Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρk,ϕk)-Hilfer Fractional Integro-Differential Equations

Abstract

In this paper, we establish the existence and stability results for the (ρk,ϕk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρk,ϕk)-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.

Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

“…After some computations, using the Mathematica program, we obtain λ = 17 10 , µ = 10 7 , a ≈ −1831.69757626, b ≈ −449.04583604, c ≈ 10.3109466, d ≈ 37.64671349, and ∆ ≈ −64327.3062105 = 0. So, assumption (H1) is satisfied.…”

“…Although the techniques employed in demonstrating our primary findings in Section 3 are conventional, their adaptation to address our problem (1) and ( 2) is innovative. For more recent investigations concerning Hadamard, Hilfer, and Hilfer-Hadamard fractional differential equations and their applications, we recommend the monograph [8] and the following papers: [1,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].…”

On a (k;chi)-Hilfer fractional system with coupled nonlocal boundary conditions including various fractional derivatives and Riemann–Stieltjes integrals

“…Boundary value problems for fractional differential equations with different kinds of boundary conditions have been studied by many researchers. For example, fractional boundary value problems with integral and multipoint boundary conditions were studied in [30], with antiperiodic boundary conditions in [2], with integral boundary conditions with sequential Riemman-Liouville and Caputo fractional derivatives in [18], with multipoint strip boundary conditions in [6], for sequential hybrid ψ-Hilfer-type fractional differential equations in [9], for nonlinear impulsive (ρ k , φ k )-Hilfer fractional integro-differential equations with nonlocal multipoint fractional integral boundary conditions in [17] and separated boundary conditions in [29]. For a variety of results on boundary value problems for fractional differential equations and inclusions, we refer to the recent monograph [4].…”