×

A fixed point approach in the study of the solution sets of Lipschitzian functional-differential inclusions. (English) Zbl 0735.34016

Given topological and Banach spaces \((\Lambda,{\mathcal T})\) and \((Z,\|\cdot\|)\) a set-valued function \(F: \Lambda\to N(Z)\) is said to have a retractive representation if there exist a set \(Y\in N(Z)\) and a continuous mapping \(f:\Lambda \times Y\to Z\) such that for every \((\lambda,y)\in \Lambda\times Y\) one has \(f(\lambda,y)\in F(\lambda)\) and \(f(\lambda,y)=y\) if and only if \(y\in F(\lambda)\). A retractive representation \(f\) is said to be equi-continuous, and when \(\Lambda\) is a uniform space, uniformly continuous or equi-uniformly continuous if the mappings \(f(\cdot,y)\), \(y\in Y\), are equi-continuous and uniformly or equi-uniformly continuous, respectively. The present paper contains sufficient conditions implying that the solution set mappings of some functional-differential inclusions in Banach spaces depending on parameters \(\lambda\in\Lambda\) have retractive and equi-uniformly continuous retractive representations. Here and in the present paper \(N(Z)\) denotes the family of all nonempty subsets of \(Z\).

MSC:

34A60 Ordinary differential inclusions
34K05 General theory of functional-differential equations
Full Text: DOI

References:

[1] Filippov, A. F., Classical solutions of differential equations with multivalued right-hand sides, SIAM J. Control Optim., 5, 609-621 (1967) · Zbl 0238.34010
[2] Hermes, H., The generalized differential equation \(x\)′ \(ϵR(t, x)\), Adv. in Math., 4, 149-169 (1970) · Zbl 0191.38803
[3] Himmelberg, C. J.; Van Vleck, F. S., Lipschitzean generalized differential equations, (Rend. Sem. Mat. Univ. Padova, 48 (1973)), 159-169 · Zbl 0289.49009
[4] Markin, J. T., Continuous dependence of fixed point sets, (Proc. Amer. Math. Soc., 38 (1973)), 545-547 · Zbl 0278.47036
[5] Kisielewicz, M., Nonlinear Functional-Differential Equations of Neutral Type (1984), PWN
[6] Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0538.34007
[7] Stassinopoulos, G. I.; Vinter, R. B., Continuous dependence of solutions of a differential inclusion on the right-hand side with applications to stability of optimal control problems, SIAM J. Control Optim., 17, 432-449 (1979) · Zbl 0442.49025
[8] Tolstonogov, A. A., O strukture množestva rešenij differencial’nych vklučenij v banachovom prostranstve, Math. Sb., 118, 3-18 (1982) · Zbl 0514.34053
[9] Tolstonogov, A. A.; Čugunov, P. I., O mnozěstve resěnij differencial’nogo vklučenija v banachovom prostranstve I, Sibirsk Math. Zh., 24, 144-159 (1983) · Zbl 0537.34011
[10] Čugunov, P. I., O zavisimosti resěnij differencial’nogo vključenija ot nacal’nych uslovij i parametra, Differentsial’nye Uravneniya, 17, 1426-1433 (1981)
[11] Bogatyrev, A. V., Nepodvižnyje tocki i svojstva rešenij differencial’nych vklučenij, Izv. Akad. Nauk SSR, 47, 895-909 (1983) · Zbl 0537.34013
[12] Bulgakov, A. I.; Ljapin, L. N., O svjaznosti množestv rešenij funkcionalnych vklučenij, Mat. Sb., 119, 295-300 (1982) · Zbl 0511.47016
[13] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions, (Lecture Notes in Math., Vol. 580 (1977), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0346.46038
[14] Haddad, G., Topological properties of the sets of solutions for functional differential inclusions, Nonlinear Anal. Theory Methods Appl., 5, 1349-1366 (1981) · Zbl 0496.34041
[15] 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
[16] Himmelberg, C. J.; Van Vleck, F. S., A note on the solution sets of differential inclusions, Rocky Mountain J. Math., 12, 621-625 (1982) · Zbl 0531.34007
[17] Lim, T. C., On fixed point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl., 110, 436-441 (1985) · Zbl 0593.47056
[18] Papageorgiou, N. S., A stability result for differential inclusions in Banach spaces, J. Math. Anal. Appl., 118, 232-246 (1986) · Zbl 0594.34016
[19] Ricceri, B., Une propriété topologique de l’ensemble des points fixes d’une contraction multivoque à valeurs convexes, Rend. Acad. Nat. Lincei, 84, 283-286 (1987) · Zbl 0666.47030
[20] Ricceri, O. Naselli, \(A\)-fixed points of multi-valued contractions, J. Math. Anal. Appl., 135, 406-418 (1988) · Zbl 0662.54030
[21] Ricceri, O. Naselli; Ricceri, B., Differential inclusions depending on a parameter, Bull. Polish Acad. Sci. Math., 37, 665-671 (1989) · Zbl 0755.34014
[22] Ricceri, O. Naselli, A continuous dependence theorem for differential inclusions in Banach spaces, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 12, 45-52 (1988) · Zbl 0679.34071
[23] Rybiński, L., Representation of multivalued mappings, (Differential Equations and Optimal Control, Proc. V Regional Sci. Session Math.. Differential Equations and Optimal Control, Proc. V Regional Sci. Session Math., Zielona Góra (1985)), 75-79 · Zbl 0653.54017
[24] Tsukada, M., Convergence of best approximations in a smooth Banach space, J. Approx. Theory, 40, 301-309 (1984) · Zbl 0545.41042
[25] Francaviglia, S.; Lechicki, A.; Levi, S., Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl., 112, 347-370 (1985) · Zbl 0587.54003
[26] Kuratowski, K., Topologie, II (1968), Academic Press: Academic Press Orlando, FL
[27] Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71, 152-182 (1951) · Zbl 0043.37902
[28] Michael, E., Continuous selections, I, Ann. of Math., 63, 361-382 (1956) · Zbl 0071.15902
[29] Ekeland, I.; Valadier, M., Representation of set-valued mappings, J. Math. Anal. Appl., 35, 621-629 (1971) · Zbl 0246.54018
[30] Ioffe, A. D., Single-valued representation of set-valued mappings. II. Application to differential inclusions, SIAM J. Control Optim., 21, 641-651 (1983) · Zbl 0539.49009
[31] Rybiński, L., On Carathéodory type selections, Fund. Math., 125, 187-193 (1985) · Zbl 0614.28005
[32] Kuratowski, K.; Ryll-Nardzewski, C., A general theorem on selectors, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys., 397-403 (1965) · Zbl 0152.21403
[33] Itoh, S., A random fixed point theorem for a multivalued contraction mapping, Pacific J. Math., 68, 85-90 (1977) · Zbl 0335.54036
[34] Michael, E., A note on paracompact spaces, (Proc. Amer. Math. Soc., 4 (1953)), 831-838 · Zbl 0052.18701
[35] Drewnowski, L.; Kisielewicz, M.; Rybiński, L., Minimal selections and fixed point sets of multivalued contraction mappings in uniformly convex Banach spaces, Comment. Math., 29, 43-50 (1989) · Zbl 0705.47046
[36] Rybiński, L., Multivalued contraction with parameter, Ann. Polon. Math., 45, 275-282 (1985) · Zbl 0604.47035
[37] Köthe, G., Topological Vector Spaces, I (1969), Springer-Verlag: Springer-Verlag Berlin/Heidelberg · Zbl 0179.17001
[38] Kisielewicz, M., Multivalued differential equations in separable Banach spaces, J. Optim. Theory Appl., 37, 231-249 (1982) · Zbl 0458.34008
[39] Orlicz, W.; Szufla, S., On some classes of nonlinear Volterra integral equations in Banach spaces, Bull. Polish Acad. Sci. Math., 30, 239-250 (1982) · Zbl 0501.45013
[40] Brooks, J. K.; Dinculeanu, N., Weak compactness in spaces of Bochner integrable functions and applications, Adv. in Math., 24, 172-188 (1977) · Zbl 0354.46026
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.