Convolutions in \((\mu,\nu)\)-pseudo-almost periodic and \((\mu,\nu)\)-pseudo-almost automorphic function spaces and applications to solve integral equations
- David Békollè dbekolle@univ-ndere.cm
- Khalil Ezzinbi ezzinbi@uca.ac.ma
- Samir Fatajou fatajou@yahoo.fr
- Duplex Elvis Houpa Danga e_houpa@yahoo.com
- Fritz Mbounja Béssémè mbounjafritz@gmail.com
Downloads
DOI:
https://doi.org/10.4067/S0719-06462021000100063Abstract
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 \(\left( \mu,\nu\right)\)- pseudo-almost periodic (respectively, \(\left( \mu,\nu\right)\)- pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.
Keywords
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.
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.
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.
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.
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.
C. Corduneanu, Almost Periodic Functions, Wiley, New York, 1968, (Reprinted, Chelsea, New York, 1989).
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.
T. Diagana, “Double weighted pseudo-almost periodic functions”, Afr. Diaspora J. Math., vol. 12, pp. 121–136, 2011.
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.
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.
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.
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.
A. Haraux, Syst`emes dynamiques et dissipatifs et applications, Recherches en Mathématiques Appliquées, Masson, Paris, 1991.
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.
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.
G. M. N‘Guérékata, Topics in Almost automorphy, Springer, New York, Boston, London, Moscow, 2005.
H. L. Royden, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988.
Most read articles by the same author(s)
- Abdelhai Elazzouzi, Khalil Ezzinbi, Mohammed Kriche, On the periodic solutions for some retarded partial differential equations by the use of semi-Fredholm operators , CUBO, A Mathematical Journal: Vol. 23 No. 3 (2021)
- Toka Diagana, Khalil Ezzinbi, Mohsen Miraoui, Pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory , CUBO, A Mathematical Journal: Vol. 16 No. 2 (2014): CUBO, A Mathematical Journal
- Mouffak Benchohra, Omar Bennihi, Khalil Ezzinbi, Existence Results for Some Neutral Partial Functional Differential Equations of Fractional order with State-Dependent Delay , CUBO, A Mathematical Journal: Vol. 16 No. 3 (2014): CUBO, A Mathematical Journal
- Khalil Ezzinbi, Valerie Nelson, Gaston N‘Gu´er´ekata, ð¶â½â¿â¾-Almost Automorphic Solutions of Some Nonautonomous Differential Equations , CUBO, A Mathematical Journal: Vol. 10 No. 2 (2008): CUBO, A Mathematical Journal
- Taoufik Chitioui, Khalil Ezzinbi, Amor Rebey, Existence and stability in the α-norm for nonlinear neutral partial differential equations with finite delay , CUBO, A Mathematical Journal: Vol. 15 No. 1 (2013): CUBO, A Mathematical Journal
Similar Articles
- Toka Diagana, Khalil Ezzinbi, Mohsen Miraoui, Pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory , CUBO, A Mathematical Journal: Vol. 16 No. 2 (2014): CUBO, A Mathematical Journal
- Wolfgang Rump, The tree of primes in a field , CUBO, A Mathematical Journal: Vol. 12 No. 2 (2010): CUBO, A Mathematical Journal
- Pertti Mattila, Search for geometric criteria for removable sets of bounded analytic functions , CUBO, A Mathematical Journal: Vol. 6 No. 4 (2004): CUBO, A Mathematical Journal
- Rafael del Rio, Asaf L. Franco, Jose A. Lara, An approach to F. Riesz representation Theorem , CUBO, A Mathematical Journal: Vol. 20 No. 2 (2018)
You may also start an advanced similarity search for this article.