Existence and stability results for Langevin equations with Hilfer fractional derivative

Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

“…If ρ → 0 and φ(t) = t, we obtain the results of [48] and [52]. Furthermore, if ρ → 0 we obtain the Ulam-Hyers and generalized Ulam-Hyers stability for the implicit fractional pantograph differential equations with φ-Hilfer fractional derivatives [52,58] and if q = 0 we obtain [51].…”

“…If t ∈ [a, b] as defined in paper [58], the function f (t, x(t), x(λt)) is not well-defined for some choice of 0 < λ < 1. Thus, our results modify and improve the above cited remarks and can be considered as the development of the qualitative analysis of fractional differential equations.…”

“…Recently Harikrishman et. al [58] established existence and uniqueness of nonlocal initial value problem for fractional pantograph differential equation involving ψ-Hilfer fractional derivative of the form:…”

“…Motivated by the papers [21,47,48] and some familiar results on fractional pantograph differential equations [16,52,55,58]. We discuss the existence and uniqueness of the solution of the implicit pantograph fractional differential equations involving φ-Hilfer fractional derivatives.…”

Further results on existence of positive solutions of generalized fractional boundary value problems

“…Also, the problem ⎧ ⎨ ⎩ C D ı 0+ (κ) + g(κ, (κ)) = 0, κ ∈ (0, 1), (0) + (0) = 0, (1) + (1) = 0, was discussed in [48], where 1 < ı ≤ 2, and C D ı 0+ is the CF operator. For some recent findings on GFDs with respect to another function κ, see [2,3,14,15,25,34,36,42,43,46,47].…”