Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

Abdulkadir Doğan

doi:10.3906/mat-1810-97

Abstract

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].

Highlights

The investigation of dynamic equations on time scales goes back to its discoverer Stefan Hilger [19], and it is a new field of theoretical research in mathematics
The topic is inspired by the conception that dynamic equations on time scales can establish connections between continuous and discontinuous mathematics
In [2], Anderson studied the existence of one positive solution of the following dynamic equation on time scales: u∆∇(t) + a(t)f (u(t)) = 0, t ∈ (0, T )T, u(0) = 0, αu(η) = u(T ), where a ∈ Cld(0, T ) is nonnegative, f : [0, ∞) → [0, ∞) is continuous, η ∈ (0, ρ(T )), and 0 < α < T /η

Summary

Introduction

The investigation of dynamic equations on time scales goes back to its discoverer Stefan Hilger [19], and it is a new field of theoretical research in mathematics. In [2], Anderson studied the existence of one positive solution of the following dynamic equation on time scales: u∆∇(t) + a(t)f (u(t)) = 0, t ∈ (0, T )T, u(0) = 0, αu(η) = u(T ), where a ∈ Cld(0, T ) is nonnegative, f : [0, ∞) → [0, ∞) is continuous, η ∈ (0, ρ(T )), and 0 < α < T /η.

Results

Conclusion