# Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space

## TL;DR

This paper proves the existence and uniqueness of a strongly continuous, self-adjoint, nonnegative solution to a nonautonomous operator Riccati integral equation in reflexive Banach spaces, advancing the mathematical understanding of such equations.

## Contribution

It establishes the solvability and uniqueness of solutions to operator Riccati equations in reflexive Banach spaces, a significant extension of existing theory.

## Key findings

- Unique strongly continuous self-adjoint nonnegative solution exists
- Solution resides in the space of bounded linear operators from X to its dual
- Advances the mathematical theory of operator Riccati equations in Banach spaces

## Abstract

We show that if $X$ is a reflexive Banach space, then a nonautonomous operator Riccati integral equation has a unique strongly continuous self-adjoint nonnegative solution $P(t)\in\mathcal{L}(X,X^*)$