On existence and approximation of solution of nonlinear Hilfer fractional differential equation

Abstract

This paper gives the existence and uniqueness results for solution of fractional differential equations with Hilfer derivative. Using some new techniques and generalizing restrictive conditions imposed on considered function, the iterative scheme for uniformly approximating the solution is established. An example is included to show the applicability of our theoretical results.

On Ulam type stability for nonlinear implicit fractional differential equations

“…This proves the inequality ( 10) is true for j = r + 1. The principle of mathematical induction completes the proof of inequality (10). Again from the inequality (10), we obtain…”

“…Although, the local stability and Mittag-Leffler stability results are obtained in the literature by fixed point theory [4,9,12,31,32], to the best of our knowledge, there are very rare works on the Ulam stability for nonlinear implicit fractional differential equations by using method of successive approximations [5,10,11,21]. Integral inequalities plays a major role in the analysis of various functional equations and their qualitative properties.…”

Local existence and uniqueness of solutions for fractional differential problem with Hilfer-Hadamard fractional derivative

“…They established the equivalence of initial value problem (IVP) (2) with following Volterra integral equation of second kind:…”

“…Further, some attractivity and Ulam stability results are obtained by applying the fixed point theory. Authors in [2]- [4] obtained the existence, uniqueness and continuations results by using both successive approximations and fixed point techniques for the solution of fractional IVP involving Hilfer fractional derivative defined in [5].…”