Positive Solutions And Mann Iterative Schemes For A Nonlinear Two-Dimentional Differential System

The purpose of this paper is to cope with a nonlinear two-dimentional differential system where Ï„a>0,k∈N,t0∈(?)\{0),{fa)a∈Λ2∈C([t0,+∞)×(?)k,(?)),{ba}a∈Λ2,{Ca)a∈Λ2∈([to,+(?)),(?)),{rj}j∈Λk,{Sj}j∈Λk∈C([t0,+∞),(?)+) with limtâ†'ral(t)=limtâ†'+∞Sal(t)=+∞,(l,a)∈Λk×Λ2.In the light of different codomain of the function{ba(t)}a∈Λ2,firstly,the full text is separated into ten theorems,and by means of B anach fixed point theorem to study the existence of uncountable many positive solutions.In each theorem,two mappings and Mann iterative approach are constructed,and then the mapings are proved to satisfy the conditions of Banach fixed point theorem which is used in the theorem and Mann iterative schemes is convergent.Finally,In the light of Banach fixed point theorem,the conditions to guarantee the existence of uncountable many positive solutions and the convergence of Mann iterative schemes of the nonlinear two-dimentional differential system are got.In order to illustrate that ten results presented in this paper generate and improve essentially the results in this field.Four nontrivial examples are constructed as applications of our results.The final six theorems of this paper is combinated by fhe range of the function {ba(t)}a∈Λ2.Thus the cortesponding examples are omitted.In a word,the results of this paper is meaningful.