A solution of nonlinear fractional random differential equation via random fixed point technique

Document Type: Special Issue on Fixed Point Theory

Authors

1 Department of Mathematics, Faculty of Science, Assuit University, Assuit 71516, Egypt

2 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

Abstract

In this paper, we investigate a new type of random $F$-contraction and obtain a common random fixed point theorem for a pair of self stochastic mappings in a separable Banach space. The existence of a unique solution for nonlinear fractional random differential equation is proved under suitable conditions.

Keywords

Main Subjects


[1] S. Abbas, M. Benchohra, G. M. N’Gue´er´ekata, Topics in Fractional Differential Equations, Springer, New York, 2012.

[2] M. Abbas, B. Ali, S. Romaguera, Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl. (2013), 2013:243.

[3] I. Arandjelovic, Z. Radenovic, S. Radenovic, Boyd-Wong-type common fixed point results in cone metric spaces, Appl. Math. Comput. 217 (2011), 7167-7171.

[4] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3 (1922), 133-181.

[5] R. Batra, S. Vashistha, Fixed points of an F -contraction on metric spaces with a graph, Int. J. Comput. Math. 91 (2014), 1-8.

[6] R. Batra, S. Vashistha, R. Kumar, A coincidence point theorem for F-contractions on metric spaces equipped with an altered distance, J. Math. Comput. Sci. 4 (2014), 826-833.

[7] A. T. Bharucha-Reid, Random Integral Equations, Mathematics in Science and Engineering, Academic Press, New York, 1972.

[8] D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464.

[9] M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat. 28 (2014), 715-722.

[10] M. A. Darwish, J. Henderson, D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011), 539-553.

[11] J. Dugundji, A. granas, Fixed Point Theory, Monografie Matematycne, Warsazawa, 1982.

[12] O. Hans, Reduzierende zuf˘allige transformationen, Czechoslov. Math. J. 7 (1957), 154-158.

[13] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

[14] S. Itoh, Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), 261-273.

[15] M. C. Joshi, R. K. Bose, Some Topics in Nonlinear Functional Analysis, Wiley Eastern Ltd, New Delhi, 1984.

[16] W. A. Kirk, B. Sems, Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, Iowa City and Newcastle, 2001.

[17] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[18] E. Rakotch, A note in contractive mappings, Proc. Amer. Math. Soc. 13 (1962), 459-465.

[19] R. A. Rashwan, D. M. Albaqeri, A common random fixed point theorem and application to random integral equations, Int. J. Appl. Math. Reser. 3 (1) (2014), 71-80.

[20] R. A. Rashwan, H. A. Hammad, Random fixed point theorems with an application to a random nonlinear integral equation, J. Linear. Topological. Algebra. 5 (2016), 119-133.

[21] R. A. Rashwan, H. A. Hammad, Random common fixed point theorem for random weakly subsequentially continuous generalized contractions with application, Int. J. Pure Appl. Math. 109 (2016), 813-826.

[22] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001.

[23] N. Shahzad, Random fixed points of discontinuous random maps, Math. Comput. Modelling. 41 (2005), 1431-1436.

[24] N. Shahzad, S. Latif, Random fixed points for several classes of 1-Ball-contractive and 1-set-contractive random maps, J. Math. Anal. Appl. 237 (1999), 83-92.

[25] A. Skorohod, Random Linear Operators, Reidel, Boston, 1985.

[26] A. Spacek, Zufallige Gleichungen, Czechoslovak Math. J. 5 (1955), 462-466.

[27] E. Tarafdar, An approach to fixed-point theorems on uniform spaces, Trans. Amer. Math. Soc. 191 (1974), 209-225.

[28] V. E. Tarasov, Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Medi, Springer, Heidelberg, 2010.

[29] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. (2012), 2012:94.