Existence and Hyers–Ulam stability for three-point boundary value problems with Riemann–Liouville fractional derivatives and integrals

Abstract

This paper is concerned with a class of two-term fractional differential equations. Three-point boundary value problems with mixed Riemann-Liouville fractional differential and integral boundary conditions are discussed. The Green's function is investigated and the existence results are obtained based on some fixed point theorems. The Hyers-Ulam stability is also studied for null boundary conditions. As an auxiliary result, a Gronwall type inequality of fractional order integral is obtained.

On Hyers–Ulam stability of a multi-order boundary value problems via Riemann–Liouville derivatives and integrals

“…In fact, stability of physical phenomena has an old history, and one can find a lot of works in the literature not only in the last century but also before it [27][28][29][30][31][32][33][34][35][36]. During recent decades, considerable attention has been given to the study of the Hyers-Ulam stability of functional differential and integral equations [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54].…”

“…under some conditions [39]. They considered two-term class of three-point boundary value problems with Riemann-Liouville fractional derivatives and integrals [39].…”

“…under some conditions [39]. They considered two-term class of three-point boundary value problems with Riemann-Liouville fractional derivatives and integrals [39]. Now, by using and mixing the idea of the above-mentioned works, we consider a new category of boundary value problem including multi-order Riemann-Liouville fractional equation supplemented with different linear integro-derivative boundary conditions as follows:…”

Existence and Ulam–Hyers stability of coupled sequential fractional differential equations with integral boundary conditions

“…Numerous forms of stabilities have been studied in literature which are Mittag-Leffler stability, exponential stability, Lyapunov stability, etc. For historical background of Ulam-Hyers stability and recent results, we refer to works [27][28][29][30][31][32][33][34][35][36]. To the best of our knowledge, the Ulam-Hyers stability has been very rarely studied for coupled system of fractional differential equations.…”

Multi-term fractional boundary value problems with four-point boundary conditions

“…In a recent work [21], the authors studied existence and stability results for the following three-point boundary value problem with Riemann-Liouville fractional derivatives and integrals λ D α 0 x(t) + D β 0 x(t) = f (t, x(t)), t ∈ J := [0, T ], x(0) = 0, µD…”