International Journal of Nonlinear Analysis and Applications
On fractional differential equations and fixed point theory
Document Type : Research Paper
Author
Department of Mathematics Education, Farhangian University, Tehran, Iran.
Abstract
In this work first we establish some fixed point theorems for $\bot-$Mizoguchi-Takahashi contractions mappings in the setting of orthogonal metric spaces. Next, we investigate the existence of solution for certain fractional differential equation via some integral boundary value conditions and obtained fixed point results.
Keywords
[1] B. Ahmad and S. Sivasundaram, On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Appl. Math. Comput. 217 (2010) 480–487.
[2] H. Baghani, M. Eshaghi Gordji and M. Ramezani, Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory Appl. 18 (2016) 465–477.
[3] D. Baleanu, S. Z. Nazemi and Sh. Rezapour, Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations, Adv. Differ. Equ. 368 (2013).
[4] M. Eshaghi, M. Ramezani, M. D. L. Sen and Y. J. Cho, On orthogonal sets and Banach’s fxed point theorem, Fixed Point Theory, 18 (2017) 569–578.
[5] M. A. Khamsi, W. M. Kozlowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal. 14 (1990) 935–953.
[6] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.
[7] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188.
[8] S. B. Nadler, Multivalued contraction mappings, Pac. J. Math. 30 (1969) 475-488.
[9] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[10] S. Reich, Fixed point of contractive functions, Boll Un Mat Ital. 4 (1972) 26-42.
[11] S. Reich, Some fixed point problems, Rend Lincei-Mat Appl. 57 (1974) 194–198.
[12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, Switzerland, 1993.
[13] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equation. Appl. Math. Chin. Univ. 22 (2007) 291–298.
[14] X. Su and S. Zhang, Solutions to boundary value problems for nonlinear differential equations of fractional order. Electron. J. Differ. Equ. 26 (2009) 1–15.
[15] Ch. Zhai and J. Ren, The unique solution for a fractional q-difference equation with three-point boundary conditions, Indag. Math. 29 (2018) 948–961.
[2] H. Baghani, M. Eshaghi Gordji and M. Ramezani, Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory Appl. 18 (2016) 465–477.
[3] D. Baleanu, S. Z. Nazemi and Sh. Rezapour, Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations, Adv. Differ. Equ. 368 (2013).
[4] M. Eshaghi, M. Ramezani, M. D. L. Sen and Y. J. Cho, On orthogonal sets and Banach’s fxed point theorem, Fixed Point Theory, 18 (2017) 569–578.
[5] M. A. Khamsi, W. M. Kozlowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal. 14 (1990) 935–953.
[6] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.
[7] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188.
[8] S. B. Nadler, Multivalued contraction mappings, Pac. J. Math. 30 (1969) 475-488.
[9] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[10] S. Reich, Fixed point of contractive functions, Boll Un Mat Ital. 4 (1972) 26-42.
[11] S. Reich, Some fixed point problems, Rend Lincei-Mat Appl. 57 (1974) 194–198.
[12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, Switzerland, 1993.
[13] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equation. Appl. Math. Chin. Univ. 22 (2007) 291–298.
[14] X. Su and S. Zhang, Solutions to boundary value problems for nonlinear differential equations of fractional order. Electron. J. Differ. Equ. 26 (2009) 1–15.
[15] Ch. Zhai and J. Ren, The unique solution for a fractional q-difference equation with three-point boundary conditions, Indag. Math. 29 (2018) 948–961.
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Volume 12, Issue 2
November 2021Pages 679-697
- Receive Date: 26 February 2019
- Revise Date: 22 December 2019
- Accept Date: 22 December 2019