Existence Result for Singular Second Order Dynamic Equations with Mixed Boundary Conditions

TL;DR

This paper establishes the existence of positive solutions for singular second-order dynamic equations with mixed boundary conditions on general time scales, using fixed point and approximation methods.

Contribution

It introduces a novel approach combining lower and upper solutions with Brouwer's fixed point theorem for singular dynamic equations on time scales.

Findings

Proves existence of positive solutions under given conditions

Develops a method for approximating singular problems via nonsingular equations

Extends results to general time scales with mixed boundary conditions

Abstract

We explore singular second-order boundary value problems with mixed boundary conditions on a general time scale. Using the lower and upper solutions method combined with the Brouwer fixed point theorem we demonstrate the existence of a positive solution and obtain the desired solution by using a sequence of solutions to a sequence of nonsingular second-order equations and passing to the limit.