Convex Majorants Method in the Theory of Nonlinear Volterra Equations

TL;DR

This paper develops a convex majorants method to analyze solutions of nonlinear Volterra integral equations, providing convergence proofs, estimates, and insights into solution behavior near blow-up or branching points.

Contribution

It introduces a convex majorants approach for nonlinear Volterra equations, offering new convergence and estimate results not previously established.

Findings

Successive approximations converge under the proposed method.

Explicit estimates for solutions are derived.

Conditions for solution blow-up and branching are identified.

Abstract

The main solutions in sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed. Convergence of the successive approximations is established through studies of majorant integral and majorant algebraic equations. Estimates are derived for the solutions and for the intervals on the right margin of which the solution has blow-up or solution start branching.