Hammerstein integral inclusions

in reflexive Banach spaces

Authors:
Tiziana Cardinali and Nikolaos S. Papageorgiou

Journal:
Proc. Amer. Math. Soc. **127** (1999), 95-103

MSC (1991):
Primary 47H04, 47H30, 45G10

DOI:
https://doi.org/10.1090/S0002-9939-99-04906-0

MathSciNet review:
1610932

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we examine multivalued Hammerstein integral equations defined in a separable reflexive Banach space. We prove existence theorems for both the ``convex'' problem (the multifunction is convex-valued) and the ``nonconvex'' problem (the multifunction is not necessarily convex-valued). We also show that the solution set of the latter is dense in the solution set of the former (relaxation theorem). Finally we present some examples illustrating the applicability of our abstract results.

**1.**N. U. Ahmed,*Nonlinear integral equations on reflexive Banach spaces with applications to stochastic integral equations and abstract evolution equations*, J. Integral Eqns.**1**(1979), pp. 1-15.**2.**H. Amann,*Existence theorems for equations of Hammerstein type*, Applicable Anal.**2**(1972/73), 385–397. MR**0394339**, https://doi.org/10.1080/00036817208839052**3.**J. Appell, E. De Pascale, H. T. Nguyêñ, and P. P. Zabreĭko,*Nonlinear integral inclusions of Hammerstein type*, Topol. Methods Nonlinear Anal.**5**(1995), no. 1, 111–124. Contributions dedicated to Ky Fan on the occasion of his 80th birthday. MR**1350348****4.**Alberto Bressan and Giovanni Colombo,*Extensions and selections of maps with decomposable values*, Studia Math.**90**(1988), no. 1, 69–86. MR**947921****5.**Haim Brezis and Felix E. Browder,*Existence theorems for nonlinear integral equations of Hammerstein type*, Bull. Amer. Math. Soc.**81**(1975), 73–78. MR**0407669**, https://doi.org/10.1090/S0002-9904-1975-13641-X**6.**Felix E. Browder,*Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type*, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 425–500. MR**0394340****7.**K. C. Chang,*Variational methods for nondifferentiable functionals and their applications to partial differential equations*, J. Math. Anal. App.**80**(1981), pp. 102-129.**8.**C. V. Coffmann,*Variational theory of set-valued Hammerstein operators in Banach spaces. The eigenvalue problem*, Ann. Scuola Norm. Sup. Pisa**4**(1978), pp. 633-655.**9.**K. Glashoff and J. Sprekels,*An application of Glicksberg’s theorem to set-valued integral equations arising in the theory of thermostats*, SIAM J. Math. Anal.**12**(1981), no. 3, 477–486. MR**613326**, https://doi.org/10.1137/0512041**10.**E. Klein and A. Thompson,*Theory of Correspondences*, Wiley, New York, (1984).**11.**L. Lyapin,*Hammerstein inclusions*, Diff. Equations**10**(1976), pp. 638-643.**12.**Salvatore A. Marano,*Existence theorems for a semilinear elliptic boundary value problem*, Ann. Polon. Math.**60**(1994), no. 1, 57–67. MR**1295108****13.**Donal O’Regan,*Integral equations in reflexive Banach spaces and weak topologies*, Proc. Amer. Math. Soc.**124**(1996), no. 2, 607–614. MR**1301043**, https://doi.org/10.1090/S0002-9939-96-03154-1**14.**Donal O’Regan,*Integral inclusions of upper semi-continuous or lower semi-continuous type*, Proc. Amer. Math. Soc.**124**(1996), no. 8, 2391–2399. MR**1342037**, https://doi.org/10.1090/S0002-9939-96-03456-9**15.**Nikolaos S. Papageorgiou,*Convergence theorems for Banach space valued integrable multifunctions*, Internat. J. Math. Math. Sci.**10**(1987), no. 3, 433–442. MR**896595**, https://doi.org/10.1155/S0161171287000516**16.**Nikolaos S. Papageorgiou,*On measurable multifunctions with applications to random multivalued equations*, Math. Japon.**32**(1987), no. 3, 437–464. MR**914749****17.**Nikolaos S. Papageorgiou,*Measurable multifunctions and their applications to convex integral functionals*, Internat. J. Math. Math. Sci.**12**(1989), no. 1, 175–191. MR**973087**, https://doi.org/10.1155/S0161171289000220**18.**N. S. Papageorgiou,*Existence of solutions for integral inclusions of Urysohn type with nonconvex-valued orientor field*, J. Optim. Theory Appl.**64**(1990), no. 1, 207–215. MR**1035431**, https://doi.org/10.1007/BF00940032

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Additional Information

**Tiziana Cardinali**

Affiliation:
Department of Mathematics, University of Perugia, Via Vanvitelli 1, Perugia 06123, Italy

**Nikolaos S. Papageorgiou**

Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 157 80, Greece

Email:
npapg@math.ntua.gr

DOI:
https://doi.org/10.1090/S0002-9939-99-04906-0

Keywords:
Compact operator,
measurable multifunction,
lsc and usc multifunction,
multivalued Nemitsky operator,
$h$-continuous multifunction,
Leray-Schauder alternative theorem,
relaxation theorem,
elliptic inclusions,
Green's operator.

Received by editor(s):
March 6, 1997

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1999
American Mathematical Society