Stability theory and the existence of Hilfer type fractional implicit differential equations with boundary conditions

Abstract

The aim of this work is to study the existence and Ulam stability of solution of Hilfer fractional implicit differential equation with boundary condition of the form

EDα,β

α,β

0+  x(t) = f (t, x(t), D0+ x(t)),       t ∈ [0, T],

(0.1)

I1−γ

1−γ

0+    x(0) = a,    I0+    x(T) = b,     γ = α + β − αβ.

Here Dα,β  is the Hilfer fractional derivative, I1−γ  is the left-sided mixed Riemann-Liouville integral of

0+                                                                       0+

order 1 − γ, α ∈ (0, 1), β ∈ [0, 1] and let X be the Banach space,  f  : J × X × X → X is given continuous

function. The results are established by the application of the contraction mapping principle  and

Schaefer’s fixed point theorem. An example is provided to illustrate the applicability of the results.