Abstract
In this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.
Highlights
The idea of fractional calculus is to replace the natural numbers in the derivative’s order with rational ones
Inspired by [2] and [3, 4, 9, 10, 12], we deal with the boundary value problem (BVP)
The function x ∈ E is a solution of the BVP (7) if and only if x solves the integral equation x(t) = –(T
Summary
The idea of fractional calculus is to replace the natural numbers in the derivative’s order with rational ones. Where u : J → (1, 2], f1 : J × × → is a continuous function and cDu0+(t) is the Caputo fractional derivative of variable-order u(t). The left Caputo fractional integral (CFI) of variable-order u(t) for the function f2(t) [7, 8, 11] is. The left Caputo fractional derivative (CFD) of variable-order v(t) for the function f2(t) [7, 8, 11] is cDva(+1t)f2(t) =. Lemma 2.2 ([18]) Let u : J → (1, 2] be a continuous function, for f2 ∈ Cδ(J, ) = {f2(t) ∈ C(J, ), tδf2(t) ∈ C(J, ), 0 ≤ δ ≤ 1}, the variable order fractional integral I0u+(t)f2(t) exists for any points on J. There exists a solution x ∈ C(J, ) of Eq (1) with z(t) – x(t) ≤ cf , t ∈ J
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