Positive Solutions And Mann Iterative Algorithms For A Nonlinear Two-dimensional Difference System With Multiple Delays

Difference equations are for one or more discrete variables change with time, space and other conditions, and the mathematical model of abstract symbols portrayed. Theoretical study based on difference equations have a great help for the extensive application of the cobweb model and infectious disease model, and a mathematical model to express the many practical problems are related to the difference equations, which is greatly facilitated the urgent need to study. Difference equations and its properties in many areas have played a significant role, but because of its complex and diverse forms of change, making the study of existence of solutions very difficult conditions.In recent years, the research development based on nonlinear difference equations are very quickly, many scholars are keen on research about nonlinear difference equations solvability included oscillatory, non-oscillatory, periodic solutions, non-negative solutions, positive solutions and bounded, stability, asymptotic behavior such fruitful research. Some solvability is largely related to fixed point theory, Liu[4-11] and other authors using Banach fixed point theorem, Krasnoselskii fixed point theorem, Schauder fixed point theorem proved convergence and solvability of several nonlinear difference equations.In this paper, to solve a class of fourth order nonlinear two dimensional difference system with multiple delays(1.1), by discussing the range of neutral 10{ }n n Nnb ?R? and 20{ }n n Nnb ?R?, using the Banach fixed point theorem and some new mathematical techniques demonstrate the seven theorems when its value for-1, 1 and interval for(0,1),(1,+∞),(-1,0),(-∞,-1),(-1/2,1/2), and empathy construct the three theorems when two difference equations are not the same value interval. In the different value of the neutral, the difference system(1.1) constructed out of different original function in the appropriate conditions, to prove that is a contraction self-mapping, so for this original function’s fixed point to calculate fourth order difference, proved it is the positive solution of difference system(1.1), while applied Mann iterative sequence, proved convergence that the iterative sequence with positive solutions of the difference system(1.1), building error estimate inequality, and constructed ten examples to illustrate the application and importance of the corresponding theorem, and because known results can not determine difference system solvable, so the difference system discussed in this paper is more general.