Abstract
The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.
Highlights
The fixed point theory is derived from the investigation of the solution for certain types of differential equations using the successive approximation method
This fact indicates that the advances and progress in fixed point theory can be referred back to differential equations and the integral equations
A self-mapping T, on an extended b-metric space (S, δe ), is named a Θe -contraction if there exists a function θ ∈ Θ such that: θ (δe ( Tx, Ty)) ≤ [θ (δe ( x, y))]r if δe ( Tx, Ty) 6= 0 for x, y ∈ S, where r ∈ [0, 1) such that lim sup ω
Summary
The fixed point theory is derived from the investigation of the solution for certain types of differential equations using the successive approximation method. Let S be a non-empty set endowed with the extended b-metric δe , and a sequence { xn } in S is said to:. By using the obtained results, we propose the solutions of the nonlinear integral equation and fractional differential equation via the fixed point approach, which are presented in Sections 3 and 4.
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