Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

Volume 10, Issue 6, pp 2986--2997 http://dx.doi.org/10.22436/jnsa.010.06.13

Publication Date: June 25, 2017 Submission Date: August 30, 2016

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Authors

Poom Kumam - KMUTT-Fixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science (TaCS) Center, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan. Amjad Ali - Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan. Kamal Shah - Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan. Rahmat Ali Khan - Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan.

Abstract

This paper is concerned with developing some conditions that reveal existing and stability analysis for solutions to a class of differential equations with fractional order. The required conditions are obtained by applying the technique of degree theory of topological type. The concerned problem is converted to the integral equation and then to operator equation, where the operator is defined by \(T : C[0, 1] \rightarrow C[0, 1]\). It should be noted that the assumptions on nonlinear function \(f(t, u(t))\) does not usually ascertain that the operator T being compact. Moreover, in this paper we also establish some conditions under which the solution of the considered class is Hyers-Ulam stable and also satisfies the conditions of Hyers-Ulam-Rassias and generalized Hyers-Ulam stability. Proper example is provided for the illustration of main results.

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Keywords

• Arbitrary order differential equations
• topological degree theory
• condensing mapping
• existence results
• stability analysis.

MSC

•  34A08
•  35R11
•  26A33

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