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 [1–6]. 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 [6] 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 x≤y if and only if y-x∈P. A cone P⊂E is called normal if there is a constant N>0, such that θ≤x≤y implies ∥x∥≤N∥y∥, for all x,y∈E. 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,y∈E, sup{x,y} and inf{x,y} exist. One can see [7] for the definition and the properties about the lattice.

Let D⊂E; the operator A:D→E is said to be an increasing operator if x,y∈D, x≤y, implies Ax≤Ay; the operator A:D→E is said to be a decreasing operator if x,y∈D, x≤y, implies Ay≤Ax.

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 xn≤x*(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 θ∈Ω; let P be a cone of E. Let A:P∩Ω¯→P is completely continuous. Suppose that
(1.1)x≰Ax,∀x∈P∩Ω¯.
Then i(A,P∩Ω,P)=1.

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

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

x≰Ax,for allx∈P∩Ω1¯ and Ax≰x,for allx∈P∩Ω2¯, or

Ax≰x,for allx∈P∩Ω1¯ and x≰Ax,for allx∈P∩Ω2¯.

Then A has a fixed point in P∩(Ω2∖Ω1¯).
2. Main ResultsTheorem 2.1.

Let E be a real Banach lattice, and let P⊂E be a normal cone. Suppose that A:E→E is a decreasing operator such that there exists a linear operator L:E→E with spectral radius r(L)<1 and
(2.1)Av-Au≤L(u-v),foru,v∈Ewithv≤u.
Then the operator A has a unique fixed point.

Proof.

For any u0∈E, since A:E→E, we have Au0∈E. Now we suppose the following two cases.

Case (I). Suppose that u0 is comparable to Au0. Firstly, without loss of generality, suppose that u0≤Au0. If Au0=u0, then the proof is finished. Suppose Au0≠u0. Since A is decreasing together with u0≤Au0, 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)∥≤N∥Ln(Au0-u0)∥,n∈N.
In fact, for n=1, using the fact that P is normal, we have
(2.3)∥A(u0)-A2u0∥≤N∥L(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))∥≤N∥L(An+1(u0)-An(u0))∥≤N∥Ln+1(Au0-u0)∥.
For any m,n∈N, 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-u0∥≤N(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→+∞∥Ln∥1/n=r(L)<α<1, there exists a n0∈N such that
(2.6)∥Ln∥<αn,n≥n0.
For any m,n∈N, m>n≥n0, since P is normal cone, we have
(2.7)∥Am(u0)-An(u0)∥≤N(r(Lm-1)+r(Lm-2)+⋯+r(Ln))∥Au0-u0∥≤N(αm-1+αm-2+⋯+αn)∥Au0-u0∥≤N(αn-αm1-α)∥Au0-u0∥≤N(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 u0≤Au0, we can get A2u0≤Au0.

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)u0≤A2(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*∥≤N∥L(x*-A2n(u0))∥+∥A2(n+1)(u0)-x*∥≤Nα∥x*-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 v0≤Au0 and v0≤u0. Since A is a decreasing operator, we have
(2.15)A2u0≤Av0,Au0≤Av0.
This shows that v0≤Av0. 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*∥≤Nαn∥u*-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*)∥≤Nαn∥u*-z*∥+Nαn∥z*-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 P⊂E be a normal cone. Suppose that A:P→P is a completely continuous and increasing operator such that there exists a linear operator L:E→E with spectral radius r(L)<1 and
(2.18)Au-Av≤L(u-v),foru,v∈Pwithv≤u.
Then the operator A has a unique fixed point u* in P.

Proof.

For any r>0, let Ω={x∈P:∥x∥≤r}. Now we suppose the following two cases.

Case (I). Firstly, suppose that there exists u0∈∂Ω such that u0≤Au0. If Au0=u0, then the proof is finished. Suppose Au0≠u0. Since u0≤Au0 and A is nondecreasing, we obtain by induction that
(2.19)u0≤Au0≤A2(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*∥≤N∥L(u*-An(u0))∥+∥An+1(u0)-u*∥≤Nα∥u*-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 x≰Ax for all x∈∂Ω. 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 P⊂E be a normal cone. Suppose that A:P→P is a completely continuous and increasing operator which satisfies the following assumptions:

there exists a linear operator L:E→E with spectral radius r(L)<1 and
(2.22)Au-Av≤L(u-v),foru,v∈Pwithv≤u;

S={x∈P:Ax≤x} is bounded.

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

Proof.

Firstly, for any r>0, let Ω={x∈P:∥x∥≤r}. Now we suppose the following two cases.

Case (I). Suppose that there exists u0∈∂Ω such that u0≤Au0. 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 x≰Ax for all x∈∂Ω. Now, since S is bounded there exists R>r such that Ax≰x for all x∈P 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)),t∈I=[0,T],u(0)=u0,
where T>0 and f:I×R→R is a continuous function.

Theorem 3.1.

Let f:I×R→R 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,∀y≥x.
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={u∈E|u(t)≥0,forallt∈I}, 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),t∈I=[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},t∈I.
Moreover, the mapping A is decreasing in u. In fact, by hypotheses, for u≥v,
(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),t∈I,
so A is decreasing. Besides, for u≥v,
(3.8)A(v)-A(u)=∫0teλ(s-t)[f(s,v(s)+λv(s)-f(s,u(s))-λu(s)]ds≤∫0teλ(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 t∈I, since 0<eλ(s-t)≤1, we have
(3.9)∥Lu∥=maxt∈I∫0teλ(s-t)μu(s)ds≤μ∫0teλ(s-t)ds∥u∥≤μt∥u∥,∥L2u∥=maxt∈I∫0teλ(s-t)μL(u(s))ds≤μ2∫0teλ(s-t)sds∥Lu∥≤μ22!t2∥u∥.
By mathematical induction, for any n∈N, we have
(3.10)∥Lnu∥≤μnn!tn∥u∥,t∈I.
So
(3.11)∥Ln∥≤μnn!Tn.
Since 0<μ<λ, we have
(3.12)r(L)=limn→+∞∥Ln∥1/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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