# On a multivalued differential equation with nonlocality in time

## TL;DR

This paper investigates the existence of solutions for a multivalued differential equation involving nonlocal time-dependent operators, using advanced fixed-point theorems in a complex Banach space setting.

## Contribution

It introduces a novel approach to analyze multivalued differential equations with nonlocality in time, extending fixed-point methods to non-embedded Banach spaces.

## Key findings

- Existence of solutions established under specified conditions.
- Generalization of Kakutani fixed-point theorem applied.
- Framework accommodates nonlocal and multivalued operators.

## Abstract

The initial value problem for a multivalued differential equation is studied, which is governed by the sum of a monotone, hemicontinuous, coercive operator fulfilling a certain growth condition and a Volterra integral operator in time of convolution type with exponential decay. The two operators act on different Banach spaces where one is not embedded in the other. The set-valued right-hand side is measurable and satisfies certain continuity and growth conditions. Existence of a solution is shown via a generalisation of the Kakutani fixed-point theorem.