×

On the solution sets of differential inclusions and the periodic problem in Banach spaces. (English) Zbl 1034.34072

In the first part of this interesting paper, the authors study the topological characterization of the solution set to the initial value problem for the semilinear differential inclusion \[ x'(t)\in Ax(t)+F(t,x(t)),\tag{1} \]
\[ x(0)=x_0\in D,\tag{2} \] where \(A\) is the infinitesimal generator of a linear \(C_0\)-semigroup \(\{U(t)\}_{t\geq 0},\) \(F: [0,T]\times D \multimap E\) is an upper-Carathéodory set-valued map and \(D\) is a closed subset of a Banach space \(E.\) Under appropriate assumptions concerning \(D\) and some natural boundary conditions, they prove that the set of all mild solutions to (1), (2), is an \(R_{\delta}\) set in the space of continuous maps \([0,T]\to E.\) The proofs are based on some new approximation-selection techniques for set-valued maps. In the second part of the paper, applications to periodic problems and to the existence of equilibria are given.

MSC:

34G25 Evolution inclusions
49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Akhmerov, R.R.; Kamenskii, M.I.; Potapov, A.S.; Rodkina, A.E.; Sadovskii, B.N., Measures of noncompactness and condensing operators, (1992), Birkhäuser Basel · Zbl 0748.47045
[2] Andres, J.; Gabor, G.; Górniewicz, L., Topological structure of solution sets to multivalued asymptotic problems, Z. anal. anwendungen, 19, 35-60, (2000) · Zbl 0974.34045
[3] Andres, J.; Gabor, G.; Górniewicz, L., Acyclicity of solution sets to functional inclusions, Nonlinear anal. TMA, 49, 671-688, (2002) · Zbl 1012.34011
[4] Aronszajn, N., Le correspondant topologique de l’unicite dans la théorie des équations differentielles, Ann. math., 43, 730-738, (1942) · Zbl 0061.17106
[5] Aubin, J.-P.; Frankowska, H., Set-valued analysis, (1990), Birkhäuser Basel
[6] Bader, R., Fixed point theorems for compositions of set-valued maps with single-valued maps, Ann. univ. mariae Curie-skłodowska, sect. A, lublin, LI.2, 29-41, (1997) · Zbl 1012.47043
[7] Bader, R., The periodic problem for semilinear differential inclusions in Banach spaces, Comment. math. univ. carolin., 39, 671-684, (1998) · Zbl 1060.34508
[8] Bader, R., On the semilinear multivalued flow under constraints and the periodic problem, Comment. math. univ. carolin., 41, 719-734, (2000) · Zbl 1057.34064
[9] Bader, R.; Kryszewski, W., On the solution set of constrained differential inclusions with applications, Set-valued anal., 9, 289-313, (2001) · Zbl 0991.34011
[10] Ben-El-Mechaiekh, H.; Kryszewski, W., Equilibria of set-valued maps on nonconvex domains, Trans. amer. math. soc., 349, 4159-4179, (1997) · Zbl 0887.47040
[11] Bessaga, Cz.; Pełczyński, J., Infinite dimensional topology, Monografie mat, Vol. 58, (1975), PWN Warszawa
[12] D. Bothe, Multivalued differential equations on graphs and applications, Ph.D. Dissertation, Universität Paderborn, 1992. · Zbl 0789.34013
[13] D. Bothe, Periodic solutions of nonlinear evolution problems, manuscript, 1997.
[14] Bothe, D., Multivalued perturbations of m-accretive differential inclusions, Israel J. math., 108, 109-138, (1998) · Zbl 0922.47048
[15] Bressan, A.; Staicu, V., On nonconvex perturbations of maximal monotone differential inclusions, Set-valued anal., 2, 415-437, (1994) · Zbl 0820.47072
[16] Cârjă, O.; Vrabie, I.I., Some new viability results for semilinear differential inclusions, Nonlinear differential equations appl., 4, 401-424, (1997) · Zbl 0876.34069
[17] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley-Interscience New York · Zbl 0727.90045
[18] Cornet, B.; Czarnecki, M.-O., Représentations lisses de sous-ensambles épi-lipschitziens de \(R\^{}\{n\}\), C. R. acad. sci. Paris Sér. I, 325, 475-480, (1997) · Zbl 0893.49012
[19] Ćwiszewski, A.; Kryszewski, W., Equilibria of set-valued mapsa variational approach, Nonlinear anal. TMA, 48, 707-746, (2002) · Zbl 1030.49021
[20] Deimling, K., Periodic solutions of differential equations in Banach spaces, Manuscripta math., 24, 31-44, (1978) · Zbl 0373.34032
[21] Deimling, K., Nonlinear functional analysis, (1985), Springer Berlin · Zbl 0559.47040
[22] Deimling, K., Multivalued differential equations, (1992), Walter de Gruyter Berlin · Zbl 0760.34002
[23] Diestel, J., Remarks on weak compactness in L1(μ,X), Glasgow math. J., 18, 87-91, (1977) · Zbl 0342.46020
[24] Diestel, J., Geometry of Banach spaces, Lnm, Vol. 485, (1975), Springer Berlin
[25] Diestel, J.; Ruess, W.M.; Schachermayer, W., Weak compactness in L1(μ,X), Proc. amer. math. soc., 118, 447-453, (1993) · Zbl 0785.46037
[26] Donchev, T., Semicontinuous differential inclusions, Rend. sem. univ. Padova, 101, 147-160, (1999) · Zbl 0936.34010
[27] Dragoni, R.; Macki, J.W.; Nistri, P.; Zecca, P., Solution sets of differential operators in abstract spaces, (1996), Addison Wesley Longman Ltd Harlow · Zbl 0892.34061
[28] Fitzpatrick, P.M.; Petryshyn, W.V., A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings, Trans. amer. math. soc., 194, 1-25, (1974) · Zbl 0297.47049
[29] Górniewicz, L., Homological methods in fixed point theory of multivalued maps, Diss. math. (warszawa), CXXIX, 1-66, (1976)
[30] Górniewicz, L., Topological approach to differential inclusions, (), 129-190 · Zbl 0834.34022
[31] L. Górniewicz, Topological structure of solution sets: current results, Nicholas Copernicus Univ. Preprints 3 (2000).
[32] Górniewicz, L.; Nistri, P.; Obukhovskii, V., Differential inclusions on proximate retracts of Hilbert spaces, Internat. J. non-linear differential equations, 3, 13-26, (1997)
[33] Haddad, G.; Lasry, J.M., Periodic solutions of functional differential inclusions and fixed points of σ-selectionable correspondences, J. math. anal. appl., 96, 295-312, (1983) · Zbl 0539.34031
[34] Hu, S.; Papageorgiou, N.S., On the topological regularity of the solution set of differential inclusions with constraints, J. differential equations, 107, 280-290, (1994) · Zbl 0796.34017
[35] S. Hu, N.S. Papageorgiou, Handbook of Set-Valued Analysis, Vols. I, II, Kluwer Academic Publishers, Dordrecht, 1999, 2001.
[36] Hyman, D.M., On decreasing sequences of compact absolute retracts, Fund. math., 64, 91-97, (1969) · Zbl 0174.25804
[37] Jarnik, J.; Kurzweil, J., On conditions on right hand sides of differential relations, Casopis pest. mat., 102, 334-349, (1977) · Zbl 0369.34002
[38] Kamenski, M.; Obukhovski, V.; Zecca, P., On the translation multioperator along the solutions of semilinear differential inclusions in Banach spaces, Canad. appl. math. qrt., 6, 139-154, (1998) · Zbl 0921.34017
[39] Kamenski, M.; Obukhovski, V.; Zecca, P., Condensing multivalued maps and semilinear differential inclusions in Banach spaces, (2000), Walter de Gruyter Berlin
[40] Kryszewski, W., Homotopy invariants for set-valued maps: homotopy-approximation approach, (), 269-284 · Zbl 0756.47036
[41] Kryszewski, W., Graph approximation of set-valued maps on non-compact spaces, Topology appl., 83, 1-21, (1997)
[42] Lakshmikantham, V.; Leela, S., Nonlinear differential equations in abstract spaces, (1981), Pergamon Press Oxford · Zbl 0456.34002
[43] Martin, H.M., Nonlinear operators and differential equations in Banach spaces, pure & applied mathematics, (1976), Wiley New York
[44] Motreanu, D.; Pavel, N.H., Tangency, flow invariance for differential equations, and optimization problems, (1999), Marcel Dekker Inc New York · Zbl 0937.34035
[45] Nussbaum, R.D., The fixed point index for local condensing maps, Ann. mat. pura appl., 89, 217-258, (1971) · Zbl 0226.47031
[46] Obukhovski, V.; Zecca, P., On some properties od dissipative functional differential inclusions in a Banach space, Topology meth. nonlinear anal., 15, 369-384, (2000) · Zbl 0980.34059
[47] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer Berlin · Zbl 0516.47023
[48] Petryshyn, W.V., Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces, Trans. amer. math. soc., 182, 323-352, (1973) · Zbl 0277.47033
[49] Plaskacz, S., Periodic solutions of differential inclusions on compact subsets of \(R\^{}\{n\}\), J. math. anal. appl., 148, 202-212, (1990) · Zbl 0705.34040
[50] Prüss, J., Periodic solutions of semilinear evolution equations, Nonlinear anal., 3, 601-612, (1979) · Zbl 0419.34061
[51] Rockafellar, R.T., Clarke’s tangent cones and boundaries of closed sets in \(R\^{}\{n\}\), Nonlinear anal., 3, 145-154, (1979) · Zbl 0443.26010
[52] Rzeżuchowski, T., Scorza-dragoni type theorems for upper semicontinuous multivalued functions, Bull. acad. polon. sci., 8, 61-67, (1980) · Zbl 0459.28007
[53] Vrabie, I.I., Compactness methods for nonlinear evolutions, (1987), Pitman London · Zbl 0721.47050
[54] Yosida, K., Functional analysis, (1966), Springer Berlin · Zbl 0152.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.