Existence Results of Mild Solutions for Second Order Perturbed Pseudo Integro-Differential Evolution Equations with Infinite State-Dependent delay in Fréchet Spaces

Abibssi, Imane; Baghli-Bendimerad, Selma

doi:10.2478/tmmp-2025-0023

Abstract

This paper is concerned with the existence of mild solutions in Fréchet spaces combined with semigroup theory, using Avramescu’s nonlinear alternative for the sum of compact operators and contraction maps, to the second order perturbed pseudo integro-differential evolution equations with state-dependent delay.

References

ABADA, N.—AGARWAL, R. P.—BENCHOHRA, M.—HAMMOUCHE H.: Existence results for nondensely defined impulsive semilinear functional differential equations with state-dependent delay, Asian-Eur. J. Math. 1 (2008), no. 4, 449–468.
Search in Google Scholar Back to article
ABIBSSI, I.—BAGHLI-BENDIMERAD, S.: On nonlocal second order perturbed pseudo integro-differential evolution equations with infinite state-dependent delay, Math. Appl.: Functional Analysis Applications 52 (2024), no. 2, 271–298.
Search in Google Scholar Back to article
ABIBSSI, I.—BAGHLI-BENDIMERAD, S.: On second order perturbed evolution equations with infinite state-dependent delay, Appl. Math. 52 (2025), no. 1, 77–99.
Search in Google Scholar Back to article
ABIBSSI, I.—BOUTEFFAL, Z.—BAGHLI-BENDIMERAD, S.: Mild solutions for second-order partial functional and neutral functional perturbed evolution equations with state-dependent delays and nonlocal conditions in Fréchet spaces, El Khayyam (2025), accepted.
Search in Google Scholar Back to article
AHMED, N. U.: Semigroup Theory with Applications to Systems and Control. Pitman Res. Notes Math. Ser. Vol. 246. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1991.
Search in Google Scholar Back to article
AOUAD, DJ.—BAGHLI-BENDIMERAD, S.: Mild solutions for perturbed evolution equations with infinite state-dependent delay, Electron. J. Qual. Theory Differ. Equ. 59 (2013), 1–24.
Search in Google Scholar Back to article
ARTHI, G.—BALACHANDRAN, K.: Controllability of second-order impulsive evolution systems with infinite delay, Nonlinear Anal. Hybrid Syst. 11 (2014), 139–153.
Search in Google Scholar Back to article
AVRAMESCU, C.: Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ. 5 (2003), 1–15.
Search in Google Scholar Back to article
BAGHLI-BENDIMERAD, S.: Global mild solution for functional evolution inclusions with state-dependent delay, J. Adv. Res. Dynam. Control Syst. 5 (2013), no. 4, 1–19.
Search in Google Scholar Back to article
BAGHLI-BENDIMERAD, S.—ABIBSSI, I.: The existence of mild solutions for nonlocal perturbed evolution equations with infinite state-dependent delay in Fréchet spaces, Arch. Control Sci. 35 LXXI (2025), no. 2, 321–349.
Search in Google Scholar Back to article
BAGHLI-BENDIMERAD, S.—BOUDEFLA, CH.: Controllability of Mild Solutions for Fractional Neutral Evolution Equations with State-Dependent Delay, Arch. Control Sci. (2025), accepted.
Search in Google Scholar Back to article
BAGHLI, S.—BENCHOHRA, M.: Multivalued evolution equations with infinite delay in Fréchet spaces, Electron. J. Qual. Theory Differ. Equ. 2008 (2008), no. 33, 24 pp.
Search in Google Scholar Back to article
BALACHANDRAN K.–ANTHONI, S. M.: Existence of solutions of second order neutral functional differential equations, Tamkang J. Math. 30 (1999), 299–309.
Search in Google Scholar Back to article
BALACHANDRAN, K.—PARK, D. G.—ANTHONI, S. M.: Existence of solutions of abstract nonlinear second-order neutral functional integro-differential equations, Comput. Math. Appl. 46 (2003), 1313–1324.
Search in Google Scholar Back to article
BÉLAIR, J.: Population models with state-dependent delays, Lect. Notes Pure Appl. Math. 131 (1990), 165–176.
Search in Google Scholar Back to article
BENCHOHRA, M.—N’GUŔÉKATA, G. M.—SALIM, A.: Advanced Topics on Semilinear Evolution Equations In: Series on Concrete and Applicable Mathematics. Vol. 25, Worls Scientific, 2025.
Search in Google Scholar Back to article
BENCHOHRA, M.—SALIM, A.—ZHOU, Y.: Integro-Differential equations: Analysis, Stability and Controllability, De Gruyter, Berlin, Boston, 2024.
Search in Google Scholar Back to article
BENDIMERAD, S. E.—AYAD, A.—BETTACHE, A.—BAGHLI S. —AYAD, A. N. E. I.: A novel formula for calculating repulsive forces in horizontally aligned nonferrous cylindrical particles within an Eddy current separator, Powder Technology, 120300 (2024), 10 pp. Doi: 10.1016/j.powtec.2024.120300.
Search in Google Scholar Back to article
BENDIMERAD, S. E.—BAGHLI, S.—AYAD, A.—TILMATINE, A.: Mutual inductance of two coaxial rectangular PCB coils incorporating magnetic layer, Journal of Electrical Engineering: The Journal of Slovak University of Technology, 72 (2021), no. 5, 287–296. Doi: 10.2478/jee-2021-0041.
Search in Google Scholar Back to article
BENSALEM, A.—SALIM, A.—BENCHOHRA, M.: Ulam-Hyers-Rassias stability of neutral functional integro-differential evolution equations with non-instantaneous impulses on an unbounded interval, Qual. Theory Dyn. Syst. 22 (2023), Art. no. 88.
Search in Google Scholar Back to article
BOUDEFLA, CH.—BAGHLI-BENDIMERAD, S.: Nonlocal controllability of mild solutions for evolution equations with state-dependent delay in Fréchet spaces, Int. J. Appl. Math. 37 (2024), no. 1, 53–71.
Search in Google Scholar Back to article
BOUDEFLA CH.—BAGHLI-BENDIMERAD, S.: Nonlocal controllability of mild solutions for neutral evolution equations with state-dependent delay in Fréchet spaces, Arch. Control Sci. 34 LXX (2024), no. 1, 117–148.
Search in Google Scholar Back to article
BOUDEFLA, CH.—SAHRAOUI, F.—BAGHLI-BENDIMERAD, S.: Controllability of mild solutions for second-order neutral evolution equations with state-dependent delay, Complex Anal. Oper. Theory 18 (2024), Art. no. 80, 24 pp.
Search in Google Scholar Back to article
DRIVER, R. D.—NORRIS, M. J.: Note on uniqueness for a one-dimensional two-body problem of classical electrodynamics, Ann. Phys. 42 (1967), 347–351.
Search in Google Scholar Back to article
HALE, J.—KATO, J.: Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj 21 (1978), 11–41.
Search in Google Scholar Back to article
HERNÀNDEZ, E.: A remark on second order differential equations with nonlocal conditions, Cadernos Mat. 4 (2003), 299–309.
Search in Google Scholar Back to article
HERNÀNDEZ, E.—MCKIBBEN, M. A.: Some comments on: Existence of solutions of abstractnonlinear second-order neutral functional integro-differential equations, Comput. Math. Appl. 50 (2005), 655–669.
Search in Google Scholar Back to article
HERNÀNDEZ, E.—SAKTHIVEL, R,—AKI, S. T.: Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations 28 (2008), 1–11.
Search in Google Scholar Back to article
HINO, Y.—MURAKAMI, S.—NAITO, T.: Functional differential equations with infinite delay, Lecture Notes in Math. Vol. 1473, Springer-Verlag, Berlin, 1991.
Search in Google Scholar Back to article
LACHACHI-MERAD, N.—BAGHLI-BENDIMERAD, S.—BENCHOHRA, M.: Unique Mild Solutions for Caputo’s Fractional Perturbed Evolution Equations with State-Dependent Delay, Evol. Equ. Control Theory, 13 (2024), no. 1, 160–172.
Search in Google Scholar Back to article
LACHACHI-MERAD, N.—BAGHLI-BENDIMERAD, S.—BENCHOHRA, M. — AOUAD, DJ.: Unique mild solution for fractional partial and neutral evolution equations with state-dependent delay, Rocky Mountain J. Math., 55 (2025), no. 5, 1351–1359.
Search in Google Scholar Back to article
LACHACHI-MERAD, N.—BAGHLI-BENDIMERAD, S.—BENCHOHRA, M. — KARAPINAR, E.: Nonlocal partial fractional evolution equations with state-dependent delay, Proyecciones, 42 (2023), no. 5, 1191–1210.
Search in Google Scholar Back to article
KOZAK, M.: A fundamental solution of a second-order differential equation in a Banach space, Univ. Iagel. Acta Math. 32 (1995), 275–289.
Search in Google Scholar Back to article
MACKEY, M. C.—MILTON, J.: Feedback delays and the origin of blood cell dynamics, Comm. Theor. Biol. 1 (1990), 299–327.
Search in Google Scholar Back to article
MEBARKI, A.—BAGHLI-BENDIMERAD, S.: Neutral multi-valued integro-differential evolution equations with infinite state-dependent delay, Turkish J. Math. 44 (2020), no. 6, 2312–2329.
Search in Google Scholar Back to article
PAZY, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983.
Search in Google Scholar Back to article
SALIM, A.—MESRI, F.—BENCHOHRA, M.—TUNÇ, C.: Controllability of second order semilinear random differential equations in Fréchet spaces, Mediterr. J. Math. 20, (2023), Art. no. 84.
Search in Google Scholar Back to article
WILLÉ, D. R.—BAKER, C. T. H: Stepsize control and continuity consistency for state-dependent delay-differential equations, J. Comput. Appl. Math. 53 (1994), no. 2, 163–170.
Search in Google Scholar Back to article
YOSIDA, K.: Functional Analysis. 6th edition. Springer-Verlag, Berlin, 1980.
Search in Google Scholar Back to article

Existence Results of Mild Solutions for Second Order Perturbed Pseudo Integro-Differential Evolution Equations with Infinite State-Dependent delay in Fréchet Spaces

Abstract

Paradigm

My account