Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator

Abstract

This paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach’s fixed point theorem to examine the existence and uniqueness of solutions, and we also explore Hyers–Ulam (H–U) and Hyers–Ulam–Rassias (H–U–R) stability in the space C([0,β],R). Furthermore, three solution sets Uσ,λ, Uθ,1 and U1,1 are constructed for σ>0, λ>0, and θ∈(0,1), and then we obtain local stability of the solutions with some ideal conditions and by using Schauder fixed point theorem on these three sets, respectively. Also, to achieve the goal, we choose the parameters for the NVFIE as δ∈(12,1), ρ∈(0,1), γ>0. Three examples are provided to clarify the results.

Keywords

• Erdélyi–Kober fractional integral operator
• Hyers–Ulam–Rassias stability
• Hyers–Ulam stability
• Local stability
• Fixed point theorem