Some Applications Related to Differential Inclusions Based on the Use of a Weighted Space

Abstract

In this paper, we present an existence theorem for the problem of discontinuous dynamical system related to ordinary differential inclusion, based on the use of the concepts related to weighted spaces introduced by Gorka and Rybka, without using any fixed point theorem. The solution concept in this theorem is considered to belong to the weighted space. For comparison with the classical case and as an application of the theorem, we give an example problem that has such a solution but no continuously differentiable solution.

Keywords

• Differential inclusion
• Weighted space
• Existence

References

1. [1] M. Abbas, A.A. Majid, A.I.M. Ismail and A. Rashid, The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Appl. Math. Comput., 239 (2014), 74–88.
2. [2] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, (1990).
3. [3] P. Bettiol and H. Frankowska, Regularity of solution maps of differential inclusions under state constraints, Set-Valued Var. Anal., 15(1) (2007), 21-45.
4. [4] H. Duru, Fonksiyonel Analiz, Nobel, Ankara, (2023).
5. [5] A.F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control, 5 (1967), 609-621.
6. [6] H. Frankowska and T. Lorenz, Filippov’s theorem for mutational inclusions in a metric space, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 24(2) (2023), 1053-1094.
7. [7] A. Fryszkowski and J. Sadowski, Filippov lemma for measure differential inclusion, Math. Nachr. 294(3), (2021), 580-602.
8. [8] R. Gama and G. Smirnov, Stability and optimality of solutions to differential inclusions via averaging method, Set-Valued Var. Anal. 22(2), (2014), 349-374.

9. [9] C. Glocker, Set-Valued Force Laws: Dynamics of Non-Smooth Systems, Springer-Verlag, Berlin, (2001).
10. [10] P. Gorka and P. Rybka, Existence and uniqueness of solutions to singular ODE’s, Arch. Math., 94 (2010), 227-233.
11. [11] A. Iqbal, N.N. Abd Hamid, A.I.M. Ismail and M. Abbas, Galerkin approximation with quintic B-spline as basis and weight functions for solving second order coupled nonlinear Schrodinger equations ¨ , Math. Comput. Simul., 187 (2021), 1–16.
12. [12] M.K. Iqbal, M. Abbas, T. Nazir and N. Ali, Application of new quintic polynomial B-spline approximation for numerical investigation of Kuramoto-Sivashinsky equation, Adv. Diff. Eqs., 2020 (2020), 558.
13. [13] N. Khalid, M. Abbas and M.K. Iqbal, Non-polynomial quintic spline for solving fourth-order fractional boundary value problems involving product terms, Appl. Math. Comput., 349 (2019), 393–407.
14. [14] N. Khalid, M. Abbas, M.K. Iqbal, J. Singh and A.I.M. Ismail, A computational approach for solving time fractional differential equation via spline functions, Alexandria Eng. J., 59(1) (2020), 3061–3078.
15. [15] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Springer Dordrecht, (1999).
16. [16] A. Majeed, M. Kamran, M. Abbas and M.Y.B. Misro, An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model, Phys. Scr., 96(8) (2021), 084002.
17. [17] M. D. P. M. Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhauser, Basel, (1993).
18. [18] T. Nazir, M. Abbas, A.I.M. Ismail, A.A. Majid and A. Rashid, The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach, Appl. Math. Model., 40 (2016), 4586–4611.
19. [19] T. Nazir, M. Abbas and M.K. Iqbal, New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations, Eng. Comput., 38(1) (2020), 83–106.
20. [20] D. Repovs and P. Semenov, ˇ Continuous Selections of Multivalued Mappings, Springer Dordrecht, (1998).
21. [21] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, A.M.S., Providence, Rhode Islands, (2002).
22. [22] R.B. Vinter, Optimal Control, Birkhauser, Boston, (2000).
23. [23] M. Yaseen, M. Abbas, T. Nazir and D. Baleanu, A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation, Adv. Differ. Equ., 2017(1) (2017), 1-18.