Unpredictable solutions of differential equations

TL;DR

This paper introduces a new approach using the topology of convergence on compact sets to define and analyze unpredictable solutions of differential equations, proving existence and uniqueness with practical examples.

Contribution

It presents a novel application of the topology of convergence on compact sets to establish the existence and uniqueness of unpredictable solutions in differential equations.

Findings

Proved existence and uniqueness of unpredictable solutions for delay differential equations.

Extended the results to quasilinear ordinary differential equations.

Provided simulation examples illustrating the theoretical findings.

Abstract

We apply the topology of convergence on compact sets to define unpredictable functions [5, 6]. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation is proved. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. Examples with simulations illustrate the obtained results.