A New Theorem on N-th Order Differential Equation with Retarded Argument

TL;DR

This paper derives a new inequality for n-th order differential equations with retarded arguments using the mean-value theorem, which simplifies to a key inequality for ordinary differential equations and aids in proving existence theorems.

Contribution

It introduces a novel inequality for n-th order differential equations with retarded arguments, extending classical results and facilitating existence proofs.

Findings

Derived a new inequality for n-th order differential equations with retarded argument

Simplifies to an inequality for ordinary differential equations when retarded argument vanishes

Provides a tool for proving existence theorems in differential equations

Abstract

In this work, using the well-known mean-value theorem (Lagrange's theorem) we obtain an inequality for n-th order differential equations with retarded argument. If the retarded argument vanishes then the inequality turns to an inequality for n-th order ordinary differential equations which plays a vital role in the proof of existence theorem for n-th order ordinary differential equations.