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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 107276, 18 pages
Existence of Positive Solutions for Multi-Point Boundary Value Problems on Infinite Intervals in Banach Spaces
School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
Received 18 July 2012; Accepted 12 September 2012
Academic Editor: Xinguang Zhang
Copyright © 2012 Zhaocai Hao and Liang Ma. 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.
We investigate the positive solutions of a class of second-order nonlinear singular differential equations with multi-point boundary value conditions on an infinite interval in Banach spaces. The tools we used are the cone theory and Mönch fixed point theorem and a monotone iterative technique. An example is also given to demonstrate the applications of our results, which include and extend some existing results.
Let be a real Banach space and let be a cone in . is said to be normal if there exists a positive constant such that implies . is said to be regular (fully regular) if , implies that their exists such that . Let . In what follows, we always assume that . Let . Obviously, for any . When , we write , that is, .
In this paper, we will consider the following boundary value problems (BVPs) for multipoint singular differential equations of mixed type on an unbounded domain in a real Banach space ( where , , , , , ,, . Here the nonlinear term may be singular at and . By singularity, we mean that as or .
Second-order boundary value problems (BVPs) on infinite intervals, arising from the study of radially symmetric solutions of nonlinear elliptic equation and models of gas pressure in a semi-infinite porous medium, have received much attention. We can see papers [1, 2] and the references therein. In a recent paper, Liu  investigated the existence of solutions of the following second-order two-point boundary value problems on the half-line: Lian and Ge  studied the solvability of the three-point BVP where . With the help of the established Green function and the Leray-Schauder continuation theorem, suitable conditions imposed on are presented for the existence of solutions. Yan et al.  established the results of existence and multiplicity of positive solutions to the BVP on the half-line by using the lower and upper solutions technique. Zhang  researched the problem by using the fixed point theorem and the monotone iterative technique.
We note that these works [3–6] are all in real space. To the best of our knowledge, very few literatures are available for the computation of positive solutions for mutlipoint BVP on the half-line in Banach space. There are two papers we should present here. Liu  discussed the existence of solutions of the following second-order two-point BVP on infinite intervals in a Banach space : where . The main tool is the Sadovskii’s fixed point theorem. Zhang  concerned the existence of solutions of the following singular problems where . The nonlinear term may be singular at and/or . They used the Mönch fixed point theorem.
Motivated by the above papers, we also use the Mönch fixed point theorem to give the existence of a positive solutions of the more general BVP (1.1) for integrodifferential equations on infinite intervals in a Banach spaces. The main features of the present paper are as follows. Firstly, comparing with [3–6], the space in this paper is Banach space. The equation we discussed here is more general than those of [3–8] because the function of (1.1) has new terms , and the boundary value conditions are more complicated. Moreover, the singularity of nonlinear term in this paper is more complex than [2, 7, 9–11]. Furthermore, an iterative sequence for the solution under some normal type conditions is established which makes it very important and convenient in applications. [3–7, 9–11] did not obtain this kind of result.
The rest of this paper is organized as follows. In Section 2, we give several important Lemmas. The main theorems are formulated and proved in Section 3, followed by an example in Section 4 to demonstrate the application of our results.
2. The Preliminary and Several Lemmas
Let Evidently, . It is easy to see that is a Banach space with norm and is also a Banach space with norm , where .
Some conditions to be used throughout the rest of the paper are listed below. In this case, let , for any and there exist and such that uniformly for , and For any , and countable sets , there exist such that where and . imply
In what follows, we write Evidently, is a closed convex set in . We will reduce the BVP (1.1) to an integral equation in . To this end, we first consider the operator defined by where
Lemma 2.2. If conditions and hold, then the operator defined by (2.11) is a continuous operator from into .
Proof. Firstly, we will show . Let
By virtue of condition , there exists a such that
Let . We have by (2.16) and Lemma 2.1 which together with condition implies the convergence of the infinite integral Thus, we have (2.11), (2.19), and tell us that Therefore, Differentiating (2.11), we get Hence, It follows from (2.21) and (2.23) that So, . On the other hand, it can be easily seen that So, . Thus, maps into and (2.24) holds.
Secondly, we will show that is continuous. Let . Then and . Similar to (2.21) and (2.23), it is easy to say So we get Then we know from (2.17) that It follows from (2.27) and (2.28) and the dominated convergence theorem that It follows from (2.26) and (2.29) that as , and the continuity of is proved.
Lemma 2.3. Let conditions and be satisfied. Then is a solution of the BVP (1.1) if and only if is a solution of the following integral equation:
Proof. Integrating the differential equation in (1.1) from to , one has
Then, integrating (2.31) from 0 to , we have
By Lemma 2.2, we know that is convergent. Since , we can compute those coefficients , , and then obtain (2.30).
Conversely, if is a solution of integral equation (2.30), then direct differentiation gives the proof.
Lemma 2.4 (Mönch Fixed Point Theorem ). Let be a closed convex set of and . Assume that the continuous operator has the following property: is relatively compact. Then has a fixed point in .
Lemma 2.5. Let be a bounded set in . Suppose that is equicontinuous on any finite subinterval of and as uniformly for . Then where and denote the Kuratowski measure of noncompactness of bounded sets in and , respectively.
Proof. The proof is similar to [11, Lemma 7], we omit it.
Lemma 2.6. If condition is satisfied, then imply that .
3. Main Results
In the following, we will give the main results of this paper.
Theorem 3.1. Assume that hold, then the BVP (1.1) has a positive solution satisfying for .
Proof. By Lemma 2.2, the operator defined by (2.11) is a continuous operator from into . And, by Lemma 2.3, we only need to show that has a fixed point in .
Choose and let . Obviously, is a bounded closed convex set in the space . It is easy to see that is not empty since . It follows from (2.24) and (3.1) that implies , that is, maps into .
Now, we are in position to show that is relatively compact. Let satisfying for some . Then . So we have by (2.11) and (2.22) that So for any , we have that (3.3) implies that is equicontinuous on any finite subinterval of .
On the other hand, by (2.17), (2.20), (3.2) we can obtain which implies that as uniformly for . Hence, by Lemma 2.5 we have where , and .
It follows from (2.18) that the infinite integral , is convergence uniformly for . So, for any , we can choose a large such that holds for any . Then, by [6, Theoerm ], (3.6), and , we obtain By (3.7) and noting the fact that is arbitrary, we see that On the other hand, . Then , that is, is relatively compact in .
Hence, the Mönch fixed point theorem implies that has a fixed point in and this theorem is proved.
Theorem 3.2. Let cone be normal and let conditions – be satisfied. Then the BVP (1.1) has a positive solution which is minimal in the sense that for any positive solution . Moreover, , where and there exists a monotone iterative sequence such that as uniformly on and as for any , where
Proof. From (3.10), we can find that and
By (3.10) and (3.12), we have and
(2.18) implies that as . Hence, , , which implies . From (2.11) and (3.11) we get
By Lemma 2.2, we get and
By Lemma 2.6 and (3.15), we have
It follows from (3.15), by induction, that
Let . So, is a bounded closed convex set in the space and operator maps into . Clearly, is not empty since .
Let and . Obviously, and . Similarly, as the proof in Theorem 3.1, we can obtain , that is, is relatively compact in . So there exist a and a subsequence such that converges to uniformly on . Since that is normal and is nondecreasing on account of (3.16), it is easy to see that the entire sequence converges to uniformly on . By and is a closed convex set in space , we have . Let be arbitrarily fixed. It is clear that and, by (3.17), we have It follows from (3.18) and (3.19) and the dominated convergence theorem that which implies that , as for any . Hence, By virtue of (3.17) and Lemma 2.1, we have Now, noting (3.21) and (3.22) and taking limit as in (3.11), we obtain which follows from Lemma 2.3 that and is positive solution of the BVP (1.1). Differentiating (3.11) twice, we get Hence, by (3.21), we obtain
Let be any other positive solution of the BVP (1.1). By Lemma 2.3, we have and , for . It is clear that for any . So, by Lemma 2.6, we can have for any . Assume that for any . Then, it follows from Lemma 2.6 that for any , that is, for any . Hence, by induction, we get Now, taking limits as in (3.26), we get for , and this completes the proof.
Theorem 3.3. Let cone be fully regular and let conditions be satisfied. Then the conclusion of Theorem 3.2 holds also.
Proof. The proof is almost the same as that of Theorem 3.2. The only difference is that, instead of using condition , the conclusion was implied directly by (3.16) and (3.17) and full regularity of and Lemma 2.5.
4. An Example
Example 4.1. Consider the following infinite system of scalar second-order multipoint singular integrodifferential equation:
Conclusion. Infinite system (4.2) has a minimal positive solution satisfying for , and this minimal solution can be obtained by taking limits from some iterative sequences.
Proof. Let with the norm . Choose . It is easy to see that is weakly sequence complete, and is a normal cone in . Thus, is fully regular.
Now we consider the infinite system (4.2), which can be regarded as the BVP (1.1) with , , , , . So we have In this situation, in which Let . Then for any and the condition holds for and . It is clear that for any . By (4.4) we get So, the condition is satisfied for , and It is easy to see that holds. Thus, our conclusion follows from Theorem 3.3 immediately.