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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

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 & 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,, \emph{Journal of Differential Equations}, 220 (2006), 434.  doi: 10.1016/j.jde.2005.04.007.  Google Scholar

[2]

A. Ćwiszewski, Degree theory for perturbations of m-accretive operators generating compact semigroups with constraints,, \emph{Journal of Evolution Equations}, 7 (2007), 1.  doi: 10.1007/s00028-006-0225-3.  Google Scholar

[3]

A. Ćwiszewski, Positive periodic solutions of parabolic evolution problems: A translation along trajectories approach,, \emph{Central European Journal of Mathematics}, 9 (2011), 244.  doi: 10.2478/s11533-011-0010-6.  Google Scholar

[4]

A. Ćwiszewski, Forced oscillations in strongly damped beam equation,, \emph{Topol. Methods Nonlinear Anal.}, 37 (2011), 259.   Google Scholar

[5]

A. Ćwiszewski, Averaging principle and hyperbolic evolution equations,, \emph{Nonlinear Analysis: Theory, 75 (2012), 2362.  doi: 10.1016/j.na.2011.10.034.  Google Scholar

[6]

A. Ćwiszewski and P. Kokocki, Krasnosel'skii type formula and translation along trajectories method for evolution equations,, \emph{Discrete Continuous Dynam. Systems - B}, 22 (2008), 605.  doi: 10.3934/dcds.2008.22.605.  Google Scholar

[7]

A. Ćwiszewski and P. Kokocki, Periodic solutions of nonlinear hyperbolic evolution systems,, \emph{Journal of Evolution Equations}, 10 (2010), 677.  doi: 10.1007/s00028-010-0066-y.  Google Scholar

[8]

J. W. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems,, \emph{London Mathematical Society Lectures Note Series}, 278 (2000).  doi: 10.1017/CBO9780511526404.  Google Scholar

[9]

K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations,, \emph{Graduate Texts in Mathematics}, 194 (2000).   Google Scholar

[10]

M. Furi and M. P. Pera, Global bifurcation of fixed points and the Poincaré translation operator on manifolds,, \emph{Annali di Matematica pura ed applicata}, 173 (1997), 313.  doi: 10.1007/BF01783474.  Google Scholar

[11]

M. Furi and M. P. Pera, A continuation principle for forced oscillations on differentiable manifolds,, \emph{Pacific Journal of Mathematics}, 121 (1986), 321.   Google Scholar

[12]

R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations,, {Lecture Notes in Mathematics}, 586 (1977).   Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[14]

E. Hille and R. Phillips, Functional Analysis and Semi-Groups,, American Mathematical Society, (1957).   Google Scholar

[15]

M. Kamenskii, O. Makarenkov and P. Nistri, A continuation principle for a class of periodically perturbed autonomous systems,, \emph{Mathematische Nachrichten}, 281 (2008), 42.  doi: 10.1002/mana.200610586.  Google Scholar

[16]

P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance,, \emph{Nonlinear Analysis: Theory, 85 (2013), 253.  doi: 10.1016/j.na.2013.02.030.  Google Scholar

[17]

B. Laloux and J. Mawhin, Multiplicity, Leray-Schauder formula, and bifurcation,, \emph{Jourbal of Differential Equations}, 24 (1977), 309.   Google Scholar

[18]

J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems,, Amer. Math. Soc., (1979).   Google Scholar

[19]

J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations,, in \emph{Topological methods in differential equations and inclusions}, (1995).   Google Scholar

[20]

J. Mawhin, Continuation theorems for nonlinear operator equations: the legacy of Leray and Schauder,, \emph{Travaux mathmatiques}, (1999).   Google Scholar

[21]

J. Mawhin, Topological bifurcation theory: old and new,, \emph{Progress in variational methods}, (2011).   Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, VEB Deutscher Verlag der Wissenschaften, (1978).   Google Scholar

show all references

References:
[1]

A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups,, \emph{Journal of Differential Equations}, 220 (2006), 434.  doi: 10.1016/j.jde.2005.04.007.  Google Scholar

[2]

A. Ćwiszewski, Degree theory for perturbations of m-accretive operators generating compact semigroups with constraints,, \emph{Journal of Evolution Equations}, 7 (2007), 1.  doi: 10.1007/s00028-006-0225-3.  Google Scholar

[3]

A. Ćwiszewski, Positive periodic solutions of parabolic evolution problems: A translation along trajectories approach,, \emph{Central European Journal of Mathematics}, 9 (2011), 244.  doi: 10.2478/s11533-011-0010-6.  Google Scholar

[4]

A. Ćwiszewski, Forced oscillations in strongly damped beam equation,, \emph{Topol. Methods Nonlinear Anal.}, 37 (2011), 259.   Google Scholar

[5]

A. Ćwiszewski, Averaging principle and hyperbolic evolution equations,, \emph{Nonlinear Analysis: Theory, 75 (2012), 2362.  doi: 10.1016/j.na.2011.10.034.  Google Scholar

[6]

A. Ćwiszewski and P. Kokocki, Krasnosel'skii type formula and translation along trajectories method for evolution equations,, \emph{Discrete Continuous Dynam. Systems - B}, 22 (2008), 605.  doi: 10.3934/dcds.2008.22.605.  Google Scholar

[7]

A. Ćwiszewski and P. Kokocki, Periodic solutions of nonlinear hyperbolic evolution systems,, \emph{Journal of Evolution Equations}, 10 (2010), 677.  doi: 10.1007/s00028-010-0066-y.  Google Scholar

[8]

J. W. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems,, \emph{London Mathematical Society Lectures Note Series}, 278 (2000).  doi: 10.1017/CBO9780511526404.  Google Scholar

[9]

K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations,, \emph{Graduate Texts in Mathematics}, 194 (2000).   Google Scholar

[10]

M. Furi and M. P. Pera, Global bifurcation of fixed points and the Poincaré translation operator on manifolds,, \emph{Annali di Matematica pura ed applicata}, 173 (1997), 313.  doi: 10.1007/BF01783474.  Google Scholar

[11]

M. Furi and M. P. Pera, A continuation principle for forced oscillations on differentiable manifolds,, \emph{Pacific Journal of Mathematics}, 121 (1986), 321.   Google Scholar

[12]

R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations,, {Lecture Notes in Mathematics}, 586 (1977).   Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[14]

E. Hille and R. Phillips, Functional Analysis and Semi-Groups,, American Mathematical Society, (1957).   Google Scholar

[15]

M. Kamenskii, O. Makarenkov and P. Nistri, A continuation principle for a class of periodically perturbed autonomous systems,, \emph{Mathematische Nachrichten}, 281 (2008), 42.  doi: 10.1002/mana.200610586.  Google Scholar

[16]

P. Kokocki, Averaging principle and periodic solutions for nonlinear evolution equations at resonance,, \emph{Nonlinear Analysis: Theory, 85 (2013), 253.  doi: 10.1016/j.na.2013.02.030.  Google Scholar

[17]

B. Laloux and J. Mawhin, Multiplicity, Leray-Schauder formula, and bifurcation,, \emph{Jourbal of Differential Equations}, 24 (1977), 309.   Google Scholar

[18]

J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems,, Amer. Math. Soc., (1979).   Google Scholar

[19]

J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations,, in \emph{Topological methods in differential equations and inclusions}, (1995).   Google Scholar

[20]

J. Mawhin, Continuation theorems for nonlinear operator equations: the legacy of Leray and Schauder,, \emph{Travaux mathmatiques}, (1999).   Google Scholar

[21]

J. Mawhin, Topological bifurcation theory: old and new,, \emph{Progress in variational methods}, (2011).   Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, VEB Deutscher Verlag der Wissenschaften, (1978).   Google Scholar

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