PC-Asymptotically almost automorphic mild solutions for impulsive integro-differential equations with nonlocal conditions

Nadia Ikhlef, Abdelhamid Bensalem, Abdelkrim Salim, Mouffak Benchohra, Sara Litimein

Abstract


In this article, we study the existence of PC-asymptotically almost automorphic mild solutions of integro-differential equations with nonlocal  conditions via resolvent operators in Banach space. Further, we give sufficient conditions for the solutions to depend continuously on the initial condition. Finally, an example is given to validate the theory part.

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Ahmad, B., Boumaaza, M., Salim, A., Benchohra, M., Random solutions for generalized Caputo periodic and non-local boundary value problems, Foundations, 3(2)(2023), 275- 289.

Appell, J., Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl., 83(1)(1981), 251-263.

Bainov, D., Simeonov, P., Integral Inequalities and Applications, Springer Science, Business Media, 57, 1992.

Bana s, J., Goebel, K., Measure of Noncompactness in Banach Spaces, in: Lecture Notes in Pure and Applied Math., vol. 60, Marcel Dekker, New York, 1980.

Benchohra, M., Abbas, S., Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics, 39, Springer, Cham, 2015.

Benchohra, M., Henderson, J., Ntouyas, S.K., Impulsive Differential Equations and inclusions, Hindawi Publishing Corporation, New York, 2006.

Benchohra, M., Karap nar, E., Lazreg, J.E., Salim, A., Advanced Topics in Fractional Differential Equations: A Fixed Point Approach, Springer, Cham, 2023.

Benchohra, M., N'Gu er ekata, G.M., Rezoug, N., asymptotically almost automorphic mild solutions for second order nonautonomous semilinear evolution equations, J. Comput. Anal. Appl., 29(3)(2021), 468-493.

Benchohra, M., Ntouyas, S.K., Nonlocal Cauchy problems for neutral functional differential and integrodi erential inclusions in Banach spaces, J. Math. Anal. Appl., 258(2001), 573-590.

N. Ikhlef, A. Bensalem, A. Salim, M. Benchohra and S. Litimein

Benkhettou, N., Aissani, K., Salim, A., Benchohra, M., Tunc, C., Controllability of fractional integro-differential equations with in finite delay and non-instantaneous impulses, Appl. Anal. Optim., 6(2022), 79-94.

Benkhettou, N., Salim, A., Aissani, K., Benchohra, M., Karap nar, E., Non-instantaneous impulsive fractional integro-di erential equations with state-dependent delay, Sahand Commun. Math. Anal., 19(2022), 93-109.

Bensalem, A., Salim, A., Benchohra, M., Ulam-Hyers-Rassias stability of neutral functional integrodi erential evolution equations with non-instantaneous impulses on an unbounded interval, Qual. Theory Dyn. Syst., 22(2023), 88.

Bensalem, A., Salim, A., Benchohra, M., N'Gu er ekata, G.M., Functional integro-differential equations with state-dependent delay and non-instantaneous impulsions: Existence and qualitative results, Fractal Fract., 6(2022), 615.

Bochner, S., A new approach in almost-periodicity, Proc. Nat. Acad. Sci. USA, 48(1962), 2039-2043.

Bugajewski, D., Diagana, T., Almost automorphy of the convolution operator and applications to differential and functional-differential equations, Nonlinear Stud., 13(2006), 129-140.

Bugajewski, D., N'Gu er ekata, G.M., On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces, Nonlinear Anal., 59(2004), 1333-1345.

Byszewski, Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem, Appl. Math. Stoch. Anal., 12(1999), 91-97.

Cao, J., Huang, Z., N'Gu er ekata, G.M., Existence of asymptotically almost automorphic mild solutions for nonautonomous semilinear evolution equations, Electron. J. Differ. Equ., 2018(2018), No. 37, 1-16.

Darbo, G., Punti uniti in transformazioni a condominio non compatto, Rend. Sem. Math. Univ. Padova, 24(1955), 84-92.

Desch, W., Grimmer, R.C., Schappacher, W., Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104(1984), 219-234.

Diagana, T., Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013.

Diagana, T., N'Gu er ekata, G.M., Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett., 20(2007), 462-466.

Diop, M.A., Ezzinbi, K., Ly, M.P., Nonlocal problems for integrodi erential equation via resolvent operators and optimal control, Di er. Incl. Control Optim., 42(2022), 5-25.

Dudek, S., Fixed point theorems in Fr echet algebras and Fr echet spaces and applications to nonlinear integral equations, Appl. Anal. Disc. Math., 11(2017), 340-357.

Ezzinbi, K., N'Gu er ekata, G.M., Almost automorphic solutions for some partial functional di erential equations, J. Math. Anal. Appl., 328(2007), 344-358.

Grimmer, R.C., Resolvent opeators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273(1982), 333-349.

Grimmer, R.C., Pritchard, A.J., Analytic resolvent operators for integral equations in a Banach space, J. Di er. Equ., 50(1983), 234-259.

Hao, X., Liu, L., Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces, Math. Methods Appl. Sci., 40(2017), 4832-4841.

Liang, J., Zhang, J., Xiao, T., Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl., 340(2008), 1493-1499.

Monch, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4(1980), 985-999.

N'Guerekata, G.M., Sur les solutions presque automorphes d'equations differentielles abstraites, Ann. Sci. Math. Quebec, 1(1981), 69-79.

N'Guerekata, G.M., Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, Plenum Publishers, New York, London, Moscow, 2001.

N'Gu er ekata, G.M., Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005.

N'Gu er ekata, G.M., Pankov, A., Integral operators in spaces of bounded, almost periodic and almost automorphic functions, Differential Integral Equations, 21(2008), 1155-1176.

Xue, X., Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces, Electron. J. Differential Equations, 64(2005), 1-7.




DOI: http://dx.doi.org/10.24193/subbmath.2025.1.09

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