On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem

Volume 9, Issue 5, pp 3348--3353 http://dx.doi.org/10.22436/jnsa.009.05.120

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Authors

Samer Al Ghour - Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan. Ahmad Al Omari - Department of Mathematics, Faculty of Science, Al al-Bayt University, P. O. Box 7, Mafraq 61710, Jordan.

Abstract

We consider the nonlocal Cauchy problem for the semilinear functional differential equation with non- integer order: \[u^\alpha(t) = Au(t) + f(t; u_t) \,\texttt{where}\, \alpha \in (0; 1] \, \texttt{and} \, t \in (0; a],\] \[u(\tau_k + 0) = Q_ku(\tau_k) \equiv u(\tau_k) + I_ku(\tau_k); k = 1; 2; ... ;K,\] \[u(t) + (g(u_{t_1} ,..., u_{t_p}))(t) = \phi(t), \, \texttt{where} \, t \in [-r; 0].\] Under suitable conditions we prove the existence and uniqueness of a mild solution to the equation.

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ISRP Style

Samer Al Ghour, Ahmad Al Omari, On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 5, 3348--3353

AMA Style

Al Ghour Samer, Al Omari Ahmad, On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem. J. Nonlinear Sci. Appl. (2016); 9(5):3348--3353

Chicago/Turabian Style

Al Ghour, Samer, Al Omari, Ahmad. "On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem." Journal of Nonlinear Sciences and Applications, 9, no. 5 (2016): 3348--3353

Keywords

Cauchy problem
mild solution
impulsive functional
fractional differential equation
fixed point.

MSC

34A37
34G20
34K30
34K99

References

[1] J. H. Barrett, Differential equations of non-integer order, Canadian J. Math., 6 (1954), 529-541.

[2] L. Byszewski, H. Akca, On a mild solution of a semlinear functional differential evolution nonlocal problem, Int. J. Stoch. Anal., 10 (1997), 265-271.

[3] M. Caputao , Linear models of dissipation whose Q is almost frequency independent, Geophys. J. R. Astr. Soc., 13 (1967), 529-539.

[4] D. Delbosco, L. Rodino , Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609-625.

[5] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167 (1998), 57-68.

[6] J. H. He, Some applications of nonliear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (1999), 86-90 .

[7] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing Co., Inc., Teaneck, NJ (1989)

[8] F. Mainardi , Fractional calculus: Some basic problem in continuum and statistical mechanics, Springer, Vienna (1997)

[9] K. S. Miller, B. Ross, An introduction to the Fractional calculus and Fractional Differential Equations, John Wiley and Sons Inc., New York (1993)

[10] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific Publishing Co., Inc., River Edge, New Jersey (1995)

[11] H. Ye, G. D. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2006), 1075-1081.