Shirish Prabhakarrao Kulkarni
Gooty Rohan
Soni Pathak
- Assistant Professor Ajeenkya D.Y. Patil University (SOE), Charholi Budruk, Pune Maharashtra India
- Professor Chhatrapati Shivaji Maharaj University, Panvel, Navi Mumbai Maharashtra India
- Assistant Professor Ajeenkya D.Y. Patil University (SOE), Charholi Budruk, Pune Maharashtra India
Abstract
In the present research, advanced results on the fixed point theorem are applied to typical boundary value problems. Finding a differential equation’s solution under certain boundary conditions is the goal of typical boundary value topics. The article appears to extend new fixed point results to a new context, likely involving fractional operators with unique kernels, notably the Caputo-type fractional operator. This extension involves applying the fixed point theorem to a broader set of problems. Caputo fractional derivatives are a generalization of ordinary derivatives to non-integer orders, commonly used in fractional calculus. This extends the scope to fractional boundary value problems, where the differential equation involves fractional derivatives, and the conditions are given in terms of fractional order. Integral type boundary conditions suggest that the conditions involve integrals, which could be part of the fractional differential equation or the boundary conditions. We have used inequalities on a triplet (U, d, T) and quatern (U, d, T, θ). We have developed a novel class of mappings and investigated a fixed point criterion for them, using Geraghty contraction as inspiration. In addition, we demonstrated two applications: one with singular kernels for fractional derivatives in a system of nonlinear differential equations and the other with a two-point boundary value problem of a second order ordinary differential equations.
Keywords: Fractional differential equations, Ordinary differential Equations, Generalized α-p- -contractions, Weakly contractive mapping, Geraghty function, θ-orbital permissible
[This article belongs to Recent Trends in Mathematics(rtm)]
References
1. G. A. Anastassiou, I. K. Argyros, Approximating fixed points with applications in fractional calculus, J. Comput. Anal. Appl. 21 (2016), 1225–124
2. Muhammad Nazam, “On solution of a system of differential equations via fixed point theorem” J. computational analysis and applications, vol. 27, no.3, (2019)
3. Jarad, F., Abdeljawad, T., Hammouch, Z.: On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative. Chaos Solitons Fractals 117, 16–20 (2018)
4. Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 226, 3457–3471 (2017)
5. Abdeljawad, T., Jarad, F., Alzabut, J.: Fractional proportional differences with memory. Eur. Phys. J. Spec. Top. 226,3333–3354 (2017)
6. Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux équations intégrales. Fundam.Math. 3, 133–181 (1922)
7. Geraghty, M.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)
8. Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)
9. Jaggi, D.S.: Some unique fixed point theorems. Indian J. Pure Appl. Math. 8, 223–230 (1977)
10. Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. 47(4), 2683–2693 (2001)
11. Dass, B.K., Gupta, S.: An extension of Banach contraction principle through rational expressions. Indian J. Pure Appl.Math. 6, 1455–1458 (1975)
12. Alqahtani, B., Hamzehnejadi, J., Karapınar, E., Lashkaripour, R.: Best proximity point for certain proximal contraction type mappings. J. Math. Anal. 9(5), 1–15 (2018)
13. Hamzehnejadi, J., Lashkaripour, R.: Best proximity points for generalized α-φ-Geraghty proximal contraction mappings. Fixed Point Theory Appl. 2016, 72 (2016)
14. Dutta, P.N., Choudhury, B.S.: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl., 2008,Article ID 406368 (2008)
15. Popescu, O.: Some new fixed point theorems for α-Geraghty contractive type maps in metric spaces. Fixed Point Theory Appl. 2014, 190 (2014)
16. Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 75,2154–2165 (2012)
17. Karapınar, E., Samet, B.: A note on ψ-Geraghty type contractions. Fixed Point Theory Appl. 2014, 26 (2013)
18. Cho, S., Bae, J., Karapınar, E.: Fixed point theorems for α-Geraghty contraction type maps in metric spaces. Fixed Point Theory Appl. 2013, 329 (2013)
19. Karapınar, E.: A discussion on α-ψ-Geraghty contraction type mappings. Filomat 28(4), 761–766 (2014)
20. Hilfer, R.: Applications of Fractional Calculus in Physics. Word Scientific, Singapore (2000)
21. Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54),3413–3442 (2003)
22. Kilbas, A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204 (2006)
23. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Danbury (2006)
24. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1,73–85 (2015)
25. Abdeljawad, T., Baleanu, D.: Monotonicity results for fractional difference operators with discrete exponential kernels.Adv. Differ. Equ. 2017, 78 (2017)
26. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math.Phys. 80(1), 11–27 (2017)
27. Atangana, A., Baleanu, D.: New fractional derivative with non-local and non-singular kernel. Therm. Sci. 20, 757–763(2016)
28. Alsaedi, A., Baleanu, D., Etemad, S., Rezapour, S.: On coupled systems of time-fractional differential problems by using a new fractional derivative. J. Funct. Spaces 2016, Article ID 4626940 (2016)
29. Agarwal, R., Baleanu, D., Hedayati, V., Rezapour, S.: Two fractional derivative inclusion problems via integral boundary condition. Appl. Math. Comput. 257, 205–212 (2015)
30. Baleanu, D., Rezapour, S., Mohammadi, H.: Some existence results on nonlinear fractional differential equations.Philos. Trans. R. Soc. 371, 20120144 (2013)
31. Baleanu, D., Mohammadi, H., Rezapour, S.: The existence of solutions for a nonlinear mixed problem of singular fractional differential equations. Adv. Differ. Equ. 2013, 359 (2013)
32. Abdeljawad, T.: Meir–Keeler α-contractive fixed and common fixed point theorems. Fixed Point Theory Appl. 2013,19 (2013)
33. Patel, D.K., Abdeljawad, T., Gopal, D.: Common fixed points of generalized Meir–Keeler α-contractions. Fixed Point Theory Appl. 2013, 260 (2013)
34. Karapınar, E., Kumam, P., Salimi, P.: On α-ψ-Meir–Keeler contractive mappings. Fixed Point Theory Appl. 2013, 94 (2013)
35. Karapınar, E., Samet, B.: Generalized α-ψ-contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)
36. Aydi, H., Karapınar, E., Erhan, I.M., Salimi, P.: Best proximity points of generalized almost ψ-Geraghty contractive non-self-mappings. Fixed Point Theory Appl. 2014, 164 (2014). https://doi.org/10.1186/1687-1812-2014-32
37. Bilgili, N., Karapınar, E., Sadarangani, K.: A generalization for the best proximity point of Geraghty-contractions.J. Inequal. Appl. 2013, 286 (2013).
| Volume | 01 |
| Issue | 01 |
| Received | January 24, 2024 |
| Accepted | March 1, 2024 |
| Published | June 3, 2024 |