Existence, Uniqueness and Stability of Solutions of a Variable-Order Nonlinear Integro-differential Equation in a Banach Space

This article studies some important results in a Banach space for non-discrete nonlinear integro-differential equations with variable order \(0<\sigma (\theta )<1\)

$$\begin{aligned}{} & {} D^{\sigma (\theta )}_{0,\theta } \vartheta (\theta ) =\eta (\theta ,\vartheta (\theta ))+\vartheta (\theta ) \int _{0}^{\theta } \kappa (\theta ,a,\vartheta (a)){\textrm{d}}a,\quad \theta \in \aleph =[0,\Theta ],\quad \Theta >0, \\{} & {} \vartheta (0)=\vartheta _0. \end{aligned}$$

The contraction mapping principle and Krasnoselskii fixed-point theorem are employed to investigate the results, and Ulam–Hyers definitions are used for stability theory. Further, we have discussed the maximal and minimal solutions with the continuation theorem for \(\sigma (\theta ) \rightarrow 1\).