Nonresonance problems

for differential inclusions

in separable Banach spaces

Authors:
Zouhua Ding and Athanassios G. Kartsatos

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2357-2365

MSC (1991):
Primary 34A60

MathSciNet review:
1340383

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real separable Banach space. The boundary value problem

is studied on the infinite interval Here, the closed and densely defined linear operator generates an evolution operator The function is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either is bounded and is compact for or is compact and is equicontinuous. The mapping is a bounded linear operator and is fixed. The nonresonance problem is solved by using Ma's fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in

**[1]**Giuseppe Anichini,*Nonlinear problems for systems of differential equations*, Nonlinear Anal.**1**(1976/77), no. 6, 691–699. MR**0592963****[2]**Jean-Pierre Aubin and Arrigo Cellina,*Differential inclusions*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR**755330****[3]**Avner Friedman,*Partial differential equations*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR**0445088****[4]**C. J. Himmelberg,*Measurable relations*, Fund. Math.**87**(1975), 53–72. MR**0367142****[5]**Shou Chuan Hu and Nikolaos S. Papageorgiou,*On the existence of periodic solutions for a class of nonlinear evolution inclusions*, Boll. Un. Mat. Ital. B (7)**7**(1993), no. 3, 591–605 (English, with Italian summary). MR**1244409****[6]**Athanassios G. Kartsatos,*The Leray-Schauder theorem and the existence of solutions to boundary value problems on infinite intervals*, Indiana Univ. Math. J.**23**(1973/74), 1021–1029. MR**0340697****[7]**Athanassios G. Kartsatos,*Nonzero solutions to boundary value problems for nonlinear systems*, Pacific J. Math.**53**(1974), 425–433. MR**0377164****[8]**Athanassios G. Kartsatos,*Locally invertible operators and existence problems in differential systems*, Tôhoku Math. J. (2)**28**(1976), no. 2, 167–176. MR**0430385****[9]**Athanassios G. Kartsatos,*Boundary value problems for abstract evolution equations*, Nonlinear Anal.**3**(1979), no. 4, 547–554. MR**537341**, 10.1016/0362-546X(79)90072-5**[10]**A. G. Kartsatos,*A compact evolution operator generated by a nonlinear time-dependent -accretive operator in a Banach space*, Math. Ann.**302**(1995), 473--487. MR**93c:47104****[11]**Athanassios G. Kartsatos and Ki-Yeon Shin,*Solvability of functional evolutions via compactness methods in general Banach spaces*, Nonlinear Anal.**21**(1993), no. 7, 517–535. MR**1241826**, 10.1016/0362-546X(93)90008-G**[12]**Tsoy-wo Ma,*Topological degrees of set-valued compact fields in locally convex spaces*, Dissertationes Math. Rozprawy Mat.**92**(1972), 43. MR**0309103****[13]**N. S. Papageorgiou,*A stability result for differential inclusions in Banach spaces*, J. Math. Anal. Appl.**118**(1986), 232--246**[14]**Nikolaos S. Papageorgiou,*Boundary value problems for evolution inclusions*, Comment. Math. Univ. Carolin.**29**(1988), no. 2, 355–363. MR**957404****[15]**N. S. Papageorgiou,*Boundary value problems and periodic solutions for semilinear evolution inclusions*, Comment. Math. Univ. Carolinae**35**(1994), 325--336. CMP**94:15****[16]**B. Przeradzki,*The existence of bounded solutions for differential equations in Hilbert spaces*, Ann. Polon. Math.**56**(1992), no. 2, 103–121. MR**1159982****[17]**James R. Ward Jr.,*Boundary value problems for differential equations in Banach space*, J. Math. Anal. Appl.**70**(1979), no. 2, 589–598. MR**543596**, 10.1016/0022-247X(79)90067-2**[18]**Pietro Zecca and Pier Luigi Zezza,*Nonlinear boundary value problems in Banach spaces for multivalue differential equations on a noncompact interval*, Nonlinear Anal.**3**(1979), no. 3, 347–352. MR**532895**, 10.1016/0362-546X(79)90024-5

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Additional Information

**Zouhua Ding**

Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Email:
ding@chuma.usf.edu

**Athanassios G. Kartsatos**

Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Email:
hermes@gauss.math.usf.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03439-9

Keywords:
Boundary value problem on an infinite interval,
differential inclusion,
upper semicontinuous function,
compact evolution operator

Received by editor(s):
December 16, 1994

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1996
American Mathematical Society