Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

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November 2015, 14(6): 2315-2334. doi: 10.3934/cpaa.2015.14.2315

Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

Piotr Kokocki ^1,

1.	Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Received January 2015 Revised May 2015 Published September 2015

Abstract
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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.

Keywords: evolution equation, Topological degree, periodic solution..

Mathematics Subject Classification: Primary: 37B30, 47J35; Secondary: 35B10, 47J1.

Citation: Piotr Kokocki. Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2315-2334. doi: 10.3934/cpaa.2015.14.2315

References:

[1]	A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups, Journal of Differential Equations, 220 (2006), 434-477. doi: 10.1016/j.jde.2005.04.007.
[2]	A. Ćwiszewski, Degree theory for perturbations of m-accretive operators generating compact semigroups with constraints, Journal of Evolution Equations, 7 (2007), 1-33. doi: 10.1007/s00028-006-0225-3.
[3]	A. Ćwiszewski, Positive periodic solutions of parabolic evolution problems: A translation along trajectories approach, Central European Journal of Mathematics, 9 (2011), 244-268. doi: 10.2478/s11533-011-0010-6.
[4]	A. Ćwiszewski, Forced oscillations in strongly damped beam equation, Topol. Methods Nonlinear Anal., 37 (2011), 259-282.
[5]	A. Ćwiszewski, Averaging principle and hyperbolic evolution equations, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), 2362-2375 doi: 10.1016/j.na.2011.10.034.
[6]	A. Ćwiszewski and P. Kokocki, Krasnosel'skii type formula and translation along trajectories method for evolution equations, Discrete Continuous Dynam. Systems - B, 22 (2008), 605-628. doi: 10.3934/dcds.2008.22.605.
[7]	A. Ćwiszewski and P. Kokocki, Periodic solutions of nonlinear hyperbolic evolution systems, Journal of Evolution Equations, 10 (2010), 677-710. doi: 10.1007/s00028-010-0066-y.
[8]	J. W. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lectures Note Series, 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.
[9]	K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
[10]	M. Furi and M. P. Pera, Global bifurcation of fixed points and the Poincaré translation operator on manifolds, Annali di Matematica pura ed applicata, 173 (1997), 313-331. doi: 10.1007/BF01783474.
[11]	M. Furi and M. P. Pera, A continuation principle for forced oscillations on differentiable manifolds, Pacific Journal of Mathematics, 121 (1986), 321-338.
[12]	R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, 586, Springer-Verlag, Berlin, 1977.
[13]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
[14]	E. Hille and R. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1957.
[15]	M. Kamenskii, O. Makarenkov and P. Nistri, A continuation principle for a class of periodically perturbed autonomous systems, Mathematische Nachrichten, 281 (2008), 42-61. doi: 10.1002/mana.200610586.
[16]	P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance, Nonlinear Analysis: Theory, Methods and Applications, 85 (2013), 253-278. doi: 10.1016/j.na.2013.02.030.
[17]	B. Laloux and J. Mawhin, Multiplicity, Leray-Schauder formula, and bifurcation, Jourbal of Differential Equations, 24 (1977), 309-322.
[18]	J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Amer. Math. Soc., Providence, R.I., 1979.
[19]	J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations, in Topological methods in differential equations and inclusions, Kluwer Acad. Publ., Dordrecht, 1995.
[20]	J. Mawhin, Continuation theorems for nonlinear operator equations: the legacy of Leray and Schauder, Travaux mathmatiques, Centre Univ. Luxembourg, Luxembourg, 1999.
[21]	J. Mawhin, Topological bifurcation theory: old and new, Progress in variational methods, World Sci. Publ., Hackensack, 2011.
[22]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[23]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

show all references

References:

[1]	A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups, Journal of Differential Equations, 220 (2006), 434-477. doi: 10.1016/j.jde.2005.04.007.
[2]	A. Ćwiszewski, Degree theory for perturbations of m-accretive operators generating compact semigroups with constraints, Journal of Evolution Equations, 7 (2007), 1-33. doi: 10.1007/s00028-006-0225-3.
[3]	A. Ćwiszewski, Positive periodic solutions of parabolic evolution problems: A translation along trajectories approach, Central European Journal of Mathematics, 9 (2011), 244-268. doi: 10.2478/s11533-011-0010-6.
[4]	A. Ćwiszewski, Forced oscillations in strongly damped beam equation, Topol. Methods Nonlinear Anal., 37 (2011), 259-282.
[5]	A. Ćwiszewski, Averaging principle and hyperbolic evolution equations, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), 2362-2375 doi: 10.1016/j.na.2011.10.034.
[6]	A. Ćwiszewski and P. Kokocki, Krasnosel'skii type formula and translation along trajectories method for evolution equations, Discrete Continuous Dynam. Systems - B, 22 (2008), 605-628. doi: 10.3934/dcds.2008.22.605.
[7]	A. Ćwiszewski and P. Kokocki, Periodic solutions of nonlinear hyperbolic evolution systems, Journal of Evolution Equations, 10 (2010), 677-710. doi: 10.1007/s00028-010-0066-y.
[8]	J. W. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lectures Note Series, 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.
[9]	K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
[10]	M. Furi and M. P. Pera, Global bifurcation of fixed points and the Poincaré translation operator on manifolds, Annali di Matematica pura ed applicata, 173 (1997), 313-331. doi: 10.1007/BF01783474.
[11]	M. Furi and M. P. Pera, A continuation principle for forced oscillations on differentiable manifolds, Pacific Journal of Mathematics, 121 (1986), 321-338.
[12]	R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, 586, Springer-Verlag, Berlin, 1977.
[13]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
[14]	E. Hille and R. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1957.
[15]	M. Kamenskii, O. Makarenkov and P. Nistri, A continuation principle for a class of periodically perturbed autonomous systems, Mathematische Nachrichten, 281 (2008), 42-61. doi: 10.1002/mana.200610586.
[16]	P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance, Nonlinear Analysis: Theory, Methods and Applications, 85 (2013), 253-278. doi: 10.1016/j.na.2013.02.030.
[17]	B. Laloux and J. Mawhin, Multiplicity, Leray-Schauder formula, and bifurcation, Jourbal of Differential Equations, 24 (1977), 309-322.
[18]	J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Amer. Math. Soc., Providence, R.I., 1979.
[19]	J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations, in Topological methods in differential equations and inclusions, Kluwer Acad. Publ., Dordrecht, 1995.
[20]	J. Mawhin, Continuation theorems for nonlinear operator equations: the legacy of Leray and Schauder, Travaux mathmatiques, Centre Univ. Luxembourg, Luxembourg, 1999.
[21]	J. Mawhin, Topological bifurcation theory: old and new, Progress in variational methods, World Sci. Publ., Hackensack, 2011.
[22]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[23]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

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