Nonlocal boundary value problems for integro-differential Langevin equation via the generalized Caputo proportional fractional derivative

Abstract

Results reported in this paper study the existence and stability of a class of implicit generalized proportional fractional integro-differential Langevin equations with nonlocal fractional integral conditions. The main theorems are proved with the help of Banach’s, Krasnoselskii’s, and Schaefer’s fixed point theorems and Ulam’s approach. Finally, an example is given to demonstrate the applicability of our theoretical findings.

Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

“…Furthermore, taking a limit as m → ∞ in (20) for t ∈ (a, b] it follows that the function u(t) satisfies the integral equality (18) which is equivalent to (15). In addition, from the limit conditions in (20) and Remark 2 it follows that the function u(t) satisfies the initial condition in (15). Remark 3.…”

“…Later, Jarad et al [10] introduced a new generalized proportional derivative which is well-behaved and has several advantages over the classical derivatives such as meaning that it generalizes formerly known derivatives in the literature. For recent contributions relevant to fractional differential equations via generalized proportional derivatives, see [11][12][13][14][15][16].…”

“…Consider the PIVP(15) in the partial case λ = 1, f (t) = e (ρ−1)( t ρ ) , a = 0, b = ∞. Then applying t 0 (t − s) α−1 E α,α ( − s) αk−1 ds ρ α(k−1) Γ(kα)…”

Coupled Nonlocal Boundary Value Problems for Fractional Integro-differential Langevin System via Variable Coefficient

“…Langevin equations (introduced by Langevin in 1908) are used to model the evolution of physical phenomena in fluctuating environments (see [17]). Recently, the generalisation of the Langevin equations (fractional Langevin equations) has been considered by authors and researchers, for more details we give the following references [18][19][20][21][22][23][24][25].…”

On a Coupled System of Fractional Differential Equations via the Generalized Proportional Fractional Derivatives

“…In detail, fractional differential equations with generalized proportional derivatives have seen significant contributions from an interested researcher. For instance, we refer to works of Abbas and Ragusa and Hristova and Abbas [29,30] and Khaminsou et al [31,32], and the references existing therein.…”