Existence And Uniqueness Of Solutions For Boundary Value Problems Of P-Laplacian Dynamic Equations On Time Scales

In recent years, the nonlinear boundary value problems have been appeared in the individual application subject, so the research of the differential equation boundary value problem contributes to the research related field.This paper is divided into the following four chapters:Chapter1Preference, the main basic concepts and theorems need to introduce in this paper.Chapter2In this chapter, we will study the existence of positive solutions to differential equation boundary value problem on time scales as followings, where T is a time scale:0, α∈T.φp(s) is a p-Laplacian operator, That is, And the nonlinear term f satisfies the following conditions:(A11) The function f:R+â†'R+is continuous;(A12) The function h:Tâ†'[0,+∞] is right dense continuous, and in any interval of [0, α]T, h does not vanish identically, where T is the time scale;(A13) B0(Ï…) is a continuous function in Real number set, exist A, B(A≥B>0) such that BÏ…<Bo(v)<AÏ… for all (?)Ï…>0.Through the conditions to limit f, and then we prove that the BVP has at least one positive solution or three positive solutions by using the fixed point theorem in nonlinear functional. Lastly, an example is given to demonstrate the application of our main result. Chapter3We will study the existence of positive solutions with following nonlinear boundary value problems: whereThedifferential equations satisfy the following conditions:and it does not vanish identically on any closed subinterval of [0,α]T, where denotes the set of all left dense continuous functions from [0,α]T to [0,+∞); That is, y(tκ+) and y(tκ-)denotes respectively the right limit and left limit of y(t) in the point t=tκ,κ=1,2,..., m; and (?)σ∈[0,1) such thatThrough the use of Leray Schauder fixed point theorem and non-linear second theorem, we get the differential equation of the boundary value problem, the existence of at least one positive solution and apply-ing a new fixed point theorem, we get at least three of the existence of positive solutions.Chapter4We will study existence of positive solutions to one di-mensional p-Laplacian non-local differential equation boundary value problem on time scales as follows: where λ>0is a constant; t∈[a, b]T (T is a time scale, a, b∈T and where is a continuous function; The function f:[α,b]T x Râ†'[0,+∞) is also continuous.Based on the nonlinear term f(t,y(t)) and conditions of nonlinear boundary value on certain conditions, and using the Krasnosel’skii fixed point theorem prove the existence of at least one positive solutions to such boundary value problem.