Research Article | Open Access
Existence and Uniqueness Theorems of Ordered Contractive Map in Banach Lattices
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  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 be a Banach space and a cone of . We define a partial ordering with respect to by if and only if . A cone is called normal if there is a constant , such that implies , for all . The least positive constant satisfying the above inequality is called the normal constant of .
Let be a Riesz space equipped with a Riesz norm. We call a Banach lattice in the partial ordering , if is norm complete. For arbitrary , and exist. One can see  for the definition and the properties about the lattice.
Let ; the operator is said to be an increasing operator if , , implies ; the operator is said to be a decreasing operator if , , implies .
Lemma 1.1 (see ). Let be a normal cone in a real Banach space . Suppose that is a monotone sequence which has a subsequence converging to , then also converges to . Moreover, if is an increasing sequence, then ; if is a decreasing sequence, then .
Lemma 1.2 (see ). Let be a bounded open set in a real Banach space such that ; let be a cone of . Let is completely continuous. Suppose that Then .
Lemma 1.3 (see ). Let be a real Banach space, and let be a cone. Assume and are two bounded open subsets of with and , and let is completely continuous. Suppose that either and , or and .Then has a fixed point in .
2. Main Results
Theorem 2.1. Let be a real Banach lattice, and let be a normal cone. Suppose that is a decreasing operator such that there exists a linear operator with spectral radius and Then the operator has a unique fixed point.
Proof. For any , since , we have . Now we suppose the following two cases.
Case (I). Suppose that is comparable to . Firstly, without loss of generality, suppose that . If , then the proof is finished. Suppose . Since is decreasing together with , we obtain by induction that and are comparable, for every . Using the contractive condition (2.1), we can obtain by induction that In fact, for , using the fact that is normal, we have Suppose that (2.2) is true when then when , we obtain For any , , since is normal cone, we have Here is the normal constant.
Given a such that , since , there exists a such that For any , , since is normal cone, we have This implies that is a Cauchy sequence in . The complete character of implies the existence of such that Next, we prove that is a fixed point of in . Since is decreasing and , we can get .
So then It is easy to know that is increasing and By induction, we obtain that Hence, the sequence has an increasing Cauchy subsequence and a decreasing Cauchy subsequence such that Thus Lemma 1.1 implies that , .
Since is a Cauchy sequence, we can get that .
Moreover Thus . That is . Hence is a fixed point of in .
Case (II). On the contrary, suppose that is not comparable to .
Now, since is a Banach lattice, there exists such that . That is and . Since is a decreasing operator, we have This shows that . Similarly as the proof of case (I), we can get that has a fixed point in .
Finally, we prove that has a unique fixed point in . In fact, let and be two fixed points of in .(1)If is comparable to , is comparable to for every , and which implies .(2)If is not comparable to , there exists either an upper or a lower bound of and because is a Banach lattice, that is, there exists such that or . Monotonicity implies that is comparable to and , for all , and This shows that when . Hence has a unique fixed point in .
Theorem 2.2. Let be a real Banach lattice, and let be a normal cone. Suppose that is a completely continuous and increasing operator such that there exists a linear operator with spectral radius and Then the operator has a unique fixed point in .
Proof. For any , let . Now we suppose the following two cases.
Case (I). Firstly, suppose that there exists such that . If , then the proof is finished. Suppose . Since and is nondecreasing, we obtain by induction that Similarly as the proof of Theorem 2.1, we can get that is a Cauchy sequence in . Since is complete, by Lemma 1.1, there exists such that Next, we prove that is a fixed point of , that is, . In fact Now, by the convergence of to , we can get . This proves that is a fixed point of .
Case (II). On the contrary, suppose that for all . 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 has a unique fixed point in .
Theorem 2.3. Let be a real Banach lattice, and let be a normal cone. Suppose that is a completely continuous and increasing operator which satisfies the following assumptions:(i)there exists a linear operator with spectral radius and
(ii) is bounded.
Then the operator has a unique nonzero fixed point in .
Proof. Firstly, for any , let . Now we suppose the following two cases.
Case (I). Suppose that there exists such that . Similarly as proof of Theorem 2.1, we get that has a nonzero fixed point in .
Case (II). On the contrary, suppose that for all . Now, since is bounded there exists such that for all with . 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 has a unique non-zero fixed point in .
In this section, we use Theorem 2.1 to show the existence of unique solution for the first-order initial value problem where and is a continuous function.
Theorem 3.1. Let be continuous, and suppose that there exists , such that Then (3.1) has a unique solution .
Proof. It is easy to know that is a Banach space with maximum norm , and it is also a Banach lattice with maximum norm . Let , and is a normal cone in Banach lattice . Equation (3.1) can be written as
This problem is equivalent to the integral equation
Define operator as the following:
Moreover, the mapping is decreasing in . In fact, by hypotheses, for ,
so is decreasing. Besides, for ,
where . Since is decreasing, then is positive linear operator.
Now, let us prove that the spectral radius . For , since , we have By mathematical induction, for any , we have So Since , we have So the condition of Theorem 2.1 holds, and Theorem 3.1 is proved.
The first author was supported financially by the NSFC (71240007), NSFSP (ZR2010AM005).
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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.