Nonconvex Fractional Partial Hyperbolic Differential Inclusions with State-Dependent Delay
Keywords:
Partial functional differential inclusion, fractional order, solution, left-sided mixed Riemann-Liouville integral, Caputo fractional derivative, state-dependent delay, fixed point.Abstract
In this paper we investigate the existence of solutions of initial value problem for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with state-dependent delay when the right-hand side is nonconvex valued by using the fixed-point theorem for contraction multivalued maps due to Covitz and Nadler.
Downloads
References
[1] R. P. Agarwal, M. Belmekki, and M. Benchohra. A survey on semilinear differential equations and inclusions involving the riemann–liouville fractional derivative. Advances in Difference Equations, pages 1–47, 2009. Article ID 981728.
[2] R. P. Agarwal, M. Benchohra, and S. Hamani. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Applicandae Mathematicae, 109(3):973–1033, 2010.
[3] J. P. Aubin and A. Cellina. Differential Inclusions. Springer, Berlin, 1984.
[4] J. P. Aubin and H. Frankowska. Set-Valued Analysis. Birkh¨auser, Boston, 1990.
[5] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo. Fractional Calculus: Models and Numerical Methods. World Scientific, New York, 2012.
[6] M. Benchohra and M. Helal. A global uniqueness result for fractional partial hyperbolic differential equations with state dependent delay. Annales Polonici Mathematici, 110(3):259–281, 2014.
[7] C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions, volume 580 of Lecture Notes in Mathematics. Springer, Berlin, 1977.
[8] H.-H. Chu, S. Rashid, Z. Hammouch, and Y. Chu. New fractional estimates for hermite–hadamard–mercer type inequalities. Alexandria Engineering Journal, 59(5):3079–3089, 2020.
[9] H. Covitz and S. B. Nadler. Multivalued contraction mappings in generalized metric spaces. Israel Journal of Mathematics, 8:5–11, 1970.
[10] T. Czlapinski. On the darboux problem for partial differential–functional equations with infinite delay in derivatives. Nonlinear Analysis, 44:389–398, 2001.
[11] K. Deimling. Multivalued Differential Equations. de Gruyter, Berlin, 1992.
[12] L. G´orniewicz. Topological Fixed Point Theory of Multivalued Mappings, volume 495 of Mathematics and Its Applications. Kluwer, Dordrecht, 1999.
[13] J. Hale and J. Kato. Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj, 21:11–41, 1978.
[14] J. K. Hale and S. M. Verduyn Lunel. Introduction to Functional Differential Equations, volume 99 of Applied Mathematical Sciences. Springer, New York, 1993.
[15] M. Helal. Fractional partial hyperbolic differential inclusions with state-dependent delay. Fractional Calculus and Applied Analysis, 10(1):179–196, 2019.
[16] M. Helal. Differential inclusions of fractional order on unbounded domains with state-dependent delay. Nonlinear Studies, 27(1):69–85, 2020.
[17] M. Helal. Fractional-order differential inclusions on unbounded domains with infinite delay. Mathematica (Cluj), 62(2):167– 178, 2020.
[18] M. Helal. The darboux problem for fractional partial hyperbolic differential inclusions on unbounded domains with delay. Studia Universitatis Babes-Bolyai Mathematica, 67(4):703–716, 2022.
[19] M. Helal. Nonconvex fractional partial hyperbolic differential inclusions with unbounded delay. Advanced Studies in European Mathematics Journal, 18(1):227–239, 2025.
[20] M. Helal and S. Sivasundaram. Darboux problem for perturbed hyperbolic fractional order differential inclusions with finite delay. Nonlinear Analysis: Real World Applications, 86:1–8, 2025.
[21] R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.
[22] S. Hu and N. S. Papageorgiou. Handbook of Multivalued Analysis. Vol. I: Theory. Kluwer, Dordrecht, 1997.
[23] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of Fractional Differential Equations, volume 204 of North-Holland Mathematics Studies. Elsevier, Amsterdam, 2006.
[24] M. Kisielewicz. Differential Inclusions and Optimal Control. Kluwer, Dordrecht, 1991.
[25] V. Lakshmikantham, S. Leela, and J. V. Devi. Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge, 2009.
[26] F. Mainardi. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London, 2010.
[27] S. Rashid, H. Kalsoom, Z. Hammouch, and Y. Chu. New multi-parameterized estimates in fractional calculus. Symmetry, 12(2):222, 2020.
[28] V. E. Tarasov. Fractional Dynamics. Springer, Heidelberg, 2010.
[29] N. Valliammal, R. Chokkalingam, Z. Hammouch, and H. M. Baskonus. A new investigation on fractional-order neutral differential systems with state-dependent delay. International Journal of Nonlinear Sciences and Numerical Simulation, 20(7-8):803–809, 2019.
[30] A. N. Vityuk. Existence of solutions of partial differential inclusions of fractional order. Izvestiya VUZ Matematika, (8):13–19, 1997.
[31] A. N. Vityuk and A. V. Golushkov. Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscillations, 7(3):318–325, 2004.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Mohamed Helal
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
