DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2014/241942 241942 Research Article Fixed Point Theorems on Nonlinear Binary Operator Equations with Applications Qiao Baomin Ciepliński Krzysztof Department of Mathematics Shangqiu Normal College Shangqiu 476000 China sqnc.edu.cn 2014 1962014 2014 18 04 2014 11 06 2014 11 06 2014 19 6 2014 2014 Copyright © 2014 Baomin Qiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone and partial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some corresponding results are obtained. Finally, the applications of our results are given.

1. Introduction

In recent years, more and more scholars have studied binary operator equations and have obtained many conclusions; see . In this paper, we will discuss solutions for these equations which associated with an ordinal symmetric contraction operator and obtain some results which generalized and improved those of . Finally, we apply our conclusions to two-point boundary value problem with two-degree super-linear ordinary differential equations.

In the following, let E always be a real Banach space which is partially ordered by a cone P, let P be a normal cone of E, N is normal constant of P, partial order ≤ is determined by P and θ denotes zero element of E. Let u,vE,u<v,D=[u,v]={xE:uxv} denote an ordering interval of E.

For the concepts of normal cone and partially order, mixed monotone operator, coupled solutions of operator equations, and so forth see [1, 5].

Definition 1.

Let A:D×DE be a binary operator. A is said to be L-ordering symmetric contraction operator if there exists a bounded linear and positive operator L:EE, where spectral radius r(L)<1 such that A(y,x)-A(x,y)L(y-x) for any x,yD,  xy, where L is called a contraction operator of A.

2. Main Results Theorem 2.

Let A:D×DE be L-ordering symmetric contraction operator, and there exists a α[0,1) such that (1)A(x2,y2)-A(x1,y1)-α(x2-x1),ux1x2v,uy2y1v. If condition (H1) uA(u,v),A(v,u)v-α(v-u) or (H2) u+α(v-u)A(u,v),A(v,u)v holds, then the following statements hold.

(C1) A(x,x)=x has a unique solution x*D, and for any coupled solutions x,yD,  x=y=x*.

(C2) For any x0,y0D, we construct symmetric iterative sequences: (2)xn=1α+1[A(xn-1,yn-1)+αxn-1],yn=1α+1[A(yn-1,xn-1)+αyn-1],hhhn=1,2,3,. Then xnx*,  ynx*(n), and for any β(r(L),1), there exists a natural number m; and if nm, we get error estimates for iterative sequences (2): (3)xn(yn)-x*2N(α+βα+1)nu-v.

Proof.

Set B(x,y)=(1/(α+1))[A(x,y)+αx], and if condition (H1) or (H2) holds, then it is obvious that (4)uB(u,v),B(v,u)v. By (1), we easily prove that B:D×DE is mixed monotone operator, and for any x,yD,  uxyv, (5)θB(y,x)-B(x,y)H(y-x), where H=(1/(α+1))(L+αI) is a bounded linear and positive operator and  I, is identical operator.

By the mathematical induction, we easily prove that (6)θBn(y,x)-Bn(x,y)Hn(y-x),uxyv, where Bn(x,y)=B(Bn-1(x,y),Bn-1(y,x)),  x,yD,  n2.

By the character of normal cone P, it is shown that (7)Bn(y,x)-Bn(x,y)NHny-x,uxyv.

For any β(r(L),1), since limn||Hn||1/n=r(H)(α+r(L))/(α+1)<(α+β)/(α+1)<1, there exists a natural number m, and if nm, we have ||Hn||<((α+β)/(α+1))n, and N||Hm||<1. Considering mixed monotone operator Bm and constant N||Hm||, Bm(x,x)=x has a unique solution x* and for any coupled solution x,yD, such that x=y=x* by Theorem 3 in .

From Bm(B(x*,x*),B(x*,x*))=B(Bm(x*,x*),Bm(x*,x*))=B(x*,x*), and the uniqueness of solution with Bm(x,x)=x, then we have B(x*,x*)=x* and A(x*,x*)=x*.

We take note of that A(x,x)=x and B(x,x)=x have the same coupled solution; therefore, a coupled solution for B(x,x)=x must be a coupled solution for Bm(x,x)=x; consequently, (C1) has been proved.

Considering iterative sequence (2), we construct iterative sequences: (8)un=B(un-1,vn-1),vn=B(vn-1,un-1), where u0=u, v0=v, it is obvious that (9)xn=B(xn-1,yn-1),yn=B(yn-1,xn-1),θvn-unHn(v-u), by the mathematical induction and characterization of mixed monotone of B; then (10)unx*vn,unxnvn,unynvn. Hence, (11)xn(yn)-unNvn-un,x*-unNvn-un,n=1,2,3,. Moreover, if nm, we get (12)xn(yn)-x*2Nvn-un2NHnv-u2N(α+βα+1)nu-v. Consequently, xnx*,ynx*(n).

Remark 3.

When α=0, Theorem 1 in  is a special case of this paper Theorem 2 under condition (H1) or (H2).

Corollary 4.

Let A:D×DE be L-ordering symmetric contraction operator; if there exists a α[0,1) such that A satisfies condition of Theorem 2, the following statement holds.

(C3) For any β(r(L),1) and α+β<1, we make iterative sequences: (13)un=A(un-1,vn-1),vn=A(vn-1,un-1)+α(vn-1-un-1),hhhn=1,2,3,, or (14)un=A(un-1,vn-1)-α(vn-1-un-1),vn=A(vn-1,un-1),hhhn=1,2,3,, where u0=u,  v0=v.

Thus, unx*,vnx*  (n), and there exists a natural number m, and if nm, we have error estimates for iterative sequences (13) or (14): (15)un(vn)-x*N(α+β)nu-v.

Proof.

By the character of mixed monotone of A, then (1) and (C1), (C2) [in (1), (C2) where α=0] hold.

In the following, we will prove (C3).

Consider iterative sequence (13); since ux*v, we get (16)u1=A(u,v)A(x*,x*)=x*A(v,u)=v1-α(v-u)v1. By the mathematical induction, we easily prove unx*vn, n1, hence (17)θx*-unvn-un,θvn-x*vn-un.

It is clear that (18)θvn-un(L+αI)(vn-1-un-1)=(L+αI)n(v-u),n1.

For any β(r(L),1), α+β<1, since (19)limn(L+αI)n1/n=r(L+αI)r(L)+α<α+β<1, there exists a natural number m, if nm, such that (20)(L+αI)n<(α+β)n.

Moreover, (21)un(vn)-x*N(L+αI)nu-vN(α+β)nu-v,(nm). Consequently, unx*, vnx*, (n).

Similarly, we can prove (14).

Theorem 5.

Let A:D×DE be a L-ordering symmetric contraction operator; if there exists a α[0,1) such that (1-α)uA(u,v),A(v,u)(1-α)v, then the following statements hold.

(C4) Operator equation A(x,x)=(1-α)x has a unique solution x*D, and for any coupled solutions x,yD, x=y=x*.

(C5) For any x0,y0,w0,z0D, we make symmetric iterative sequences (22)xn=11-αA(xn-1,yn-1),yn=11-αA(yn-1,xn-1),hhn=1,2,3,,(23)wn=A(wn-1,zn-1)+αwn-1,zn=A(zn-1,wn-1)+αzn-1,hhln=1,2,3,. Then xnx*, ynx*, wnx*, znx*  (n), and for any β(r(L),1),α+β<1, there exists a natural number m, and if nm, then we have error estimates for iterative sequences (22) and (23), respectively, (24)xn(yn)-x*2N(β1-α)nu-v,wn(zn)-x*2N(α+β)nu-v.

Proof.

Set B(x,y)=(1/(1-α))A(x,y) or C(x,y)=A(x,y)+αx; we can prove that this theorem imitates proof of Theorem 2.

Similarly, we can prove the following theorems.

Theorem 6.

Let A:D×DE be L-ordering symmetric contraction operator; if there exists a α[0,1) such that u+αvA(u,v),  A(v,u)v+αu, then the following statements hold.

(C6) Equation A(x,x)=(1+α)x has a unique solution x*D, and for any coupled solutions x,yDx=y=x*.

(C7) For any x0,y0D, we make symmetric iterative sequence: (25)xn=11+αA(xn-1,yn-1),yn=11+αA(yn-1,xn-1),hhn=1,2,3,. Then xnx*, ynx*  (n); moreover, β(r(L),1), and there exists natural number m, and if nm, then we have error estimates for iterative sequence (25): (26)xn(yn)-x*2N(βα+1)nu-v,

(C8) For any β(r(L),1)(α+β<1),w0,z0D, we make symmetry iterative sequence wn=A(wn-1,zn-1)-αzn-1,zn=A(zn-1,wn-1)-αwn-1,  n1; then wnx*,  znx*  (n), and there exists a natural number m, and if nm, we have error estimates for iterative sequence (24).

Remark 7.

When α=0, Corollary 2 in  is a special case of this paper Theorems 26.

Remark 8.

The contraction constant of operator in  is expand into the contraction operator of this paper.

Remark 9.

Operator A of this paper does not need character of mixed monotone as operator in .

3. Application

We consider that two-point boundary value problem for two-degree super linear ordinary differential equations: (27)x′′+a(t)xm+11+b(t)x=0,t[0,1],(m2)x(0)=x(1)=0.

Let k(t,s) be Green function with boundary value problem (23); that is, (28)k(t,s)=min{t,s}={t,tss,s<t.

Then the solution with boundary value problem (23) and solution for nonlinear integral equation with type of Hammerstein(29)x(t)=01k(t,s){a(s)[x(s)]m+11+b(s)x(s)}ds are equivalent, where maxt[0,1]01k(t,s)ds=1/2.

Theorem 10.

Let a(t), b(t) be nonnegative continuous function in [0,1],  p=maxt[0,1]a(t), q=maxt[0,1]b(t). If p<1,mp+q<2, then boundary value problem (23) has a unique solution x*(t) such that 0x*(t)1(t[0,1]). Moreover, for any initial function x0(t),y0(t), such that (30)0x0(t)1,0y0(t)1(t[0,1]), we make iterative sequence: (31)xn(t)=01k(t,s){a(s)[xn-1(s)]m+11+b(s)yn-1(s)}ds,yn(t)=01k(t,s){a(s)[yn-1(s)]m+11+b(s)xn-1(s)}ds,hhhhhhhhhlln=1,2,3,. Then xn(t) and yn(t) are all uniformly converge to x*(t) on [0,1], and we have error estimates: (32)|xn(t)(yn(t))-x*(t)|2(mp+q2)n,hlt[0,1],n=1,2,3,.

Proof.

Let E=C[0,1],  P={xEx(t)0,t[0,1]},  x=maxt[0,1]|x(t)| denote norm of; then E has become Banach space, P is normal cone of E, and its normal constant N=1. It is obvious that integral Equation (24) transforms to operator equation A(x,x)=x, where (33)A(x,y)(t)=01k(t,s){a(s)[x(s)]m+11+b(s)y(s)}ds,t[0,1].

Set u=u(t)0,  v=v(t)1; then D=[0,1] denote ordering interval of E, A:D×DE is mixed monotone operator, and 0A(0,1),A(1,0)(1+p)/2<1.

Set (34)Lx(t)=01k(s,t)[ma(s)+b(s)]x(s)ds,t[0,1].

Then L:EE is bounded linear operator, its spectral radius r(L)(mp+q)/2<1, and for any x,yE,0x(t)y(t)1 such that 0A(y,x)(t)-A(x,y)(t)L(y-x)(t), A is L-ordering symmetric contraction operator, by Theorem 2 (where α=0); then Theorem 10 has been proved.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the NSF of Henan Education Bureau (2000110019) and by the NSF of Shangqiu (200211125).

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