×

Multiple bounded solutions of differential inclusions: The Nielsen theory approach. (English) Zbl 0940.34008

Summary: The generalized Nielsen number is defined for self-maps, which are composed by operators with \(R_\delta\)-values, on compact connected ANRs. Then it is applied to Carathéodory differential inclusions with constraints for obtaining multiplicity criteria. More precisely, such problems are transformed to those for the lower estimate of fixed points of the related operators with the given properties on bounded, compact, connected neighbourhood retracts of Fréchet spaces. In this way, multiple solutions can be proved e.g. for the multivalued initial value problem on the half-line or, in the single-valued case, for boundary value problems to ordinary differential equations.

MSC:

34A60 Ordinary differential inclusions
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Andres, J., A target problem for differential inclusions with state-space constraints, Demonstratio Math., 30, 783-790 (1997) · Zbl 0910.34026
[2] J. Andres, Almost periodic and bounded solutions of Carathéodory differential inclusions, Differential Integral Equations, in press.; J. Andres, Almost periodic and bounded solutions of Carathéodory differential inclusions, Differential Integral Equations, in press.
[3] J. Andres, G. Gabor, and, G. Górniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc, in press.; J. Andres, G. Gabor, and, G. Górniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc, in press. · Zbl 0936.34023
[4] J. Andres, L. Górniewicz, and, J. Jezierski, A generalized Nielsen number and multiplicity results for differential inclusions, Topology Appl, in press.; J. Andres, L. Górniewicz, and, J. Jezierski, A generalized Nielsen number and multiplicity results for differential inclusions, Topology Appl, in press.
[5] Aubin, J.-P.; Cellina, A., Differential Inclusions (1982), Springer-Verlag: Springer-Verlag Berlin
[6] Bader, R., A sufficient condition for the existence of multiple periodic solutions of differential inclusions, (Geba, K.; Górniewicz, L., Topology in Nonlinear Analysis. Topology in Nonlinear Analysis, Banach Center Publ., 35 (1996), PWN: PWN Warsaw), 129-138 · Zbl 0847.34015
[7] Bader, R.; Kryszewski, W., Fixed point index for compositions of set-valued maps with the proximally ∞-connected values on arbitrary ANR’s, Set-Valued Anal., 2, 459-480 (1994) · Zbl 0846.55001
[8] Jiang, Boju, Nielsen Fixed Point Theory. Nielsen Fixed Point Theory, Contemp. Math., 14 (1983), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0512.55003
[9] Borsuk, K., Theory of Retracts (1967), PWN: PWN Warsaw · Zbl 0153.52905
[10] Borisovich, A. Yu.; Kucharski, Z.; Marzantowicz, W., Nielsen numbers and lower estimates for the number of solutions to a certain system of nonlinear integral equations, Appl. Aspects of Global Analysis. Appl. Aspects of Global Analysis, New Developments in Global Analysis Series (1994), Voronezh Univ. Press, p. 3-10 · Zbl 0853.45009
[11] Brooks, R. B.S.; Brown, R. F.; Pak, J.; Taylor, D. H., Nielsen numbers of maps of tori, Proc. Amer. Math. Soc., 52, 398-400 (1975) · Zbl 0309.55005
[12] Brown, R. F., The Lefschetz Fixed Point Theorem (1971), Scott, Foresman: Scott, Foresman Glenview · Zbl 0216.19601
[13] Brown, R. F., Multiple solutions to parametrized nonlinear differential systems from Nielsen fixed point theory, (Rassias, Th. M., Nonlinear Analysis (1987), World Scientific: World Scientific Singapore), 89-98 · Zbl 0676.47038
[14] Brown, R. F., Topological identification of multiple solutions to parametrized nonlinear equations, Pacific J. Math., 131, 51-69 (1988) · Zbl 0615.47042
[15] Brown, R. F., Nielsen fixed point theory and parametrized differential equations, Contemp. Math. (1988), Amer. Math. Soc: Amer. Math. Soc Providence, p. 33-45 · Zbl 0665.34050
[16] Brown, R. F.; Zezza, P., Multiple local solutions to nonlinear control processes, J. Optim. Theory Appl., 67, 463-485 (1990) · Zbl 0697.49041
[17] Dzedzej, Z., Fixed point index theory for a class of nonacyclic multivalued maps, Dissertationes Math., 253, 1-58 (1985) · Zbl 0592.55001
[18] Fečkan, M., Nielsen fixed point theory and nonlinear equations, J. Differential Equations, 106, 312-331 (1993) · Zbl 0839.47041
[19] Fečkan, M., Multiple perturbed solutions near nondegenerate manifolds of solutions, Comment. Math. Univ. Carolin., 34, 635-643 (1993) · Zbl 0795.58006
[20] Fečkan, M., Multiple periodic solutions of small vector fields on differentiable manifolds, J. Differential Equations, 113, 189-200 (1994) · Zbl 0821.34039
[21] Górniewicz, L.; Granas, A.; Kryszewski, W., On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighbourhood retracts, J. Math. Anal. Appl., 161, 457-473 (1991) · Zbl 0757.54019
[22] Górniewicz, L., Homological method in fixed-point theory of multivalued maps, Dissertationes Math., 129, 1-71 (1976) · Zbl 0324.55002
[23] Granas, A., The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France, 100, 209-228 (1972) · Zbl 0236.55004
[24] Jezierski, J., The Nielsen relation for multivalued maps, SERDICA, 13, 174-181 (1987) · Zbl 0652.55004
[25] Jarnı́k, J.; Kurzweil, J., Integral of multivalued mappings and its connection with differential relations, Cas. Pěst. Mat., 108, 8-28 (1983) · Zbl 0536.28006
[26] Tsai-han, Kiang, The Theory of Fixed Point Classes (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0676.55001
[27] Krasnosel’skii, M. A., The Operator of Translation along the Trajectories of Differential Equations. The Operator of Translation along the Trajectories of Differential Equations, Transl. Math. Monogr., 19 (1968), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 1398.34003
[28] Kryszewski, W.; Miklaszewski, D., The Nielsen number of set-valued maps: An Approximation approach, SERDICA, 15, 336-344 (1989) · Zbl 0712.55003
[29] McCord, C. K., Nielsen Theory and Dynamical Systems. Nielsen Theory and Dynamical Systems, Contemp. Math., 152 (1993), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0780.00035
[30] Musielak, J., Introduction to Functional Analysis (1976), PWN: PWN Warsaw
[31] Rybinski, L., On Carathéodory type selectors, Fund. Math., 125, 187-193 (1985) · Zbl 0614.28005
[32] Schirmer, H., A Nielsen number for fixed points and near points of small multifunctions, Fund. Math., 88, 145-156 (1975) · Zbl 0306.55008
[33] H. Schirmer, An index and a Nielsen number for \(n\); H. Schirmer, An index and a Nielsen number for \(n\) · Zbl 0543.55003
[34] Scholz, K., The Nielsen fixed point theory for non-compact spaces, Rocky Mountain J. Math., 4, 81-87 (1974) · Zbl 0275.55013
[35] Spanier, E., Algebraic Topology (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0145.43303
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.