Lower and upper estimates of semi-global and global solutions to mixed-type functional differential equations

J Diblík,G Vážanová

doi:10.1515/anona-2021-0218

Abstract

AbstractIn the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered and the results are illustrated by selected examples.

Highlights

In the paper, the existence is considered of semi-global and global solutions to what is called mixed-type functional di erential equations
The existence of global solutions to various classes of functional di erential equations has been investigated for some time, most of the papers only deal with semi-global solutions of delayed equations or advanced equations, or with mixed-type equations on nite intervals
By Theorem 3, we prove that there exist nontrivial left semi-global solutions to equation (3.5)

Summary

Introduction

The existence is considered of semi-global and global solutions to what is called mixed-type (or advanced-delayed) functional di erential equations. Let us prove the existence of a right semi-global positive solution to the nonlinear equation y (t) = (sin t) e−t y(t − )y(t + ). All the hypotheses of Theorem 1 hold and, by (2.8) with i = j = , there exists a right semi-global solution y : J+− → R

Results

Conclusion