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

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] → 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.