# Solvability of inclusions of Hammerstein type

## TL;DR

This paper develops a general continuation theorem for admissible multimaps and applies it to solve a wide range of operator inclusions of Hammerstein type in Lebesgue-Bochner spaces, with diverse applications.

## Contribution

It introduces a universal rule for solving Hammerstein-type operator inclusions using a new continuation theorem for admissible multimaps.

## Key findings

- Established a general continuation theorem of Leray-Schauder type.
- Applied the theorem to various differential and integral inclusions.
- Demonstrated the approach's validity through multiple examples.

## Abstract

A fairly general continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps is set forth. This result is then used to establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of $m$-accretive mutivalued differential equations, the anti-periodic problem for semilinear differential inclusions, abstract integral inclusions of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusions.