AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 191675 10.1155/2012/191675 191675 Research Article Existence and Uniqueness Theorems of Ordered Contractive Map in Banach Lattices Li Xingchang Wang Zhihao Su Yongfu Center for Economic Research Harbin University of Commerce Harbin 150028 China hrbcu.edu.cn 2012 2 12 2012 2012 18 09 2012 05 11 2012 2012 Copyright © 2012 Xingchang Li and Zhihao Wang. 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.

This paper presents some existence and uniqueness theorems of the fixed point for ordered contractive mapping in Banach lattices. Moreover, we prove the existence of a unique solution for first-order ordinary differential equations with initial value conditions by using the theoretical results with no need for using the condition of a lower solution or an upper solution.

1. Introduction and Preliminaries

Existence of fixed points in partial ordered complete metric spaces has been considered further recently in . Many new fixed point theorems are proved in a metric space endowed with partial order by using monotone iterative technique, and their results are applied to problems of existence and uniqueness of solutions for some differential equation problems. In  the existence of a minimal and a maximal solution for a nonlinear problem is presented by constructing an iterative sequence with the condition of a lower solution or an upper solution.

In this paper, the theoretical results of fixed points are extended by using the theorem of cone and monotone iterative technique in Banach lattices. But the iterative sequences can be constructed with no need for using the condition of a lower solution or an upper solution. To demonstrate the applicability of our results, we apply them to study a problem of ordinary differential equations in the final section of the paper, and the existence and uniqueness of solution are obtained.

Let E be a Banach space and P a cone of E. We define a partial ordering with respect to P by xy if and only if y-xP. A cone PE is called normal if there is a constant N>0, such that θxy implies xNy, for all x,yE. The least positive constant N satisfying the above inequality is called the normal constant of P.

Let E be a Riesz space equipped with a Riesz norm. We call E a Banach lattice in the partial ordering , if E is norm complete. For arbitrary x,yE, sup{x,y} and inf{x,y} exist. One can see  for the definition and the properties about the lattice.

Let DE; the operator A:DE is said to be an increasing operator if x,yD, xy, implies AxAy; the operator A:DE is said to be a decreasing operator if x,yD, xy, implies AyAx.

Lemma 1.1 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Let P be a normal cone in a real Banach space E. Suppose that {xn} is a monotone sequence which has a subsequence {xni} converging to x*, then {xn} also converges to x*. Moreover, if {xn} is an increasing sequence, then xnx*    (n=1,2,3,); if {xn} is a decreasing sequence, then x*xn(n=1,2,3,).

Lemma 1.2 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let Ω be a bounded open set in a real Banach space E such that θΩ; let P be a cone of E. Let A:PΩ¯P is completely continuous. Suppose that (1.1)xAx,xPΩ¯. Then i(A,PΩ,P)=1.

Lemma 1.3 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let E be a real Banach space, and let PE be a cone. Assume Ω1 and Ω2 are two bounded open subsets of E with θΩ1Ω2 and Ω1¯Ω2, and let A:P(Ω2¯Ω1)P is completely continuous. Suppose that either

xAx,for all  xPΩ1¯ and Axx,for all  xPΩ2¯, or

Axx,for all  xPΩ1¯ and xAx,for all  xPΩ2¯.

Then A has a fixed point in P(Ω2Ω1¯).

2. Main Results Theorem 2.1.

Let E be a real Banach lattice, and let PE be a normal cone. Suppose that A:EE is a decreasing operator such that there exists a linear operator L:EE with spectral radius r(L)<1 and (2.1)Av-AuL(u-v),for  u,vE  with  vu. Then the operator A has a unique fixed point.

Proof.

For any u0E, since A:EE, we have Au0E. Now we suppose the following two cases.

Case (I). Suppose that u0 is comparable to Au0. Firstly, without loss of generality, suppose that u0Au0. If Au0=u0, then the proof is finished. Suppose Au0u0. Since A is decreasing together with u0Au0, we obtain by induction that {An+1(u0)} and {An(u0)} are comparable, for every n=0,1,2,. Using the contractive condition (2.1), we can obtain by induction that (2.2)An+1(u0)-An(u0)NLn(Au0-u0),nN. In fact, for n=1, using the fact that P is normal, we have (2.3)A(u0)-A2u0NL(Au0-u0). Suppose that (2.2) is true when n=k then when n=k+1, we obtain (2.4)An+2(u0)-An+1(u0)=A(An+1(u0))-A(An(u0))NL(An+1(u0)-An(u0))NLn+1(Au0-u0). For any m,nN, m>n, since P is normal cone, we have (2.5)Am(u0)-An(u0)=(Am(u0)-Am-1(u0))++(An+1(u0)-An(u0))N(Lm-1+Lm-2++Ln)(Au0-u0)Nr((Lm-1+Lm-2++Ln))Au0-u0N(r(Lm-1)+r(Lm-2)++r(Ln))Au0-u0. Here N is the normal constant.

Given a α such that r(L)<α<1, since limn+Ln1/n=r(L)<α<1, there exists a n0N such that (2.6)Ln<αn,nn0. For any m,nN, m>nn0, since P is normal cone, we have (2.7)Am(u0)-An(u0)N(r(Lm-1)+r(Lm-2)++r(Ln))Au0-u0N(αm-1+αm-2++αn)Au0-u0N(αn-αm1-α)Au0-u0N(αn1-α)Au0-u0. This implies that {An(u0)} is a Cauchy sequence in E. The complete character of E implies the existence of x*P such that (2.8)limn+An(u0)=x*. Next, we prove that x* is a fixed point of A in E. Since A is decreasing and u0Au0, we can get A2u0Au0.

So (2.9)Au0-A2(u0)L(Au0-u0), then (2.10)A2u0-u0=(Au0-u0)-(Au0-A2(u0))(I-L)(Au0-u0)θ. It is easy to know that A2 is increasing and (2.11)A2(u0)A4(u0),A3(u0)A(u0). By induction, we obtain that (2.12)u0A2(u0)A2n(u0)A2n+1(u0)A3(u0)Au0. Hence, the sequence {An(u0)} has an increasing Cauchy subsequence {A2n(u0)} and a decreasing Cauchy subsequence {A2n+1(u0)} such that (2.13)limn+A2n(u0)=u*,limn+A2n+1(u0)=v*. Thus Lemma 1.1 implies that A2n(u0)u*, v*A2n+1(u0).

Since {An(u0)} is a Cauchy sequence, we can get that u*=v*=x*.

Moreover (2.14)Ax*-x*Ax*-A(A2n(u0))+A2(n+1)(u0)-x*NL(x*-A2n(u0))+A2(n+1)(u0)-x*Nαx*-A2n(u0)+A2(n+1)(u0)-x*. Thus Ax*-x*=0. That is Ax*=x*. Hence x* is a fixed point of A in E.

Case (II). On the contrary, suppose that u0 is not comparable to Au0.

Now, since E is a Banach lattice, there exists v0 such that inf{Au0,u0}=v0. That is v0Au0 and v0u0. Since A is a decreasing operator, we have (2.15)A2u0Av0,Au0Av0. This shows that v0Av0. Similarly as the proof of case (I), we can get that A has a fixed point x* in E.

Finally, we prove that A has a unique fixed point x* in E. In fact, let u* and v* be two fixed points of A in E.

If u* is comparable to v*, An(u*)=u* is comparable to An(v*)=v* for every n=0,1,2,, and (2.16)u*-v*=Anu*-Anv*Nαnu*-v*, which implies u*=v*.

If u* is not comparable to v*, there exists either an upper or a lower bound of u* and v* because E is a Banach lattice, that is, there exists z*E such that z*u*,z*v* or u*z*,u*z*. Monotonicity implies that An(z*) is comparable to An(u*) and An(v*), for all n=0,1,2,, and (2.17)u*-v*=An(u*)-An(v*)An(z*)-An(u*)+An(z*)-An(v*)Nαnu*-z*+Nαnz*-v*.

This shows that u*-v*0 when n+. Hence A has a unique fixed point x* in E.

Theorem 2.2.

Let E be a real Banach lattice, and let PE be a normal cone. Suppose that A:PP is a completely continuous and increasing operator such that there exists a linear operator L:EE with spectral radius r(L)<1 and (2.18)Au-AvL(u-v),for  u,vP  with  vu. Then the operator A has a unique fixed point u* in P.

Proof.

For any r>0, let Ω={xP:xr}. Now we suppose the following two cases.

Case (I). Firstly, suppose that there exists u0Ω such that u0Au0. If Au0=u0, then the proof is finished. Suppose Au0u0. Since u0Au0 and A is nondecreasing, we obtain by induction that (2.19)u0Au0A2(u0)A3(u0)An(u0)An+1(u0). Similarly as the proof of Theorem 2.1, we can get that {An(u0)} is a Cauchy sequence in E. Since E is complete, by Lemma 1.1, there exists u*E,An(u0)u* such that (2.20)limn+An(u0)=u*. Next, we prove that u* is a fixed point of A, that is, Au*=u*. In fact (2.21)Au*-u*Au*-A(An(u0))+An+1(u0)-u*NL(u*-An(u0))+An+1(u0)-u*Nαu*-An(u0)+An+1(u0)-u*. Now, by the convergence of {An(u0)} to u*, we can get Au*-u*=0. This proves that u* is a fixed point of A.

Case (II). On the contrary, suppose that xAx for all xΩ. Thus Lemma 1.2 implies the existence of a fixed point in this case also.

Finally, similarly as the proof of Theorem 2.1, we can get that A has a unique fixed point x* in P.

Theorem 2.3.

Let E be a real Banach lattice, and let PE be a normal cone. Suppose that A:PP is a completely continuous and increasing operator which satisfies the following assumptions:

there exists a linear operator L:EE with spectral radius r(L)<1 and (2.22)Au-AvL(u-v),for  u,vP  with  vu;

S={xP:Axx} is bounded.

Then the operator A has a unique nonzero fixed point u* in P.

Proof.

Firstly, for any r>0, let Ω={xP:xr}. Now we suppose the following two cases.

Case (I). Suppose that there exists u0Ω such that u0Au0. Similarly as proof of Theorem 2.1, we get that A has a nonzero fixed point u* in P.

Case (II). On the contrary, suppose that xAx for all xΩ. Now, since S is bounded there exists R>r such that Axx for all xP with x=R. Thus Lemma 1.3 implies the existence of a nonzero fixed point in this case.

Finally, similarly as the proof of Theorem 2.1, we can get that A has a unique non-zero fixed point u* in P.

3. Applications

In this section, we use Theorem 2.1 to show the existence of unique solution for the first-order initial value problem (3.1)u'(t)=f(t,u(t)),tI=[0,T],u(0)=u0, where T>0 and f:I×RR is a continuous function.

Theorem 3.1.

Let f:I×RR be continuous, and suppose that there exists 0<μ<λ, such that (3.2)-μ(y-x)f(t,y)+λy-[f(t,x)+λx]0,yx. Then (3.1) has a unique solution u*.

Proof.

It is easy to know that E=C(I) is a Banach space with maximum norm ·, and it is also a Banach lattice with maximum norm ·. Let P={uE|u(t)0,for  all  tI}, and P is a normal cone in Banach lattice E. Equation (3.1) can be written as (3.3)u'(t)+λu(t)=f(t,u(t))+λu(t),tI=[0,T],u(0)=u0. This problem is equivalent to the integral equation (3.4)u(t)=e-λt{u0+0teλs[f(s,u(s))+λu(s)]ds}. Define operator A as the following: (3.5)(Au)(t)=e-λt{u0+0teλs[f(s,u(s))+λu(s)]ds},tI. Moreover, the mapping A is decreasing in u. In fact, by hypotheses, for uv, (3.6)f(t,u(t))+λu(t)f(t,v(t))+λv(t) implies that (3.7)(Au)(t)=e-λt{u0+0teλs[f(s,u(s))+λu(s)]ds}e-λt{u0+0teλs[f(s,v(s))+λv(s)]ds}=(Av)(t),tI, so A is decreasing. Besides, for uv, (3.8)A(v)-A(u)=0teλ(s-t)[f(s,v(s)+λv(s)-f(s,u(s))-λu(s)]  ds0teλ(s-t)μ[u(s)-v(s)]ds=L(u-v), where Lu=0teλ(s-t)μu(s)ds. Since A is decreasing, then L is positive linear operator.

Now, let us prove that the spectral radius r(L)<1. For tI, since 0<eλ(s-t)1, we have (3.9)Lu=maxtI0teλ(s-t)μu(s)dsμ0teλ(s-t)dsuμtu,L2u=maxtI0teλ(s-t)μL(u(s))dsμ20teλ(s-t)sdsLuμ22!t2u. By mathematical induction, for any nN, we have (3.10)Lnuμnn!tnu,tI. So (3.11)Lnμnn!Tn. Since 0<μ<λ, we have (3.12)r(L)=limn+Ln1/n=0<1. So the condition of Theorem 2.1 holds, and Theorem 3.1 is proved.

Acknowledgments

The first author was supported financially by the NSFC (71240007), NSFSP (ZR2010AM005).

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