Electron. J. Differential Equations, Vol. 2016 (2016), No. 290, pp. 1-14.

Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems

Mohammed D. Kassim, Khaled M. Furati, Nasser-eddine Tatar

Abstract:
We study the boundedness and asymptotic behavior of solutions for a class of nonlinear fractional differential equations. These equations involve two Riemann-Liouville fractional derivatives of different orders. We determine fairly large classes of nonlinearities and appropriate underlying spaces where solutions are bounded, exist globally and decay to zero as a power type function. Our results are obtained by using generalized versions of Gronwall-Bellman inequality, appropriate regularization techniques and several properties of fractional derivatives. Three examples are given to illustrate our results.

Submitted July 7, 2016. Published October 31, 2016.
Math Subject Classifications: 34C11, 42B20, 34E10.
Key Words: Regularization technique; Mittag-Leffler function; power type decay; weighted space.

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Mohammed D. Kassim
King Fahd University of Petroleum and Minerals
Department of Mathematics and Statistics
Dhahran 31261, Saudi Arabia
email: dahan@kfupm.edu.sa
Khaled M. Furati
King Fahd University of Petroleum and Minerals
Department of Mathematics and Statistics
Dhahran 31261, Saudi Arabia
email: kmfurati@kfupm.edu.sa
Nasser-eddine Tatar
King Fahd University of Petroleum and Minerals
Department of Mathematics and Statistics
Dhahran 31261, Saudi Arabia
email: tatarn@kfupm.edu.sa

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