Existence results for functional perturbed differential equations of fractional order with state-dependent delay in Banach spaces

M. Helal

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 1, pp. 112-130

Résumé

In this paper, we provide sufficient conditions for the existence of solutions of initial value problem, for perturbed partial functional hyperbolic differential equations of fractional order involving Caputo fractional derivative with state-dependent delay by reducing the research to the search of the existence and the uniqueness of fixed points of appropriate operators. Our main result for this problem is based on a nonlinear alternative fixed point theorem for the sum of a completely continuous operator and a contraction one in Banach spaces due to Burton and Kirk and a fractional version of Gronwall's inequality. We should observe the structure of the space and the properties of the operators to obtain existence results. To our knowledge, there are very few papers devoted to fractional differential equations with finite and/or infinite constant delay on bounded domains. Many other questions and issues can be investigated regarding the existence in the space of weighted continuous functions, the uniqueness, the structure of the solutions set and also whether or not the condition satisfied by the operators are optimal. This paper can be considered as a contribution in this setting case. Examples are given to illustrate this work.

Bibliographie

[1] Diethelm, K. and Freed, A. D., “On the Solution of Nonlinear Fractional Order Differential Equations Used in the Modeling of Viscoplasticity”, Scientific Computing in Chemical Engineering II, Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, eds. F. Keil, W. Mackens, H. Voss, and J. Werther, Springer-Verlag, Heidelberg, 1999, 217–224 | MR

[2] Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993 | MR | Zbl

[3] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of their Applications, Academic Press, San Diego, 1999 | MR | Zbl

[4] Abbas, S. and Benchohra, M., “Partial Hyperbolic Differential Equations with Finite Delay Involving the Caputo Fractional Derivative”, Communications in Mathematical Analysis, 7:2 (2009), 62–72 | MR | Zbl

[5] Abbas, S. and Benchohra, M., “Darboux Problem for Perturbed Partial Differential Equations of Fractional Order with Finite Delay”, Elsevier Nonlinear Analysis: Hybrid Systems, 3:4 (2009), 597–604 | DOI | MR | Zbl

[6] Benchohra, M., Henderson, J., Ntouyas, S. K. and Ouahab A., “Existence Results for Functional Differential Equations of Fractional Order”, Journal of Mathematical Analysis and Applications, 338 (2008), 1340–1350 | DOI | MR | Zbl

[7] Hale, J. K. and Verduyn Lunel, S. M., Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993 | DOI | MR | Zbl

[8] Lakshmikantham, V., Wen, L. and Zhang, B., Theory of Differential Equations with Unbounded Delay, Mathematics and its Applications, 298, Kluwer Academic Publishers, Dordrecht, 1994 | MR | Zbl

[9] Corduneanu, C. and Lakshmikantham, V., “Equations with Unbounded Delay”, Nonlinear Analysis: Theory, Methods and Applications, 4:5 (1980), 831–877 | DOI | MR | Zbl

[10] Willé, D. R. and Baker, C. T. H., “Stepsize Control and Continuity Consistency for State-Dependent Delay-Differential Equations”, Journal of Computational and Applied Mathematics, 53:2 (1994), 163–170 | DOI | MR | Zbl

[11] Anguraj A., Arjunan, M. M. and Hernàndez, E. M., “Existence Results for an Impulsive Neutral Functional Differential Equation with State-Dependent Delay”, Applicable Analysis, 86:7 (2007), 861–872 | DOI | MR | Zbl

[12] Hartung, F., “Linearized Stability for a Class of Neutral Functional Differential Equations with State-Dependent Delays”, Nonlinear Analysis: Theory, Methods and Applications, 69:5–6 (2008), 1629–1643 | DOI | MR | Zbl

[13] Hernandez, E., Sakthivel, R. and Tanaka Aki, S., “Existence Results for Impulsive Evolution Differential Equations with State-Dependent Delay”, Electronic Journal of Differential Equations, 2008, no. 28, 1–11 | MR

[14] Darwish, M. A. and Ntouyas, S. K., “Semilinear Functional Differential Equations of Fractional Order with State-Dependent Delay”, Electronic Journal of Differential Equations, 2009, no. 38, 1–10 | MR | Zbl

[15] Benchohra, M. and Hellal, M., “Perturbed Partial Functional Fractional Order Differential Equations with Infinite Delay”, Journal of Advanced Research in Dynamical and Control System, 5:2 (2013), 1–15 | MR

[16] Benchohra, M. and Hellal, M., “Perturbed Partial Fractional Order Functional Differential Equations with Infinite Delay in Fréchet Spaces”, Nonlinear Dynamics and System Theory, 14:3 (2014), 244–257 | MR | Zbl

[17] Helal, M., “Perturbed Partial Functional Differential Equations on Unbounded Domains with Finite Delay”, Applied Mathematics E-Notes, 20 (2020), 82–92 | MR | Zbl

[18] Helal, M., “On Fractional Differential Perturbed Equations with State-Dependent Delay in Fréchet Spaces”, Nonlinear Studies, 28:4 (2021), 1085–1106 | MR

[19] Vityuk, A. N. and Golushkov, A. V., “Existence of Solutions of Systems of Partial Differential Equations of Fractional Order”, Nonlinear Oscillations, 7:3 (2004), 318–325 | DOI | MR

[20] Henry, D., Geometric Theory of Semilinear Parabolic Partial Differential Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin–New York, 1981 | DOI | MR

[21] Burton, T. A. and Kirk, C., “A Fixed Point Theorem of Krasnoselskii–Schaefer Type”, Mathematische Nachrichten, 189 (1998), 23–31 | DOI | MR | Zbl

[22] Hale, J. and Kato, J., “Phase Space for Retarded Equations with Infinite Delay”, Funkcialaj Ekvacioj, 21 (1978), 11–41 | MR | Zbl

[23] Hartung, F., “Differentiability of Solutions with Respect to Parameters in Neutral Differential Equations with State-Dependent Delays”, Journal of Mathematical Analysis and Applications, 324:1 (2006), 504–524 | DOI | MR | Zbl

[24] Hino, Y., Murakami, S. and Naito, T., Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl

[25] Czlapinski, T., “On the Darboux Problem for Partial Differential-Functional Equations with Infinite Delay at Derivatives”, Nonlinear Analysis: Theory, Methods and Applications, 44:3 (2001), 389–398 | DOI | MR | Zbl

[26] Czlapinski, T., “Existence of Solutions of the Darboux Problem for Partial Differential-Functional Equations with Infinite Delay in a Banach Space”, Commentationes Mathematicae, 35 (1995), 111–122 | MR | Zbl

[27] Hernández, E., Prokopczyk, A. and Ladeira, L., “A Note on Partial Functional Differential Equations with State-Dependent Delay”, Nonlinear Analysis: Real World Applications, 7:4 (2006), 510–519 | DOI | MR | Zbl