The Study Of The Existence Of Solutions Of Binary Nonlinear Operator Equations And Applications

In this thesis, we firstly use the iterative technique to study the existence and uniqueness of coupled fixed points of the counter-mixed monotone operators, obtain some existence and uniqueness results of the coupled fixed point and iterative convergence; then introduce the new conpect of Ï„-(?)-concave-convex operator, and study the existence and uniqueness of fixed point for such a class of operators, obtain some new results. As an application, we apply one of the main results to study the existence of solutions of nonlinear Hammerstein integral equations; finally, as for binary nonlinear operators, we introduce the new concept of g-comparable operator, it is a gemeralization of mixed monotone operator and counter-mixed monotone operator, and study the existence and uniqueness of g-coupled fixed point of such a class of operators. obtain some new results. In addition,we give some examples to support our results. Our main results are listed as follows:1. Let E be a real Banach space and P be a normal cone of E. Suppose that an operator A:P×Pâ†'P is counter-mixed monotone and condensing. In addition, suppose that there exist u0,v0∈P such that u0≤A(v0,u0)≤A(u0,v0)≤v0Then there exist u*,v*∈[u0,v0] such that where And A has a coupled fixed point in [u0,v0]×[u0, v0]. Moreover1) If u*=v*, then A has a unique coupled fixed point (u*,u*)in3) If u*≠v*,then A has a coupled fixed point (uâ–³,vâ–³) in P*. but (u*,v*) is not a coupled fixed point of A. And for any coupled fixed point(r,s)∈P*of A, we have(u*,v*)<(r,s)<(V*,u*).Suppose P(?) E is minimal,then for any coupled fixed point(x,y)of A,the following formula is not true:2.Let E be a real Banach space and P be a normal cone Of E.Suppose that an operator A:P×Pâ†'P is mixed monotone and τ一(?)一concave-convex.In addition,suppose that there exists h∈P+such that A(h,h)∈Ph and for any t∈(a,b) and x,y∈Ph,(?)(t,x,y)≥(?)(t,h,h).Then A has a unique fixed point x*in Ph and for any x0,y0∈Ph, constructing successively the sequences we have||xn-x*||â†'0,||yn-x*||â†'0,(nâ†'∞).3.Let(X,d)be a complete metric space and(X,≤)be a partially ordered set. Let F:X×Xâ†'X be an operator having the g—comparable property and be continuons and commutable with g,where g:Xâ†'X be a continuous mapping. Suppose F(X×x)(?)g(X)and the following conditions are satisfied:(1)There exist altering distance functions (?) and φsuch that for all x,y,u,y∈X for which g(x)is comparable with g(u)and g(y)is comparable with g(v).(2)There exist x0,y0∈Xsuch that g(x0)is comparable with F(x0,y0)and g(y0) is comparable with F(y0,x0).(3) If{xn}n=1∞is a comparable sequence of X, then for all s,l∈N+, xs is comparable with x1Then F has a g-coupled fixed point, that is, there exists (x,y)∈X×X such that g(x)=F(x,y), g(y)=F(y,x).4. Let (X,d) be a complete metric space and (X,≤) be a partially ordered set. Let F:X×Xâ†'X be an operator having the g-comparable property and be continuous and commutable with g, where g:Xâ†'>X be a continuous mapping. Suppose F(X x X)(?) g(X) and the following conditions hold:(1) There exist altering distance functions (?) and φ, such that for all x,y,u,v∈X for which g(x) is comparable with g(u) and g(y) is comparabl e with g (v).(2) There exist x0,y0∈X such that g(x0) is comparable with F(x0,y0) and g (y0) is comparable with F (y0. x0).(3) If{xn}n=1∞is a comparable sequence of X, then for all s,l∈N+, xs is comparable with x(?) Then F has a g-coupled fixed point,that is,there exists (x,y)∈X×X such that g(x)=F(x..y),g(y)=F(y,x).