Practical Stability in Terms of Two Measures with Initial Time Difference for Set Differential Equations involving Causal Operators

Abstract

In this paper, we investigate generalized variational comparison results aimed to study the stability properties in terms of two measures for solutions of Set Differential Equations (SDEs) involving causal operators, taking into consideration the difference in initial conditions. Next, we employ these comparison results in proving the theorems that give sufficient conditions for practical stability in terms of two measures with initial time difference (ITD) for the solutions of perturbed SDEs involving causal operators in regard to their unperturbed ones.

Keywords

• Causal Operators; Initial Time Difference; Lyapunov function; Practical Stability; Set Differential Equations; Two Measures
• Causal Operators; Initial Time Difference
• Lyapunov function
• Practical Stability
• Set Differential Equations
• Two Measures

References

1. [1] T. G. Bhaskar and J. V. Devi, Nonuniform stability and boundedness criteria for set differential equations, Applicable Analysis, 84(2), (2005), 131-143. doi: 10.1080/00036810410001724346
2. [2] T. G. Bhaskar and J. V. Devi, Stability criteria for set differential equations, Mathematical and Computer Modelling, 41(11-12), (2005) 1371-1378.
3. [3] F. Brauer and J. A. Nohel, The qualitative theory of ordinary differential equations: an introduction, Dover, NY, USA, 1989.
4. [4] M. C¸ ic¸ek, C. Yakar and B. O˘gur, Stability, boundedness, and Lagrange stability of fractional differential equations with initial time difference, The Scientific World Journal, Volume 2014, Article ID 939027, (2014), 1-7. doi: 10.1155/2014/939027
5. [5] C. Corduneanu, Functional equations with causal operators, CRC Press, 2002. doi: 10.1201/9780203166376
6. [6] J. V. Devi, Existence, uniqueness of solutions for set differential equations involving causal operators with memory, European Journal of Pure and Applied Mathematics, 3(4), (2010), 737-747.
7. [7] J. V. Devi, Comparison theorems and existence results for set causal operators with memory, Nonlinear Studies, 18(4), (2011), 603-610.
8. [8] J. V. Devi, Generalized monotone iterative technique for set differential equations involving causal operators with memory, International Journal of Advances in Engineering Sciences and Applied Mathematics, 3(1-4), (2011), 74–83. doi: 10.1007/s12572-011-0031-1

9. [9] J. V. Devi and Ch. A. Naidu, Stability results for set differential equations involving causal operators with memory, European Journal of Pure and Applied Mathematics, 5(2), (2012), 187-196.
10. [10] J. V. Devi and Ch. A. Naidu, Boundedness results for impulsive set differential equations involving causal operators with memory, Communications in Applied Analysis, 17(1), (2013), 9–19.
11. [11] Z. Drici, F. A. Mcrae and J. V. Devi, Stability results for set differential equations with causal maps, Dynamic Systems and Applications, 15(3), (2006), 451-464.
12. [12] M. B. G¨ucen and C. Yakar, Strict stability of fuzzy differential equations by Lyapunov functions, World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering, 12(5), (2018), 315-319. doi: 10.5281/zenodo.1316718
13. [13] V. Lakshmikantham, On the stability and boundedness of differential systems, Mathematical Proceedings of the Cambridge Philosophical Society, 58(3), (1962), 492-496. doi: 10.1017/S030500410003677X
14. [14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations. World Scientific, Singapore, 1989. doi: 10.1142/0906
15. [15] V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of set differential equations in metric spaces, Cottenham, Cambridge, Cambridge Scientific Publishers, 2006.
16. [16] V. Lakshmikantham and S. G. Deo, Method of variation of parameters for dynamic systems, Amsterdam, Netherlands, Gordon and Breach Science Publishers, 1998.
17. [17] V. Lakshmikantham and S. Leela, Differential and integral inequalities: theory and applications, New York, USA, Academic Press, 1969.
18. [18] V. Lakshmikantham, S. Leela and J. V. Devi, Stability theory for set differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 11(2-3), (2004), 181-190.
19. [19] V. Lakshmikantham, S. Leela, Z. Drici and F. A. McRae, Theory of causal differential equations, Atlantis Studies in Mathematics for Engineering and Science, 2010. doi: 10.2991/978-94-91216-25-1
20. [20] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability analysis of nonlinear systems, NY, USA: M. Dekker, 1989.
21. [21] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Practical stability of nonlinear systems, World Scientific, Singapore, 1991. doi: 10.1142/1192
22. [22] V. Lakshmikantham, S. Leela and A. S. Vatsala, Setvalued hybrid differential equations and stability in terms of two measures, International Journal of Hybrid Systems, 2(2), (2002), 169-188.
23. [23] V. Lakshmikantham and X. Z. Liu, Stability analysis in terms of two measures, World Scientific, Singapore, 1993. doi: 10.1142/2018
24. [24] V. Lakshmikantham, V. M. Matrosov and S. Sivasundaram, Vector Lyapunov functions and stability analysis of nonlinear systems, Dordrecht, Boston, USA, Kluwer Academic Publishers, 1991.
25. [25] V. Lakshmikantham and M. Rama Mohana Rao, Theory of integro-differential equations, Lausanne, Switzerland, Gordon and Breach Science Publishers, 1995.
26. [26] V. Lakshmikantham and A. S. Vatsala, Theory of differential and integral inequalities with initial time difference and applications, In: Rassias T.M., Srivastava H.M. (eds) Analytic and Geometric Inequalities and Applications, Mathematics and Its Applications, Vol 478, Dordrecht, Netherlands, Springer, 1999. doi:10.1007/978-94-011-4577-0 12
27. [27] V. Lakshmikantham and A. S. Vatsala, Differential inequalities with initial time difference and applications, Journal of Inequalities and Applications, 3(3), (1999), 233-244. doi: 10.1155/S1025583499000156
28. [28] J. LaSalle and S. Lefschetz, Stability by Liapunov’s direct methods with applications, Mathematics in Science and Engineering, V. 4. New York, Academic Press, 1961.
29. [29] X. Liu and M. D. Shaw, Boundedness in terms of two measures for perturbed systems by generalized variation of parameters, Communications in Applied Analysis, 5(4), (2001), 435-444.
30. [30] Ch. A. Naidu, Stability results for impulsive set differential equations involving causal operators with memory, Global Journal of Mathematical Sciences: Theory & Practical, 2(2), (2014), 49-53.
31. [31] Ch. A. Naidu, D. B. Dhaigude and J. V. Devi, Stability results in terms of two measures for set differential equations involving causal operators, European Journal of Pure and Applied Mathematics, 10(4), (2017), 645-654.
32. [32] M. D. Shaw and C. Yakar, Generalized variation of parameters with initial time difference and a comparison result in terms of Lyapunov-like functions, International Journal of Nonlinear Differential Equations Theory-Methods and Applications, 5(1-2), (1999), 86-108.
33. [33] M. D. Shaw and C. Yakar, Stability criteria and slowly growing motions with initial time difference, Problems of Nonlinear Analysis in Engineering Systems, 6, (2000), 50–66.
34. [34] H. Smith, An introduction to delay differential equations with applications to the life sciences, New York, USA, Springer, 2011. doi: 10.1007/978-1- 4419-7646-8
35. [35] N. N. Tu and T. T. Tung, Stability of set differential equations and applications, Nonlinear Analysis: Theory, Methods & Applications, 71(5-6), (2009), 1526-1533. doi: 10.1016/j.na.2008.12.045
36. [36] C. Yakar, Boundedness criteria in terms of two measures with initial time difference, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, Watam Press. Waterloo, (2007), pp 270–275. DCDIS 14 (S2) 1–305.
37. [37] C. Yakar and M. Arslan, Terminal value problems with causal operators, Hacettepe Journal of Mathematics and Statistics, 47(4), (2018), 897-907. doi: 10.15672/HJMS.2018.566
38. [38] C. Yakar, B. Bal and A. Yakar, Monotone technique in terms of two monotone functions in finite system, Journal of Concrete and Applicable Mathematics, 9(3), (2011), 233-239.
39. [39] C. Yakar and M. C¸ ic¸ek, Initial time difference boundedness criteria and Lagrange stability, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 18(6), (2011), 797-811.
40. [40] C. Yakar and M. C¸ ic¸ek, Theory, methods and applications of initial time difference, boundedness and Lagrange stability in terms of two measures for nonlinear systems, Hacettepe Journal of Mathematics and Statistics, 40(2), (2011), 305-330.
41. [41] C. Yakar, M. C¸ ic¸ek and M. B. G¨ucen, Boundedness and Lagrange stability of fractional order perturbed system related to unperturbed systems with initial time difference in Caputo’s sense, Advances in Difference Equations, 54, (2011), 1-14. doi: 10.1186/1687-1847-2011-54
42. [42] C. Yakar, M. C¸ ic¸ek and M. B. G¨ucen, Practical stability, boundedness criteria and Lagrange stability of fuzzy differential systems, Computers & Mathematics with Applications, 64(6), (2012), 2118-2127. doi: 10.1016/j.camwa.2012.04.008
43. [43] C. Yakar and M. B. G¨ucen, Initial time difference stability of causal differential systems in terms of Lyapunov functions and Lyapunov functionals, Journal of Applied Mathematics, Volume 2014, Article ID 832015, (2014), 1-7. doi: 10.1155/2014/832015
44. [44] C. Yakar and M. D. Shaw, A comparison result and Lyapunov stability criteria with initial time difference, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 12(6), (2005), 731-737.
45. [45] C. Yakar and M. D. Shaw, Initial time difference stability in terms of two measures and variational comparison result, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 15(3), (2008), 417-425.
46. [46] C. Yakar and M. D. Shaw, Practical stability in terms of two measures with initial time difference, Nonlinear Analysis: Theory, Methods & Applications, 71(12), (2009), e781-e785. doi: 10.1016/j.na.2008.11.039
47. [47] C. Yakar and H. Talab, Stability of perturbed set differential equations involving causal operators in regard to their unperturbed ones considering difference in initial conditions, Advances in Mathematical Physics, Volume 2021, Article ID 9794959, (2021), 1-12. doi: 10.1155/2021/9794959.