Existence and uniqueness for ψ‐Hilfer fractional differential equation with nonlocal multi‐point condition

Abstract

In this paper, we study and investigate the ψ−Hilfer fractional differential equation with nonlocal multi‐point condition of the form: eqnarrayleft center righteqnarray-1eqnarray-2eqnarray-3Da+q,p;ψu(t)=f(t,u(t),Da+q,p;ψu(t)),t∈[a,b],eqnarray-1eqnarray-2eqnarray-3Ia+1−r;ψu(a)=∑i=1mβiu(ηi),q≤r=q+p−qp<1,ηi∈[a,b], where 0<q<1,1em0≤p≤1,3.0235ptm∈double-struckN, βi∈double-struckR, i=1,2,...,m, −∞<a<b<∞, Da+q,p;ψ is the ψ− Hilfer fractional derivative, f:false[a,bfalse]×double-struckR×double-struckR→double-struckR is a continuous function, and Ia+1−r;ψ is the ψ‐Riemann‐Liouville fractional integral of order 1−r. By using Schaefer's and Banach fixed point theorems, we prove the existence, uniqueness, and stability analysis of this problem. An example is given to illustrate the applicability of our results.

Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

“…The operator T :Ē → X as provided in (14) is completely continuous by Theorem 2. Take w ∈ E on using (F 2 ), one has…”

“…On the other hand, very well performed research has been conducted on the numerical side of FODEs. In this regard plenty of research articles addressing numerical and qualitative analysis have been presented in the past few years (for instance see [11][12][13][14][15][16] and the references therein). Here we remark that stability analysis is also an important aspect of qualitative analysis.…”

On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

“…Motivated by their significance, a ton of scientists generalized these equations into different types and presented the solvability aspect of such problems both numerically and theoretically; for additional subtleties, see [40][41][42][43][44][45][46] and the references therein. Besides, some authors applied various kinds of fractional derivatives and studied the existence and stability of Ulam-Hyers, which can be found in [47][48][49][50][51]. However, not many works have been proposed for pantograph FDEs, especially those involving ABC fractional operator and nonlocal conditions.…”

Study on Krasnoselskii’s fixed point theorem for Caputo–Fabrizio fractional differential equations

“…But recently another form of derivative, called nonsingular type, has attracted much attention from the researchers. The existence theory, together with stability results, has been very well investigated for other FODEs; for details, see [8][9][10]. The considered differential operator has been introduced in 2015 by Caputo and Fabrizio [11] (in short, we write it as (CFFD)), which replaces the singular kernel by a nonsingular kernel of exponential type.…”