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Integral equations, \(\varepsilon\)-fixed points, fixed point regions, linear approximations, and a family of kernels. (English) Zbl 1454.34014

Summary: We study a nonlinear Volterra integral equation by first transforming it into an equivalent equation with kernel depending on an (arbitrary) positive number \(J\). The notion of “\(\varepsilon\)-nearly fixed point” is introduced and a family of resolvent kernels along with a family of corresponding mappings are considered. A part of the paper is devoted to finding a region in which all possible solutions of the equation must reside, yet properties of the family of the resolvents are given. An extensive discussion containing an illustrative example is provided.

MSC:

34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
45D05 Volterra integral equations
45G05 Singular nonlinear integral equations
47H10 Fixed-point theorems
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