Reduction method for functional nonconvex differential inclusions

Year 2021, Volume: 3 Issue: 1, 6 - 14, 29.04.2021

Hanane Chouial Mustapha Fateh Yarou

https://doi.org/10.47087/mjm.853437

Abstract

Our aim in this paper is to present a reduction method that solves first order functional differential inclusion in the nonconvex case. This approach is based on a discretization of the time interval, a construction of approximate solutions by reducing the problem to a problem without delay and an application of known results in this case. We generalises earlier results, the right hand side of the inclusion has nonconvex values and satisfies a linear growth condition instead to be integrably bounded. The lack of convexity is replaced by the topological properties of decomposable sets, that represents a good alternative in the absence of convexity.

Keywords

delay, linear growth condition, Lower semicontinuous, nonconvex differential inclusion

Supporting Institution

Research supported by the General direction of scientific research and technological development (DGRSDT), Algeria

Project Number

PRFU No. C00L03UN180120180001.

References

• D. Affane and M. F. Yarou; Second-order perturbed state-dependent sweeping process with subsmooth sets, In: Zeidan D., Padhi S., Burqan A., Ueberholz P. (eds) computational Mathematics and Applications. Forum for Interdisciplinary Mathematics. Springer, Singapore (2020) 147-169.
• D. Affane and M. F. Yarou; General second order functional differential inclusions driven by the sweeping process with subsmooth sets, J. Nonlin. Funct. Anal. Article ID 26 (2020) 1-18.
• D. Affane and M. F. Yarou; Unbounded perturbation to time-dependent subdifferential operators with delay, Electr. J. Math. Anal. Appl. 02(8) (2020) 209-219.
• D. Affane and M. F. Yarou; Fixed point approach for differential inclusions governed by subdifferential operators, AIP Conference Proceedings 2183, 060002 (2019); https : // doi.org /10.1063 / 1. 5136157
• N. Fetouci and M. F. Yarou; A fixed point approach for a differential inclusion governed by the subdifferential of PLN functions, AIP Conference Proceedings 2183, 060005 (2019); https://doi.org/10.1063/1.5136160
• J. P. Aubin, A. Cellina; Differential inclusions, Springer-Verlag, (1984).
• M. Bounkhel and M. F. Yarou; Existence results for first and second order nonconvex sweeping process with delay, Portug. Math. 61 (2) (2004) 207-230.
• C. Castaing, A. Salvadori and L. Thibault; Functional evolution equations governed by nonconvex sweeping process, J. Nonlin. Conv. Anal. 2(2) (2001) 217-241.
• C. Castaing and M. Valadier; Convex Analysis and Measurable Multifunctions, Lecture Note in Math. 580, Springer, Berlin, (1997).
• A. Fryszkowski; Continuous selections for a class of non-convex multivalued maps, Studia Math. 76(2) (1983) 163-174.
• A. Fryszkowski; Existence of solutions of functional-differential inclusion in nonconvex case, Anal. Polonici Math. 45(2) (1985) 121-124.
• A. Fryszkowski and L. Gorniewicz; Mixed semicontinuous mappings and their applications to differential inclusions, Set-Valued Anal. 8 (2000) 203-217.
• M. F. Yarou; Reduction approach to second order perturbed state-dependent sweeping process, Crea. Math. Infor. 28 (02) (2019) 215-221.