究上述多值微分包含问题,其中A是强连续有界线性算子族{S(t) : t∈[0,T]}的生成元, F是一个upper?Carathe′odory多值映射和g是某给定的算子.
本章中我们主要利用不动点技巧、非紧测度性质、集值分析以及微分包含理论的相关已知结果,讨论了一般Banach空间中半线性微分包含适度解的存在性(见Lemma 2.9和Theorem 2.7).行文中,引理2.9给出的不等式对于整个定理2.7的证明起着至关重要的作用.在定理2.7中,我们既没有对Banach空间附加任何条件,也没有假设半群的紧性,因此我们的结果推广了文[22, 28, 30, 88, 89,91]中的主要定理.
第三章在实Banach空间中考虑如下发展型微分包含解的存在性这里线性无界算子族{A(t)}t∈[0,d]生成一强连续发展系统U(t, s), F仍是一多值映射.
在这章中,我们首先证明了当g是全连续算子时上述发展包含适度解的存在性(见Theorem 3.5).在定理3.5中,对于包含的线性部分我们只假设其生成强连续的发展系统,既不需其紧性,甚至也不需其等度连续性.主要是在其证明中,设法构造了一个新的非紧测度,正是该正则测度便于我们寻找连续函数空间中的非空紧凸子集,从而大大降低了对发展系统的要求.因此该定理又从本质上进一步改善了第二章中给出的结果.其次讨论了当g是Lipschitz连续算子时该发展包含适度解的存在性(见Theorem 3.11).在定理3.11的证明中,我们充分利用了对非紧性测度的估计和叠加算子的性质,从而在不需要空间可分性和发展系统紧性的情形下得到了上述主要定理.因此我们的结果推广了这方面的许多工作(如文献[7, 22, 28, 30, 41, 47, 88, 91]).最后,我们应用定理3.5给出的结果讨论了半线性偏微分方程的一个例子.
第四章主要处理下列非线性非局部多值问题积分解的存在性及其渐近性态:其中耗散算子,生成压缩半群S(t), F是相应于其第二变量的弱上半连续多值映射, X~*是一致凸的Banach空间.
4.1节中首先回忆了Banach空间的一些几何性质,接着介绍了一些基本概念,并给出了非自治耗散系统积分解的存在唯一性和Be′nilan不等式.在4.2节中,我们讨论了半群S(t)是等度连续和g是全连续情形下,上述非线性微分包含积分解的存在性(见Theorem 4.15). 4.3节得到了g是Lipschitz单值算子和多值映射F是关于Hausdor?距离的Lipschitz型情形下积分解的存在性(见Theorem4.17).在最后的4.4节中,首先讨论了殆非扩张曲线的渐近性质,找寻殆非扩张曲线与我们所研究的耗散系统积分解之间的内在联系,并利用这些内在联系研究积分解u(t)在t趋于无穷时的渐近行为(见Theorem 4.23和Theorem 4.26).我们的结果改进了文[22, 58, 79, 80, 86, 87, 91, 92]中的许多已知结果.
The theory of differential inclusions in Banach spaces is a very active branchin that of nonlinear analysis. Since the 1970s, many famous mathematicians (such asV.Barbu, J.P.Aubin, T.Kato, N.H.Pavel, etc.) in US, Romania and Japan and so onhave started to be engaged in a series of studies(see [2, 9, 13, 71]) in this subject. Inrecent dozens of years, the research work in the field is extremely crucial to nonlinearproblems and control theory in modern physics and engineering technology. Volterraequations(see [2]), partial differential equations(see [9, 13, 71]) and many problems incontrol theory and optimization theory can be rewritten in the abstract form as differ-ential inclusions. Therefore, the study of the existence of solutions (including strongsolutions, weak solutions, mild solutions and integral solutions) and asymptotic prop-erties for differential inclusions is very important. In this thesis, we mainly discuss theexistence and properties of solutions for nonlocal differential inclusions with multival-ued perturbtation in Banach spaces . It consists of four chapters.
In Chapter 1, we discuss the existence of mild solutions for the following nonlocalinitial valued problem :
where F is an upper semicontinuous multifunction, g : C([0, T]; E)→E is a givenEffvalued function and the linear operator (usually unbounded) A : D(A) (?) E→E isthe densely defined generator of a compact semigroup S (t) for t > 0 in a real Banachspace E.
we prove a new result on the existence of mild solutions for the nonlocal multi-valued differential inclusion in a Banach space by using the technique of multi-valuedfixed point theorem and compactness methods (see Theorem 1.2). In our proof, we tryto construct a new and special set-valued mapping. Secondly, by careful analysis, themultivalued mapping is an upper semicontinuous and compact operator with closedand convex values on a given disc. Finally, we make full use of the properties of the set-valued mapping to manage to construct a relatively compact solution sequence inC([0, T]; E). Thus, we can obtain the main result. Moreover, under the asymptoticconditions and strong boundedness conditions on F and g, we also obtain the sameresults, respectively (see Corollary 1.6 and Remark 1.7). Our results extend the mainones in [5, 64] to the nonlocal multi-valued case. Since the Lipschitz type conditionon perturbation F is not required, our results are new even in the case that the pertur-bation F is single-valued. Finally, Using the established results, we take two examplesin order to point out the effectiveness of the abstract results proved in the former.
Chapter 2 is devoted to continuing to study the class of nonlocal multivaluedproblem when F is an upper-Carathe′odory multifunction , g : C([0, T]; E)→E is agiven operator and A : D(A) ff E→E is the densely defined infinitesimal generatorof a strongly continuous semigroup of bounded linear operators {S(t) : t∈[0, T]} ingeneral Banach spaces.
In this chapter, we deal with the existence of mild solutions for the semilineardifferential inclusions in general Banach spaces by using multi-valued fixed point the-orem on upper semicontinuity, the measure of noncompactness and some known re-sults about the theory of semilinear differential inclusions and multivalued analysis(see Lemma 2.9 and Theorem 2.7). Throughout this chapter, the new inequality inLemma 2.9 is very key to the proof of Theorem 2.7. In Theorem 2.7, we haven’t anyhypothesis for the Banach space E. Moreover, we don’t also assume the semigroupS(t) is compact. Therefore our results extend those in [22, 28, 30, 88, 89, 91] to thecase that the nonlocal multi-valued differential inclusions.
The purpose of Chapter 3 is to present the existence of mild solutions for the fol-lowing nonlocal evolution differential inclusion with the upper semicontinuous non-linearity:in a real Banach space. Here the family of linear unbounded operators {A(t)}t∈[0,d]generates a strongly continuous evolution system U(t, s) and F is a multifunction.
In this chapter, we first prove the existence result of the above evolution inclusionunder the assumption the nonlocal condition g is completely continuous (see Theorem 3.5). In Theorem 3.5, we only suppose that the linear part of the inclusion generates astrongly continuous evolution system U(t, s), which is not assumed to be equicontin-uous or compact. In proof, we introduce a new measure of noncompactness, which isvery crucial to construct a nonempty, compact and convex subset in C([0, d]; X). Thisallows us to improve considerably the results in the second chapter by replacing theequicontinuity of C0ffsemigroup by a strictly weaker condition. Subsequently, we dealwith the same problem concerning the case that g is Lipschitz continuous (see Theo-rem 3.11). It is worth mentioning that we try to make full use of the estimations to theHausdorff measure of noncompactness and the properties of the superposition operatorin our proof of Theorem 3.11. Thus, we may derive the existence of mild solutionswithout the assumption of separability on Banach space X and that of compactnesson the associated evolution system U(t, s). Therefore our results improve and extendsome known results in this field (see, for example, [7, 22, 28, 30, 41, 47, 88, 91]) andthe references therein). Finally, in the last section we discuss an example of semilinearpartial differential equations.
Chapter 4 is concerned with this existence and asymptotic properties of integralsolutions for the following nonlinear nonlocal initial value problem:in a real Banach space when A : D(A) ff X→X is a nonlinear mffdissipative operatorwhich generates a contraction semigroup S (t) and F is a weakly upper semicontinuousmultifunction with respect to its second variable in a real Banach space with uniformlyconvex dual X~*.
In this chapter, we prove the existence of integral solutions of the above Cauchyproblem . Moreover, we discuss the asymptotic properties of integral solutions. InSection 4.1, we first recall some facts about geometric properties of Banach spaces.In the sequel, we introduce some basic definitions and give the existence and unique-ness result and Be′nilan inequalities about the integral solutions of the nonautonomousdissipative system in Banach spaces. In Section 4.2, we discuss the case that g iscompletely continuous and S (t) is equicontinuous (see Theorem 4.15). The existenceresult for the above problem is stated when g is Lipschitz and F is Lipschitz-type map- ping about the Hausdorff metric in Section 4.3 (see Theorem 4.17). Finally, we firstdiscuss the asymptotic properties of almost nonexpansive curves. In the sequel, we tryto look for the relations between almost nonexpansive curves and integral solutions ofgiven dissipative system. Then, by the relations between almost nonexpansive curvesand integral solutions, we study the asymptotic behavior of integral solutions in Sec-tion 4.4 (see Theorem 4.23 and Theorem 4.26). These results extend and improvesome known results in [22, 58, 79, 80, 86, 87, 91, 92].
引文
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[51] J. S. Jung, J. S. Park, and E. H. Park. Asymptotic behaviour of generalizedalmost nonexpansive sequences and applications. Proc. Nonlinear Funct. Anal.,1:65–79, 1996.
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[57] G. Li. Asymtotic behavior for commutative semigroups of asymptotically non-expansive type mappings. Nonlinear Anal., 42:175–183, 2000.
[58] G. Li and K. K. Jong. Asymptotic behavior of non-autonomous dissipative sys-tems in Hilbert spaces. Acta Math. Sci., 18(Supp.):25–30, 1998.
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[60] G. Li and J. K. Kim. Nonlinear ergodic theorem for a general curve defined ongeneral semigroups. Nonlinear Anal., 55:1–14, 2003.
[61] G. Li and J. K. Kim. Nonlinear ergodic theorem for commutative semigroups ofnon-Lipschitzian mappings in Banach space. Houston Journal Math., 29:23–36,2003.
[62] G. Li and J. P. Ma. Nonlinear ergodic theorem for semitopological semigroupsof non-Lipschitzian mappings in Banach space. Chin. Scie. Bull., 42(1):8–11,1997.
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[69] S. Ntouyas and P. Tsamatos. Global existence for semilinear integrodifferentialequations with delay and nonlocal conditions. Anal. Appl., 64:99–105, 1997.
[70] N. S. Papageougiou. Functional differential inclusions in Banach spaces withnonconvex right hand side. Funkcial Ekvac., 32:145–156, 1989.
[71] N. H. Pavel. Nolinear Evolution Operators and Semigrouos. Lecture Notes inMath., vol.1260, Springer-Verlag, Berlin, 1987.
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[80] B. Djafari Rouhani. Asymptotic properties of some non-autonomous system inBanach spaces. J. Differential Equations, 229:412–425, 2006.
[81] G. Song and J. Ma. Asymptotic behaviour of solutions to the nonlinear evolutionequation. Science in China, Ser.A, 23:679–686, 1993.
[82] A. Tolstonogov. Differential Inclusions in a Banach Space. Math. and Its Appl.,Vol.524, Kluwer Aca., 2000.
[83] I. I. Vrabie. Compactness Methods for Nonlinear Evolutions. Longman, Harlow,1987.
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[85] X. Xue. Existence of solutions for semilinear nonlocal Cauchy problems in Ba-nach spaces. Electron. J. Differential Equations, 64:1–7, 2005.
[86] X. Xue. Nonlinear differential equations with nonlocal conditions in Banachspaces. Nonlinear Anal., 63:575–586, 2005.
[87] X. Xue and G. Song. Multivalued perturbation to nonlinear differential inclusionswith memory in Banach spaces. Chin. Annals Math., 17(B):237–244, 1996.
[88] X. M. Xue and G. Z. Song. Perturbed nonlinear evolution inclusions in Banachspaces. Acta Math. Sci., 15(2):189–195, 1995.
[89] X. M. Xue and G. Z. Song. Semilinear differential inclusions in separable Banachspaces. Northeast Math. J., 11(2):151–156, 1995.
[90] E. Zeidler. Nonlonear Functional Analysis and its Applications. /B, NolinearMonotone Operators, Springer-Verlag, New York-Berlin-Heidelberg, 1990.
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[92] L. Zhu and G. Li. On a nonlocal problem for semilinear differential equationswith upper semicontinuous nonlinearities in general Banach spaces. J. Math.Anal. Appl., 341:660–675, 2008.