Abstract
We investigate the following nonlinear first-order periodic boundary value problem on time scales: , , . Some new existence criteria of positive solutions are established by using the monotone iterative technique.
Highlights
Periodic boundary value problems PBVPs for short for dynamic equations on time scales have been studied by several authors by using the method of lower and upper solutions, fixed point theorems, and the theory of fixed point index
In this paper we are interested in the existence of positive solutions for the following first-order PBVP on time scales: xΔ tptxσtft, x t, t ∈ 0, T T, 1.1
If σ t > t, we say that t is right scattered, while if ρ t < t, we say that t is left scattered
Summary
Periodic boundary value problems PBVPs for short for dynamic equations on time scales have been studied by several authors by using the method of lower and upper solutions, fixed point theorems, and the theory of fixed point index. We refer the reader to 1–10 for some recent results. In this paper we are interested in the existence of positive solutions for the following first-order PBVP on time scales: xΔ tptxσtft, x t , t ∈ 0, T T, 1.1. X0 xσT , where σ will be defined, T is a time scale, T > 0 is fixed and 0, T ∈ T. By applying the monotone iterative technique, we obtain the existence of positive solution for the PBVP 1.1 , and give an iterative scheme, which approximates the solution. For abstract monotone iterative technique, see 11 and the references therein
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