Nonresonance problems for differential inclusions in separable Banach spaces
Authors:
Zouhua Ding and Athanassios G. Kartsatos
Journal:
Proc. Amer. Math. Soc. 124 (1996), 23572365
MSC (1991):
Primary 34A60
MathSciNet review:
1340383
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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
(58 #28782)
 [2]
JeanPierre
Aubin and Arrigo
Cellina, Differential inclusions, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 264, SpringerVerlag, Berlin, 1984. Setvalued maps
and viability theory. MR 755330
(85j:49010)
 [3]
Avner
Friedman, Partial differential equations, Holt, Rinehart and
Winston, Inc., New YorkMontreal, Que.London, 1969. MR 0445088
(56 #3433)
 [4]
C.
J. Himmelberg, Measurable relations, Fund. Math.
87 (1975), 53–72. MR 0367142
(51 #3384)
 [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 (94k:34120)
 [6]
Athanassios
G. Kartsatos, The LeraySchauder theorem and the existence of
solutions to boundary value problems on infinite intervals, Indiana
Univ. Math. J. 23 (1973/74), 1021–1029. MR 0340697
(49 #5448)
 [7]
Athanassios
G. Kartsatos, Nonzero solutions to boundary value problems for
nonlinear systems, Pacific J. Math. 53 (1974),
425–433. MR 0377164
(51 #13337)
 [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
(55 #3390)
 [9]
Athanassios
G. Kartsatos, Boundary value problems for abstract evolution
equations, Nonlinear Anal. 3 (1979), no. 4,
547–554. MR
537341 (80i:34104), http://dx.doi.org/10.1016/0362546X(79)900725
 [10]
A. G. Kartsatos, A compact evolution operator generated by a nonlinear timedependent accretive operator in a Banach space, Math. Ann. 302 (1995), 473487. MR 93c:47104
 [11]
Athanassios
G. Kartsatos and KiYeon
Shin, Solvability of functional evolutions via compactness methods
in general Banach spaces, Nonlinear Anal. 21 (1993),
no. 7, 517–535. MR 1241826
(94h:34100), http://dx.doi.org/10.1016/0362546X(93)90008G
 [12]
Tsoywo
Ma, Topological degrees of setvalued compact fields in locally
convex spaces, Dissertationes Math. Rozprawy Mat. 92
(1972), 43. MR
0309103 (46 #8214)
 [13]
N. S. Papageorgiou, A stability result for differential inclusions in Banach spaces, J. Math. Anal. Appl. 118 (1986), 232246
 [14]
Nikolaos
S. Papageorgiou, Boundary value problems for evolution
inclusions, Comment. Math. Univ. Carolin. 29 (1988),
no. 2, 355–363. MR 957404
(89k:34018)
 [15]
N. S. Papageorgiou, Boundary value problems and periodic solutions for semilinear evolution inclusions, Comment. Math. Univ. Carolinae 35 (1994), 325336. 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
(93d:34109)
 [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
(80i:34110), http://dx.doi.org/10.1016/0022247X(79)900672
 [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
(80h:34084), http://dx.doi.org/10.1016/0362546X(79)900245
 [1]
 G. Anichini, Nonlinear problems for systems of differential equations, Nonlinear Anal. TMA 6 (1977), 691699. MR 58:28782
 [2]
 J. P. Aubin and A. Cellina, Differential Inclusions, SpringerVerlag, New York, 1984. MR 85j:49010
 [3]
 A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. MR 56:3433
 [4]
 C. J. Himmelberg, Measurable selections, Fund. Math. 87 (1975), 5372. MR 51:3384
 [5]
 S. Hu and N. S. Papageorgiou, On the existence of periodic solutions for a class of nonlinear evolution inclusions, Boll. Un. Mat. Ital. B 7 (1993), 591605. MR 94k:34120
 [6]
 A. G. Kartsatos, The LeraySchauder theorem and the existence of solutions to boundary value problems on infinite intervals, Indiana Univ. Math. J. 23 (1974), 10211029. MR 49:5448
 [7]
 A. G. Kartsatos, Nonzero solutions to boundary value problems for nonlinear systems, Pacific J. Math. 53 (1974), 425433. MR 51:13337
 [8]
 A. G. Kartsatos, Locally invertible operators and existence problems in differential systems, Tôhoku Math. J. 28 (1976), 167176. MR 55:3390
 [9]
 A. G. Kartsatos, Boundary value problems for abstract evolution equations, Nonlinear Anal. TMA 3 (1978), 18. MR 80i:34104
 [10]
 A. G. Kartsatos, A compact evolution operator generated by a nonlinear timedependent accretive operator in a Banach space, Math. Ann. 302 (1995), 473487. MR 93c:47104
 [11]
 A. G. Kartsatos and K. Y. Shin, Solvability of functional evolutions via compactness methods in general Banach spaces, Nonlinear Anal. TMA 21 (1993), 517535. MR 94h:34100
 [12]
 T. W. Ma, Topological degrees of setvalued compact fields in locally convex spaces, Dissert. Math. 92 (1972), 143. MR 46:8214
 [13]
 N. S. Papageorgiou, A stability result for differential inclusions in Banach spaces, J. Math. Anal. Appl. 118 (1986), 232246
 [14]
 N. S. Papageorgiou, Boundary value problems for evolution inclusions, Comment. Math. Univ. Carolinae 29 (1988), 355363. MR 89k:34018
 [15]
 N. S. Papageorgiou, Boundary value problems and periodic solutions for semilinear evolution inclusions, Comment. Math. Univ. Carolinae 35 (1994), 325336. CMP 94:15
 [16]
 B. Przeradzki, The existence of bounded solutions for differential equations in Hilbert spaces, Ann. Polon. Math. 56 (1992), 103121. MR 93d:34109
 [17]
 J. R. Ward, Boundary value problems for differential equations in Banach spaces, J. Math. Anal. Appl. 70 (1979), 589598. MR 80i:34110
 [18]
 P. Zecca and P. Zezza, Nonlinear boundary value problems in Banach spaces for multivalued differential equations on a noncompact interval, Nonlinear Anal. TMA 3 (1979), 347352. MR 80h:34084
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Additional Information
Zouhua Ding
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 336205700
Email:
ding@chuma.usf.edu
Athanassios G. Kartsatos
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 336205700
Email:
hermes@gauss.math.usf.edu
DOI:
http://dx.doi.org/10.1090/S0002993996034399
PII:
S 00029939(96)034399
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
