Abstract
We study the existence and uniqueness of positive solutions for a class of singular m-point boundary value problems of second order differential equations on a measure chain. A sharper sufficient condition for the existence and uniqueness of positive solutions as well as positive solutions is obtained by the technique of lower and upper solutions and the maximal principle theorem.
1. Introduction
In this paper, we present the existence of positive solutions for the following second order singular -point boundary value problem on a measure chain: where , , are constants and , . We assume that belong to . We use to denote , and similar notations are used for other intervals. The function is rd-continuous, is continuous, and may be singular at and , . Observe that when or , the problem (1)-(2) reduces to boundary value problems of ordinary differential equations or difference equations.
The existence of positive solutions for boundary value problem on a measure chain has been paid more attentions by many researchers. Related problems on measure chains can be found in [1–9].
Recently, by making use of the Krasnosel’skii fixed point theorem, Goodrich [1] studied the existence of at least one positive solution for the following boundary value problem: where the nonlocal boundary condition is continuous.
Sun and Li [5] gave the existence results of the following three-point boundary value problem on time scale : where , , , , and .
By applying functional-type cone expansion-compression fixed point theorem, P. Wang and Y. Wang [6] established the existence of positive solutions of the following nonlinear boundary value problem given by the dynamic equation on time scales:
By employing the Krasnosel’skii fixed point theorem, Hao et al. [2] discussed the existence of positive solutions of the following boundary value problem on a time scale:
Inspired and motivated greatly by the work of [1, 2, 5–9], we establish the existence and uniqueness of positive solution for singular -point boundary value problem (1)-(2). By constructing lower and upper solutions and using the maximal theorem, we not only obtain a sharper sufficient condition for the existence and uniqueness of positive solution, but also prove a sufficient condition for the existence of positive solution. Our technique is different from those of [1–9] and our results naturally complement/improve their work.
We state some basic notions connected to time scales, which can be found in [9].
Definition 1. Let be a time scale. For , the forward jump operator is defined by and one defines the backward jump operator by
Definition 2. One says that is right-scattered if , and one says that is left-scattered if . Points are said to be isolated if they are both right-scattered and left-scattered.
Definition 3. One says that is right-dense if and , and one says that is left-dense if and . Points are said to be dense if they are both right-dense and left-dense. One defines the graininess function by
For convenience, one lists the following conditions which will be referred to later:(H1), , are constants satisfying , , and ;(H2) is rd-continuous, and there exists such that ;(H3) is continuous, and is nonincreasing with respect to , for all , , and , for all .
2. Preliminaries and Lemmas
Definition 4. A function is said to be a positive solution of the problem (1)-(2) if satisfies the problem (1)-(2) and , ; a positive solution of the problem (1)-(2) is said to be a positive solution if and exist, and , .
Definition 5. One says that a function is a lower solution of the problem (1)-(2) on , if and satisfies Similarly, is said to be an upper solution of the problem (1)-(2) on , if and satisfies One says is a couple of lower and upper solutions of the problem (1)-(2), if there exist a lower solution and an upper solution of the problem (1)-(2) such that
Lemma 6 (maximal principle). Suppose that (H1) is satisfied. In addition, assume that , . Let For such that and for , then for .
Proof. For all , let
then , , and , .
Integrating (14) from to , we obtain
Again integrating (16) from to and exchanging integral sequence, we get
From (17) and boundary condition (15), we obtain
where and
Consequently, from (18) and the definition of , we see that , .
Let We will use the Banach space equipped with the norm. Let where . Then is a positive cone of . Define the nonlinear operator as follows: which for notational simplicity will be written as where with and
It is easy to see that
It is clear that the existence of a positive solution of (1)-(2) is equivalent to the existence of a nontrivial fixed point of in .
Lemma 7. Suppose that (H1)–(H3) hold. Then , and is a decreasing operator.
Proof. It is obvious that , so is not empty: . For all , by the definition of , there exists a real number , such that , . From , we know that
From the definition of and (23), we find
Let . From , we see . Since is continuous on , thus
On the other hand, for all , by making use of (25), we obtain
where
Therefore is well defined on , and for all . So .
For all with , from , we see that
Therefore, and hence is a decreasing operator.
Lemma 8. Suppose that (H1)–(H3) hold. Then for any , the problem (1)-(2) has an upper solution and a lower solution , and is a couple of lower and upper solution of the problem (1)-(2).
Proof. For all , we know that By simple computation, we obtain Let Obviously , are well defined, and Since , then for any , we see that . Thus there exists positive real number , such that . From (35) and (36), we know that where . Therefore . Thus and are well defined, and Since is a nonincreasing operator, from (35), we know that From (34) and the above discussion, we know that Equation (33) implies and satisfy conditions (2). From (38) and (40), we know that is a couple of lower and upper solution of the problem (1)-(2), and . Therefore Consequently
3. Main Results
Theorem 9. Suppose that (H1)–(H3) hold. Then the problem (1)-(2) has a unique positive solution satisfying , where is a positive constant.
Proof of Theorem 9. We have the following.
(I) Existence of Positive Solution to the Problem (1)-(2). From Lemma 8, we know that the problem (1)-(2) has a couple of lower and upper solution. Let be a couple of lower and upper solution of the problem (1)-(2). Then for any , we have that , , , are strictly positive continuous function.
Define auxiliary function and operator as follows:
Obviously, we see that is continuous.
Consider the following second order differential equation -point singular boundary value problem:
It is well known that the existence of a positive solution of problem (46) is equivalent to the existence of a nontrivial fixed point of in .
Now we prove that is a completely continuous operator.
From , there exists a positive real number such that , for all . Combining (H2)~(H3) with (44) and (42), we know that
is definitely a finite real number. Denote
Let constant , for all . Thus, we obtain
Consequently, is uniformly bounded.
Now we prove that is compact. Let be a bounded set. Then there exists such that , for all . It is easy to prove that is a bounded set in . Since is continuous on , thus it is uniformly continuous. Therefore, choose , for all , and such that ; we have . Thus, for any such that , we obtain
Therefore is equicontinuous on . Then by making use of Arzela-Ascoli theorem [8] on time scale, we know that is relatively compact. Consequently is compact.
On the other hand, let . Denote . Then for any , from (49), we know that there exist such that
For , since is continuous on , consequently uniformly continuous. Thus there exists satisfying , for any , such that ; we have
Thus, from (51) and (52), for all such that , we obtain
which implies is continuous on . Consequently is a completely continuous operator.
From Schauder fixed point theorem [7] on time scale, we know that has at least one positive fixed point in , and satisfies . Thus satisfies the following differential equation -point boundary value problem:
Now we will prove that , for all . First, we will prove that , for all . In fact, if not, then there exists such that
Let , for all . Denote
Then , for all . Thus from (44), we see that . Combining (43) with (44), we get
In view of the above discussion and (54), we have . For above , there are two cases: (1) ; (2) , .
If (1) holds, then , for all which contradicts (55).
If (2) holds, from , we know that , . With the aid of increasing on it follows that , , that is, increasing on . From it follows that , which contradicts (55). Thus we have .
Similarly, we can verify that . Consequently is a positive solution of the problem (1)-(2).
(II) Unique Positive Solution of the Problem (1)-(2). Let , be two positive solutions of the problem (1)-(2), and . Without loss of generality, we assume that such that .
Let
and for all . It is easy to see that there are two cases for above , , : (i) ; (ii) , ;
From , we have , . Thus, for all .
Case (i). From and , we obtain , , which contradicts .
Case (ii). From , we know that . With the aid of increasing on , we get that , ; that is, is increasing on . From , we see that , which contradicts . Therefore the problem (1)-(2) has a unique positive solution.
Theorem 10. Suppose that (H1)–(H3) hold. In addition, one assumes that the following condition is satisfied: Then, the problem (1)-(2) has a unique positive solution , and there exist positive real numbers such that
Proof. By making use of Lemma 8, we know that the problem (1)-(2) has a couple of lower and upper solution. Applying Theorem 9 we see that the problem (1)-(2) has a unique positive solution . From , we know that is integrable on . Thus is integrable on . It follows from the fact that and exist that we see that is a positive solution of the problem (1)-(2). Clearly, may be expressed by
Thus, for any , we get
where .
On the other hand,
where . Therefore, from (62) and (63), we see that (60) holds.
Corollary 11. Suppose that (H1)–(H3) hold. In addition, assume that the following condition is satisfied: (H4), , for all .
Then the following problem
has a unique positive solution , and there exists a constant such that .
Corollary 12. Suppose that (H1)–(H3) hold. In addition, assume that the following condition is satisfied: (), for all .Then the problem (64) has a unique positive solution , and there exist positive constants such that .
If is nonsingular at or , , and for all , one has , , then the following conclusion holds.
Theorem 13. Suppose that , is continuous, , for all , and is nonincreasing with respect to , for all . Then the problem (1)-(2) has a unique positive solution.
Remark 14. Under some weaker condition, we not only establish the existence of positive solution of the problem (1)-(2), but also obtain the uniqueness of the positive solution.
Remark 15. Without the cavity of and other stronger conditions imposed on , only nonincreasing with respect to , we obtain new results. The main results hold even if the problem is nonsingular.
4. Examples
In this section, we will present two examples to illustrate the main result in this paper.
Example 1. Let . Consider the following boundary value problem:
Then, the four-point boundary value problem (65) has at least one positive solution.
Note that , for , is nonincreasing in , and is singular at and .
Obviously, and are satisfied. Moreover, for any fixed , follows immediately from
Thus, the existence of a positive solution follows from Theorem 9.
Example 2. Let . Consider the following boundary value problem:
Then, the four-point boundary value problem (65) has at least one positive solution.
Note that , for , is nonincreasing in , and is singular at and .
Obviously, and are satisfied. Moreover, for any fixed , (H3) follows immediately from
Thus, the existence of a positive solution follows from Theorem 9.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contribution
All authors contributed equally to this paper. They read and approved the final paper.
Acknowledgments
The authors are very grateful to referee for his/her valuable comments and suggestions. The first author was supported financially by the Foundation of Shanghai Natural Science (13ZR1430100) and the Foundation of Shanghai Municipal Education Commission (DYL201105) and (2013M541455). The second author was supported financially by the Natural Science Foundation of Zhejiang Province of China (Y12A01012).