The Existence Of Uncountably Many Bounded Positive Solutions For Two-dimensional Third Order Nonlinear Neutral Delay System Of Difference Equations

This paper is mainly to study the existence of uncountably many bounded positive solutions for a new two-dimensional third order nonlinear neutral delay system of difference equations, the form of the equations is as follows:For illustrating the theorems in this paper more better, this paper is divided into four parts. At first, in the first part introduction of this paper, the form of difference equation or the system of difference equations has been introduced by some scholars in recent years, and the solutions conditions of them, the new two-dimensional third order nonlinear neutral delay system of difference equations and basic conditions are given. Secondly, in the second part preparation and lemma of this paper, in order to read and understand theorems of this paper conveniently, some correspounding symbols in the theorem of the provisions are restricted, and at the same time, it also describes the lemmas which are used in the proof of the theorems. Thirdly, in the third part which is also the main part of this paper, seven theorems are introduced and proved amply, among them the proving process of partial theorems which are similar to the proof of previous theorems are left out, some theorems are also introduced in this part which are the combination of the value of coefficients of seven theorems, in the proving process of the theorems of this paper, the main researches of the proof of the theorems are using Banach fixed point theorem and the nonlinear methods of analysis, through the discussion on the value of coefficients {N}neN0and {q}nEN0, in the different intervals of (-∞q,+∞), in this part provides the restrictive conditions of other coefficients and functions, and carries out the provisions of the convergence and boundedness of some formulars, to discuss the problem of the existence of the solutions for the new two-dimensional third order difference equations in the bounded space, through the discussion on different conditions and different limited conditions, in the proof of the theorems of this paper, the existence of uncountably many bounded positive solutions for the binary three order difference equations can be observed. In the end, in the forth part of this paper, some examples are listed, the coefficients, the functions and the numbers of the difference equations are given the specific form of size, to further better display the third part.