On Fixed Points of Nonlinear Monotone and Strongly Concave Operators Acting in Normal Cones

TL;DR

This paper introduces a new class of nonlinear monotone operators with strong concavity in Banach spaces, providing fixed point existence, convergence, and uniqueness results, with applications to integral operators and nonlinear heat equations.

Contribution

It develops constructive principles for fixed points of strongly concave monotone operators and demonstrates convergence and uniqueness, extending to integral operators and PDEs.

Findings

Fixed points exist under new constructive principles.

Iterative process converges geometrically to the fixed point.

Fixed point is unique in a wide conical segment.

Abstract

We introduce and study a new class of nonlinear monotone operators acting in normal cones of real Banach spaces and possessing the property of strong concavity. We establish new constructive principles for the existence of nonzero fixed points for this class of operators. Further, we prove that the corresponding iterative process converges to the fixed point at geometric rate. We also establish the uniqueness of the fixed point in a sufficiently wide conical segment. These results are applied to Hammerstein-type and Urysohn-type nonlinear integral operators acting in non-reflexive Banach spaces, as well as to the Cauchy problem for a nonlinear heat equation.