×

Topological degree methods for perturbations of operators generating compact \(C_0\) semigroups. (English) Zbl 1086.47030

Let \(E\) be a Banach space, \(A:D(A)\subseteq E\to E\) a closed linear operator such that \(-A\) generates a compact \(C_0\)-semigroup, \(M\subset E\) a neighborhood retract and \(F:M\to E\) a locally Lipschitz map. The present paper is devoted to the construction of a topological degree theory for maps of the form \(-A+F:M\cap D(A)\to E\). By using the introduced topological degree and an abstract result concerning the branching of fixed points, the author studies the bifurcation of periodic points of a parameterized boundary value problem of the form \[ \begin{gathered} \dot{u}=- {\mathcal A}(\lambda)u+{\mathcal F}(t,u,\lambda), \\ u(t)\in M,\; u(0)=u(T).\end{gathered} \] As applications, the author considers periodic problems for some classes of partial differential equations.

MSC:

47H11 Degree theory for nonlinear operators
47J35 Nonlinear evolution equations
47J15 Abstract bifurcation theory involving nonlinear operators
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47N20 Applications of operator theory to differential and integral equations
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] Ambrosio, L.; Dancer, N., Calculus if variations and partial differential equations—topics on evolution problems and degree theory, (2000), Springer Berlin
[3] Aubin, J.-P.; Frankowska, H., Set-valued analysis, (1991), Birkhäuser Boston
[4] Bader, R.; Kryszewski, W., On the solution set of differential inclusions and the periodic problem in Banach spaces, Nonlinear anal., 54, 707-754, (2003) · Zbl 1034.34072
[5] Bostan, M., Periodic solutions for evolution equations, Elect. J. differential equations monograph, 03, (2002), http://ejde.math.swt.edu/ · Zbl 1010.34060
[6] D. Bothe, Multivalued differential equations on graphs and applications, Ph.D. Thesis, University of Paderborn, 1992. · Zbl 0789.34013
[7] Bothe, D., Periodic solutions of a nonlinear periodic problems from heterogeneous catalysis, Differential integral equations, 16, 6, 641-670, (2001) · Zbl 1032.34061
[8] R. Brown, The Lefschetz Fixed Point Theorems, Scott Foresman and Comp., 1991
[9] A. Ćwiszewski, Topological degree for perturabations of m-accretive operators generating compact semigroups, to appear. · Zbl 1124.47039
[10] A. Ćwiszewski, W. Kryszewski, Branching of periodic solutions in constraint sets, to appear. · Zbl 1188.35088
[11] Diaz, J.I.; Vrabie, I.I., Existence for reaction diffusion systems. A compatness method approach, J. math. anal. appl., 188, 521-540, (1994) · Zbl 0815.35132
[12] Dugundji, J.; Granas, A., Fixed point theory, (2004), Springer Berlin
[13] Fu, X.; Song, S., The generalized degree for multivalued compact perturbations of \(M\)-accretive operators and applications, Nonlinear anal., 43, 767-776, (2001) · Zbl 0997.47049
[14] Furi, M.; Pera, M.P., Global bifurcation of fixed points and the Poincaré translation operator on manifolds, Ann. mat. pura appl. IV,, CLXXIII, 313-331, (1997) · Zbl 0944.37024
[15] 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
[16] Z.G. Guan, A.G. Kartsatos, A degree for maximal monotone operators, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, vol. 178, Dekker, New York, 1997.
[17] Hu, S.; Papageorgiou, N., Handbook of multivalued analysis—volume II: applications, (2000), Kluwer Dordrecht · Zbl 0943.47037
[18] A.G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, Proceedings of the First World Congress of Nonlinear Analysis, Tampa, FL, 1992, Walter De Gruyter, New York, 1995. · Zbl 0849.47027
[19] Kartsatos, A.G.; Skrypnik, I.V., Topological degree theories for densely defined mappings involving operators of type \((S_+)\), Adv. differential equations, 4, 3, 413-456, (1999) · Zbl 0959.47037
[20] Martin, R.H., Nonlinear operators and differential equations in Banach spaces, (1976), Wiley New York
[21] Nussbaum, R.D., The fixed point index for local condensing maps, Ann. mat. pura appl., 84, 217-258, (1971) · Zbl 0226.47031
[22] N.H. Pavel, Differential Equations, Flow Invariance And Applications, Research Notes in Mathematics, vol. 113, Pitman Advanced Publishing Program, London, 1984.
[23] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer Berlin · Zbl 0516.47023
[24] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, vol. 49, AMS, Providence, RI, 1997. · Zbl 0870.35004
[25] Vrabie, I.I., Periodic solutions for nonlinear evolution equations in a Banach space, Proc. amer. math. soc., 19, 3, 653-661, (1990) · Zbl 0701.34074
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.