Abstract
In this paper, we study the class of boundary value problems for a nonlinear implicit fractional differential equation with periodic conditions involving a ψ-Hilfer fractional derivative. With the help of properties Mittag-Leffler functions, and fixed-point techniques, we establish the existence and uniqueness results, whereas the generalized Gronwall inequality is applied to get the stability results. Also, an example is provided to illustrate the obtained results.
Highlights
The theory of fractional differential equations is very important since their nonlocal property is appropriate to describe memory phenomena such as nonlocal elasticity, propagation in complex medium, polymers, biological tissues, earth sediments, expansion of universe too, and they have been emerging as an important area of investigation in the last few decades
For some recent results of stability analysis by different types of fractional derivative operators (FDOs), we refer the reader to a series of papers [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]
This paper is organized as following: In Section 2, we recall the basic definitions and lemmas which are used throughout this paper, we present the concepts of some fixed point theorems
Summary
The theory of fractional differential equations is very important since their nonlocal property is appropriate to describe memory phenomena such as nonlocal elasticity, propagation in complex medium, polymers, biological tissues, earth sediments, expansion of universe too, and they have been emerging as an important area of investigation in the last few decades. ΛiIζ0+, y(τi), p r = p + β − pβ, τi ∈ (0, T ], i=1 where Dp0+,βdenotes the Hilfer FD of order p ∈ (0, 1) and type β ∈ [0, 1], I10−+ r is the Reimann Liouville fractional integral of order 1 − r, r = p + β(1 − p), c < 0 by using some properties of Mittag-Leffler functions, and fixed point methods. In [36] studied the existence, uniqueness and different types of stabilities of solutions for the following problem: D0p+,βy(t) − λy(t) = f(t, y(t), Dp0+,βy(t)), t ∈ (0, T ] , I10−+ ry(0) = I10−+ ry(T ).
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