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Convolutions in \((\mu,\nu)\)-pseudo-almost periodic and \((\mu,\nu)\)-pseudo-almost automorphic function spaces and applications to solve integral equations. (English) Zbl 1460.34050

Summary: In this paper we give sufficient conditions on \(k\in L^1(\mathbb{R})\) and the positive measures \(\mu, \nu\) such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo-almost automorphic) function spaces are invariant by the convolution product \(\zeta f=k\ast f\). We provide an appropriate example to illustrate our convolution results. As a consequence, we study under Acquistapace-Terreni conditions and exponential dichotomy, the existence and uniqueness of (\(\mu,\nu\))-pseudo-almost periodic (respectively, (\(\mu,\nu\))-pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
35B15 Almost and pseudo-almost periodic solutions to PDEs
35K57 Reaction-diffusion equations
37A30 Ergodic theorems, spectral theory, Markov operators
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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[1] P. Acquistapace, F. Flandoli, and B. Terreni, “Initial boundary value problems and optimal control for nonautonomous parabolic systems”, SIAM Journal on Control and Optimization, vol. 29, pp. 89-118, 1991. · Zbl 0753.49003
[2] P. Acquistapace, and B. Terreni, “A unified approach to abstract linear nonautonomous parabolic equations”, Rendiconti del Seminario Matematico della Universit‘a di Padova, vol. 78, pp. 47-107, 1987. · Zbl 0646.34006
[3] M. Baroun, S. Boulite, G. M. N’Guérékata, and L. Maniar, “Almost automorphy of semilinear parabolic evolution equations”, Electronic Journal of Differential Equations, vol. 60, pp. 1-9, 2008. · Zbl 1170.34344
[4] J. Blot, P. Cieutat, and K. Ezzinbi, “Measure theory and almost automorphic functions: new developments and applications”, Nonlinear Analysis, vol. 75, pp. 2426-2447, 2012. · Zbl 1248.43004
[5] J. Blot, P. Cieutat, and K. Ezzinbi, “New approach for weighted pseudo almost periodic functions under the light of measure theory, basic results and applications”, Applicable Analysis, vol. 92, no. 3, pp. 493-526, 2013. · Zbl 1266.43004
[6] C. Corduneanu, Almost Periodic Functions, Wiley, New York, 1968, (Reprinted, Chelsea, New York, 1989). · Zbl 0175.09101
[7] A. Coronel, M. Pinto, and D. Sepulveda, “Weighted pseudo almost periodic functions, convolutions and abstract integral equations”, J. Math. Anal. Appl., vol. 435, pp. 1382-1399, 2016. · Zbl 1337.42003
[8] T. Diagana, “Double weighted pseudo-almost periodic functions”, Afr. Diaspora J. Math., vol. 12, pp. 121-136, 2011. · Zbl 1247.42008
[9] T. Diagana, “Existence of weighted pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations”, Nonlinear Analysis, vol. 74, pp. 600-615, 2011. · Zbl 1209.34074
[10] T. Diagana, “Pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations”, Journal of the Franklin Institute, vol. 348, pp. 2082-2098, 2011. · Zbl 1237.34132
[11] T. Diagana, K. Ezzinbi, and M. Miraoui, “Pseudo-Almost Periodic and Pseudo-Almost Automorphic solutions to Some Evolution Equations Involving Theorical Measure Theory”, CUBO, A Mathematical Journal, vol. 16, no. 2, pp. 01-31, 2014. · Zbl 1326.34075
[12] M. Fréchet, “Sur le théor‘eme ergodique de Birkhoff”, Les comptes Rendus Mathématiques de l’Académie de Sciences Paris, vol. 213, pp. 607-609, 1941. · Zbl 0027.07704
[13] A. Haraux, Syst‘emes dynamiques et dissipatifs et applications, Recherches en Mathématiques Appliquées, Masson, Paris, 1991. · Zbl 0726.58001
[14] L. Maniar, and R. Schnaubelt, “Almost periodicity of inhomogeneous parabolic evolution equations”, Lecture Notes in Pure and Applied Mathematics, vol. 234, pp. 299-318, 2003. · Zbl 1047.35078
[15] F. Mbounja Béssém“è, D. Békollè, K. Ezzinbi, S. Fatajou, and D.E. Houpa Danga, “Convolution in μ-pseudo almost periodic and μ-pseudo almost automorphic functions spaces and applications to solve Integral equations”, Nonautonomous Dynamical Systems, vol. 7, pp. 32-52, 2020. · Zbl 1509.43008
[16] G. M. N’Guérékata, Topics in Almost automorphy, Springer, New York, Boston, London, Moscow, 2005. · Zbl 1073.43004
[17] H. L. Royden, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988. · Zbl 0704.26006
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