Almost automorphic solutions of dynamic equations on time scales

Abstract In the present work, we introduce the concept of almost automorphic functions on time scales and present the first results about their basic properties. Then, we study the nonautonomous dynamic equations on time scales given by x Δ ( t ) = A ( t ) x ( t ) + f ( t ) and x Δ ( t ) = A ( t ) x ( t ) + g ( t , x ( t ) ) , t ∈ T where T is a special case of time scales that we define in this article. We prove a result ensuring the existence of an almost automorphic solution for both equations, assuming that the associated homogeneous equation of this system admits an exponential dichotomy. Also, assuming that the function g satisfies the global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear dynamic equation on time scales. Further, we present some applications of our results for some new almost automorphic time scales. Finally, we present some interesting models in which our main results can be applied.

[1]  Wan-Tong Li,et al.  Periodic solutions for dynamic equations on time scales , 2007 .

[2]  Wan-Tong Li,et al.  Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales , 2007 .

[3]  Tomás Caraballo,et al.  ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL/DIFFERENCE EQUATIONS WITHOUT FAVARD’S SEPARATION CONDITION. II , 2008 .

[4]  Carlos Lizama,et al.  Almost automorphic solutions of non-autonomous difference equations , 2013 .

[5]  J. Mesquita,et al.  Basic results for functional differential and dynamic equations involving impulses , 2013 .

[6]  K. Gopalsamy,et al.  Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons , 2002, IEEE Trans. Neural Networks.

[7]  A. Tsoularis,et al.  Analysis of logistic growth models. , 2002, Mathematical biosciences.

[8]  M. Fan,et al.  Convergence dynamics and pseudo almost periodicity of a class of nonautonomous RFDEs with applications , 2005 .

[9]  S. Bochner.  UNIFORM CONVERGENCE OF MONOTONE SEQUENCES OF FUNCTIONS. , 1961, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Ferhan Merdivenci Atici,et al.  An application of time scales to economics , 2006, Math. Comput. Model..

[11]  A. Dogan.  Higher-order singular multi-point boundary-value problems on time scales , 2011, Proceedings of the Edinburgh Mathematical Society.

[12]  Shui-Nee Chow,et al.  Existence of periodic solutions of autonomous functional differential equations , 1974 .

[13]  Guangxing Song,et al.  Extremal solutions of periodic boundary value problems for first order integro-differential equations of mixed type , 2004 .

[14]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[15]  T. Caraballo,et al.  Levitan/Bohr Almost Periodic and Almost Automorphic Solutions of Second-Order Monotone Differential Equations , 2011 .

[16]  S Bochner,et al.  A NEW APPROACH TO ALMOST PERIODICITY. , 1962, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Lili Wang,et al.  Unique Existence Theorem of Solution of Almost Periodic Differential Equations on Time Scales , 2012 .

[18]  Tomás Caraballo,et al.  Almost periodic and almost automorphic solutions of linear differential equations , 2012 .

[19]  A. Slavík.  Averaging dynamic equations on time scales , 2012 .

[20]  K. Ezzinbi,et al.  Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities , 2011 .

[21]  Jing-huai Liu,et al.  Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations , 2010 .

[22]  Damiano Brigo,et al.  Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices , 2000, Finance Stochastics.

[23]  Zhenxin Liu,et al.  Square-mean almost automorphic solutions for some stochastic differential equations , 2010, 1001.3049.

[24]  Yongkun Li,et al.  Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales , 2012, Advances in Difference Equations.

[25]  M. Bohner,et al.  Periodic solutions of functional dynamic equations with infinite delay , 2008 .

[26]  Wan-Tong Li,et al.  Asymptotic properties of the positive equilibrium of a discrete survival model , 2004, Appl. Math. Comput..

[27]  Márcia Federson,et al.  Measure functional differential equations and functional dynamic equations on time scales , 2012 .

[28]  Sanyi Tang,et al.  Optimal impulsive harvesting on non-autonomous Beverton–Holt difference equations , 2006 .

[29]  M. Saunders,et al.  Plant-Provided Food for Carnivorous Insects: a Protective Mutualism and Its Applications , 2009 .

[30]  Antonín Slavík.  Dynamic equations on time scales and generalized ordinary differential equations , 2012 .

[31]  M. Fan,et al.  Boundedness and Stability of Semi-linear Dynamic Equations on Time Scales (Progress in Qualitative Theory of Functional Equations) , 2012 .

[32]  E. Savaş.  Quasi-power increasing sequence for generalized absolute summability , 2008 .

[33]  Constantin Corduneanu,et al.  Almost periodic functions , 1968 .

[34]  M. Bohner,et al.  Periodicity of scalar dynamic equations and applications to population models , 2007 .

[35]  Jerome A. Goldstein,et al.  Almost automorphic solutions of semilinear evolution equations , 2004 .

[36]  Yongkun Li,et al.  Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales , 2011 .

[37]  A. Peterson,et al.  On the asymptotic behavior of solutions of Emden–Fowler equations on time scales , 2010, Annali di Matematica Pura ed Applicata.

[38]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[39]  A. Peterson,et al.  Belohorec‐type oscillation theorem for second order sublinear dynamic equations on time scales , 2011 .

[40]  PRODUCT INTEGRATION ON TIME SCALES , 2010 .

[41]  Gisèle M. Mophou,et al.  Almost Automorphic Functions of Order and Applications to Dynamic Equations on Time Scales , 2014 .

[42]  D. Anderson,et al.  POSITIVE PERIODIC TIME-SCALE SOLUTIONS FOR FUNCTIONAL DYNAMIC EQUATIONS , 2006 .

[43]  Gaston M. N’Guérékata,et al.  Almost Automorphic and Almost Periodic Functions in Abstract Spaces , 2001 .

[44]  Christopher C. Tisdell,et al.  Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling , 2008 .

[45]  Wei Lin,et al.  Square-mean pseudo almost automorphic process and its application to stochastic evolution equations , 2011 .

[46]  Carlos Lizama,et al.  Almost Automorphic Solutions of Difference Equations , 2009 .

[47]  Tom Fenchel,et al.  Theories of populations in biological communities , 1977, Vegetatio.

[48]  S Bochner,et al.  CONTINUOUS MAPPINGS OF ALMOST AUTOMORPHIC AND ALMOST PERIODIC FUNCTIONS. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[49]  Yongkun Li,et al.  Almost Periodic Functions on Time Scales and Applications , 2011 .

[50]  Youssef N. Raffoul,et al.  Existence results for periodic solutions of integro-dynamic equations on time scales , 2009 .

[51]  H. Qian,et al.  Remarks on discrete and continuous large-scale models of DNA dynamics. , 1998, Biophysical journal.