×

Krasnosel’skii type formula and translation along trajectories method on the scale of fractional spaces. (English) Zbl 1352.47043

Summary: We provide a global continuation principle of periodic solutions for the equation \(\dot u = - Au + F(t,u)\), where \( A:D(A) \to X\) is a sectorial operator on a Banach space \(X\) and \(F:[0, +\infty) \times X^\alpha \to X\) is a nonlinear map defined on a fractional space \(X^\alpha\). The approach that we use in this paper is based upon the theory of topological invariants that applies in the situation when the Poincaré operator associated with the equation is endowed with some form of compactness.

MSC:

47J35 Nonlinear evolution equations
37B30 Index theory for dynamical systems, Morse-Conley indices
35B10 Periodic solutions to PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
35K20 Initial-boundary value problems for second-order parabolic equations
35K90 Abstract parabolic equations
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. \'Cwiszewski, Topological degree methods for perturbations of operators generating compact \(C_0\) semigroups,, Journal of Differential Equations, 220, 434 (2006) · Zbl 1086.47030 · doi:10.1016/j.jde.2005.04.007
[2] A. \'Cwiszewski, Degree theory for perturbations of m-accretive operators generating compact semigroups with constraints,, Journal of Evolution Equations, 7, 1 (2007) · Zbl 1124.47039 · doi:10.1007/s00028-006-0225-3
[3] A. \'Cwiszewski, Positive periodic solutions of parabolic evolution problems: A translation along trajectories approach,, Central European Journal of Mathematics, 9, 244 (2011) · Zbl 1222.47132 · doi:10.2478/s11533-011-0010-6
[4] A. \'Cwiszewski, Forced oscillations in strongly damped beam equation,, Topol. Methods Nonlinear Anal., 37, 259 (2011) · Zbl 1248.47059
[5] A. \'Cwiszewski, Averaging principle and hyperbolic evolution equations,, Nonlinear Analysis: Theory, 75, 2362 (2012) · Zbl 1253.47050 · doi:10.1016/j.na.2011.10.034
[6] A. {\'C}wiszewski, Krasnosel’skii type formula and translation along trajectories method for evolution equations,, Discrete Continuous Dynam. Systems - B, 22, 605 (2008) · Zbl 1165.47056 · doi:10.3934/dcds.2008.22.605
[7] A. {\'C}wiszewski, Periodic solutions of nonlinear hyperbolic evolution systems,, Journal of Evolution Equations, 10, 677 (2010) · Zbl 1239.34039 · doi:10.1007/s00028-010-0066-y
[8] J. W. Cholewa, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lectures Note Series, 278 (2000) · Zbl 0954.35002 · doi:10.1017/CBO9780511526404
[9] K. J. Engel, One-parameter semigroups for linear evolution equations,, Graduate Texts in Mathematics, 194 (2000) · Zbl 0952.47036
[10] M. Furi, Global bifurcation of fixed points and the Poincaré translation operator on manifolds,, Annali di Matematica pura ed applicata, 173, 313 (1997) · Zbl 0944.37024 · doi:10.1007/BF01783474
[11] M. Furi, A continuation principle for forced oscillations on differentiable manifolds,, Pacific Journal of Mathematics, 121, 321 (1986) · Zbl 0554.34029
[12] R. E. Gaines, <em>Coincidence Degree and Nonlinear Differential Equations</em>,, {Lecture Notes in Mathematics, 586 (1977)} · Zbl 0339.47031
[13] D. Henry, <em>Geometric Theory of Semilinear Parabolic Equations</em>,, Springer-Verlag (1981) · Zbl 0456.35001
[14] E. Hille, <em>Functional Analysis and Semi-Groups</em>,, American Mathematical Society (1957) · Zbl 0033.06501
[15] M. Kamenskii, A continuation principle for a class of periodically perturbed autonomous systems,, Mathematische Nachrichten, 281, 42 (2008) · Zbl 1140.34017 · doi:10.1002/mana.200610586
[16] P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance,, Nonlinear Analysis: Theory, 85, 253 (2013) · Zbl 1292.34059 · doi:10.1016/j.na.2013.02.030
[17] B. Laloux, Multiplicity, Leray-Schauder formula, and bifurcation,, Jourbal of Differential Equations, 24, 309 (1977) · Zbl 0366.47029
[18] J. Mawhin, <em>Topological Degree Methods in Nonlinear Boundary Value Problems</em>,, Amer. Math. Soc. (1979) · Zbl 0414.34025
[19] J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations,, in Topological methods in differential equations and inclusions (1995) · Zbl 0834.34047
[20] J. Mawhin, Continuation theorems for nonlinear operator equations: the legacy of Leray and Schauder,, Travaux mathmatiques (1999) · Zbl 1157.01307
[21] J. Mawhin, Topological bifurcation theory: old and new,, Progress in variational methods (2011) · Zbl 1258.58001
[22] A. Pazy, <em>Semigroups of Linear Operators and Applications to Partial Differential Equations</em>,, Springer-Verlag (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[23] H. Triebel, <em>Interpolation Theory, Function Spaces, Differential Operators</em>,, VEB Deutscher Verlag der Wissenschaften (1978) · Zbl 0387.46033
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.