Abstract
In this paper, using the theory of fixed point index on a cone and the Leray-Schauder fixed point theorem, we present the multiplicity of positive solutions for the singular nonlocal boundary-value problems involving nonlinear integral conditions and the existence of at least one positive solution for the singular nonlocal boundary-value problems with sign-changed nonlinearities.
MSC:34B10, 34B15, 34B18.
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1 Introduction
Nonlocal boundary-value problems with linear and nonlinear integral conditions have seen a great deal of study lately (see [1–16], and references therein) because of their interesting theory and their applications to various problems, such as heat flow in a bar of finite length [4, 11]. In this paper, we consider the existence of positive solutions of the nonlinear boundary-value problem (BVP) of the form
with integral boundary conditions
where is a linear functional on given by
involving a Stieltjes integral with a signed measure.
In [2], Goodrich considered the following problem:
with integral boundary conditions
and deduced the existence of at least one positive solution to the BVP (1.3)-(1.4) in which has either asymptotically sublinear or asymptotically superlinear growth, and in [3] Goodrich demonstrated that if the nonlinear functional satisfies a certain asymptotic behavior, then the BVP (1.3)-(1.4) possesses at least one positive solution. For the case that H is linear and involves a signed measure, Webb and Infante discussed the multiplicity of positive solutions for nonlocal boundary-value problems [12–14]. For the case that H is linear and the Borel measure associated with the Lebesgue-Stieltjes integral is positive, we can find some results on the existence of positive solutions [7, 8, 16, 17]. The results in the above literature are obtained under the condition that is continuous on , i.e., f has no singularity at . And it is well known that study of singular two-point boundary-value problems for the second-order differential equation (1.1) (singular in the dependent variable) is very important and there are many results on the existence of positive solutions [15, 18–24]. But there are fewer results on the existence of positive solutions for the singular BVP (1.1)-(1.2) [5, 6]. One goal in this paper is to consider the existence of positive solutions under the condition that is singular at . Our paper has the following features.
Firstly, in order to overcome the difficulties of the singularity of f we establish a new cone and get the new condition (3.13) which is different from that in [5, 6]. Moreover, we get a multiplicity of positive solutions for BVP (1.1)-(1.2) different from that in [2, 3, 12–14] under the condition that or is superlinear at .
Secondly, when f is singular and sign-changed, we get the existence of at least one positive solution to the BVP (1.1)-(1.2) which is different from that in [2, 3, 5, 6, 12–14] where f is nonnegative and continuous at . Moreover, the results are different from that in [7, 8, 16, 17] where integral boundary conditions are linear and the Borel measure is positive.
Our paper is organized as follows. In Section 2, we present some lemmas and preliminaries. Section 3 discusses the existence of multiple positive solutions for the BVP (1.1)-(1.2) when f is positive. In Section 4, we discuss the existence of at least one positive solution of BVP (1.1)-(1.2) when f is singular and sign-changed.
2 Preliminaries
In this paper, the following lemmas are needed.
Lemma 2.1 (see [25])
Let Ω be a bounded open set in real Banach space E, P a cone of and continuous and compact. Suppose , , . Then
Lemma 2.2 (see [25])
Let Ω be a bounded open set in real Banach space E, P a cone of and continuous and compact. Suppose , . Then
Let E be a Banach space, , , and a continuous compact operator. If for any with and , then F has a fixed point in .
Let us begin by stating the hypotheses which we shall impose on the BVP (1.1)-(1.2).
-
(C1) Assume that there are three linear functionals such that
Moreover, assume that there exists a constant such that
holds for each , where P is the cone introduced in (2.1) below [2].
-
(C2) The functionals and are linear and, in particular, have the form
where satisfy with
and
hold, where the latter holds for each and is defined in (3.2) below [2].
-
(C3) Let be a real-valued, continuous function. Moreover, .
-
(C4)
-
(C5)
Let with norm . It is easy to see that is a Banach space.
Assume that (C2) hold. Define
It is easy to prove P is a cone of and we have the following lemma.
Lemma 2.4 (see [20])
Let (defined in (2.1)). Then
3 Multiplicity of positive solutions for the singular boundary-value problems with positive nonlinearities
In this section, we consider the existence of multiple positive solutions for the BVP (1.1)-(1.2). To show that the BVP (1.1)-(1.2) has a solution, for , we define
where
Lemma 3.1 Suppose (C1)-(C5) hold. Then is continuous and compact for all .
Proof It is easy to prove that is well defined and for all . For , we have
and so
Moreover, from (C2), we may estimate
and
Combining (3.3), (3.4), and (3.5), . A standard argument shows that is continuous and compact [9, 18, 26]. □
Define
Lemma 3.2 If and (C2) hold, there exists a such that
Proof Suppose . There are two cases to consider.
(1). Lemma 2.4 implies that
(2) . Condition (C4) guarantees that
Since , , and , we know that is concave on and for all . And from (C2), a similar argument as (3.4) and (3.5) shows that and . Then and Lemma 2.4 implies that
Let . From (3.6) and (3.7), one has
which means that
Thus
where
and (C1) guarantees that
And so
The concavity yields
The proof is complete. □
For , let
We have the following lemmas.
Lemma 3.3 Suppose that (C1)-(C5) hold and there exists an such that
uniformly on . Then, there exists an such that for all
Proof From (3.8), there exists an such that
where
Let and
Now we show
Suppose that there exists a with . Then, . Since is concave on (since ) we find from Lemma 2.4 that for . For , one has
which together with (3.9) yields
Then we have, using (3.11),
which is a contradiction. Hence equation (3.10) is true. Lemma 2.2 guarantees that
The proof is complete. □
Lemma 3.4 Suppose that (C1)-(C5) hold and
Then, there exists an such that for all
Proof From equation (3.12), there exists an such that
where
Let and
Now we show
Suppose that there exists a with . Then, . Now (C1) guarantees that
which together with equation (3.13) implies that
This is a contradiction. Hence (3.14) is true. Lemma 2.2 guarantees that
The proof is complete. □
Theorem 3.1 Suppose (C1)-(C5) hold and the following conditions are satisfied:
and
hold; here
Then the BVP (1.1)-(1.2) has at least one positive solution.
Proof From equation (3.16), choose and with such that
Let
and . For , we define as in equation (3.1). Lemma 3.1 guarantees that is continuous and compact.
Now we show that
Suppose that there is a and with , i.e., satisfies
Then on . From equation (3.17), we have , which together with implies that there exists a with , and for all . For , from equations (3.15) and (3.19), we have
We integrate equation (3.20) from () to t to obtain
and then integrate equation (3.21) from to 1 to obtain
i.e.,
which contradicts equation (3.17). Therefore, equation (3.18) is true. Lemma 2.1 implies that
which yields the result that there exists a such that
i.e., in Lemma 3.2. Now Lemma 3.2 guarantees that there exists a such that
Now we consider the set . Obviously, means that
Now we show that
There are two cases to consider.
(1) There exists a subsequence of with . Without loss of generality, we assume that , , which together with implies that there exists a satisfying that with for and for . Let . Now we show that . To the contrary, suppose that . Then there exists a subsequence of such that as . From equation (3.21), using in place of , we have
which implies that
This contradicts for all .
Let . From equation (3.22), we have
Similarly as the proof in equation (3.21), one has
which means that
For , from equation (3.21), using in place of , we have
which yields
Combining equations (3.25) and (3.26), we find that equation (3.24) holds.
(2) There exists a such that and is nonincreasing on for all . From and , there exists such that . Now implies that . Hence, from equation (3.20), using in place of , we have
and so
Then
which implies that (3.24) hold.
Now Arzela-Ascoli theorem guarantees that has a convergent subsequence. Without loss of generality, we assume that there is a such that
which together with equation (3.22) and implies that
Since () satisfies , we have
We integrate the above equation from to t to yield
and so
for and
and the Lebesgue Dominated Convergent theorem together with equation (3.27) implies that
for and
We differentiate equation (3.28) to get
which together with equations (3.27) and (3.29) means that the BVP (1.1)-(1.2) has at least one positive solution. The proof is complete. □
Theorem 3.2 Suppose the conditions of Theorem 3.1 hold and there exists an such that
uniformly on . Then the BVP (1.1)-(1.2) has at least two positive solutions.
Proof Choose as in (3.17), with , and in Lemma 3.3. Set , and
By the proof of Theorem 3.1 and Lemma 3.3, we have
and
which implies that
Then, there exist and such that
By the proof of Theorem 3.1, there exist a subsequence of and such that
And moreover, is a positive solution to the BVP (1.1)-(1.2) with , .
A similar argument shows that there exist a subsequence of and such that
And moreover, is a positive solution to the BVP (1.1)-(1.2) and equation (3.18) guarantees that . Hence, and are two positive solutions for the BVP (1.1)-(1.2). The proof is complete. □
Theorem 3.3 Suppose the conditions of Theorem 3.1 hold and
Then the BVP (1.1)-(1.2) has at least two positive solutions.
Proof Choose as in (3.17), with , and in Lemma 3.4. Set , and
By the proof of Theorem 3.1 and Lemma 3.4, we have
and
which implies that
Then, there exist and such that
A similar argument to that in Theorem 3.2 shows that the BVP (1.1)-(1.2) has at least two positive solutions. The proof is complete. □
Example 3.1 Consider
with
where
with
Then equations (3.30)-(3.31) have at least two positive solutions.
To prove that the BVP (3.30)-(3.31) has at least two positive solutions, we use Theorem 3.2. Let , , , , , . For (defined in (2.1)), we have
which means that (C1) holds. Since
and
(C2) is true. Since , we have . Then
Equation (3.32) guarantees that
Letting , we have
for all , which means that equations (3.15)-(3.16) hold. Since
we get (C4). Moreover, since
uniformly on , all conditions of Theorem 3.2 hold, which implies that equations (3.30)-(3.31) have at least two positive solutions.
Example 3.2 Consider
with
where
with
Then equations (3.33)-(3.34) have at least two positive solutions.
To prove that the BVP (3.33)-(3.34) has at least two positive solutions, we use Theorem 3.3. Let , , , , , . Since , we have . Then
Also we have
Then, letting , we get
for all , which means that equations (3.15)-(3.16) hold. Since
we get (C4). Obviously, (C1)-(C3), and (C5) hold. Moreover, since
uniformly on , all conditions of Theorem 3.3 hold, which implies that equations (3.30)-(3.31) have at least two positive solutions.
4 Positive solutions for singular boundary-value problems with sign-changing nonlinearities
-
(H1) Assume that there are three linear functionals
where satisfy ;
-
(H2) , ;
-
(H3) Let be a real-valued, continuous function. Moreover, ;
-
(H4) , there exists a decreasing function , and a nonnegative function such that and there exists a such that
-
(H5) there exist such that
and
where .
For , let . Obviously, . For , we define as
where
and
From a standard argument (see [18, 25, 26]), we have the following result.
Lemma 4.1 Suppose (H1)-(H4) hold. Then the operator is continuous and compact from to .
From (H3) and (H5), there exists such that
Choose with and let . Now we have the following lemmas.
Lemma 4.2 Suppose (H1)-(H5) hold. Then, for , there exists a with such that
Proof Let . For , we now prove that
for any .
Suppose equation (4.2) is not true. Then there exists with and such that
We first claim that for any .
Suppose there exists a with . Let and . Since and , we have , , , and for all , which implies that
and so is concave down on . This is a contradiction.
Now (H5) guarantees that
which together with means that there is a with and . Let and . Obviously, , , , , , for all and for all . Let and . It is easy to see that , for all , and for all .
Now we consider the properties of y on . We get a countable set of such that
-
1.
, ,
-
2.
, , ,
-
3.
is strictly decreasing in , (if is strictly decreasing in , put ; i.e, ).
Differentiating equation (4.3) and using the assumptions (H2) and (H4), we obtain
Integrating (4.4) from to t, we have, by the decreasing property of ,
for , ; that is to say,
It follows from equation (4.5) that
for , .
On the other hand, for any with , we can choose and such that , and . Integrating equation (4.6) from to , and from to , we have
and
Summing equation (4.7) from 1 to , we have by equation (4.8) and
Since ,
For the properties of y on , a similar argument shows that for any
Letting in (4.9), we have
which contradicts equation (4.1). Hence equation (4.2) holds.
It follows from Lemma 3.2 that has a fixed point in . Using and 1 in place of y and λ in (4.3), we obtain easily , . And satisfies
The proof is complete. □
Lemma 4.3 Suppose that all conditions of Lemma 4.2 hold and satisfies (4.11). For a fixed , let . Then .
Proof Since , we get . For any fixed natural number n ( defined in Lemma 4.2), let such that . If , there exists a countable set such that
So there exists such that , . Let . Then we have two cases.
Case 1. There exist and such that . By the same argument in Lemma 4.2, we can get , such that
and
The inequality (4.13) shows that is concave down in , which contradicts equation (4.12).
Case 2. , for any . And so we have
On the other hand, for any ,
which contradicts equation (4.14). Hence, . The proof is complete. □
Theorem 4.1 If (H1)-(H5) hold, then BVP (1.1)-(1.2) has at least one positive solution.
Proof For any natural number (defined in Lemma 4.2), it follows from Lemma 4.2 that there exist , for all satisfying (4.11). Now we divide the proof into three steps.
Step 1. There exists a convergent subsequence of in . For a natural number in Lemma 4.2, it follows from Lemma 4.3 that , for any natural numbers ; i.e., is uniformly bounded in . Since also satisfies
we have
Obviously
for . It follows from inequality (4.15) that is equicontinuous in . The Ascoli-Arzela theorem guarantees that there exists a subsequence of which converges uniformly on . Then, for , we choose a convergent subsequence of on ,
for , we choose a convergent subsequence of on ,
for , we choose a convergent subsequence of on ,
for , we choose a convergent subsequence of on ,
We may choose the diagonal sequence which converges everywhere in and it is easy to verify that converges uniformly on any interval . Without loss of generality, let be in the rest. Putting , , we have continuous in and , for any by Lemma 4.3.
Step 2. satisfies equation (1.1). Fixed , we may choose such that and
Letting in above equation, we have
Differentiating equation (4.16), we get the desired result.
Step 3. satisfies equation (1.2). Let
and
where . Then
Using , 1, in place of , λ and in Lemma 4.2, from equation (4.9); we have
and using , 1, in place of , λ and in Lemma 4.2, from equation (4.10), we obtain easily
It follows from the above inequalities that and .
-
(1)
Fixing , we get . From equation (4.9) of the proof in Lemma 4.2, one easily has
Letting in the above inequality and noticing , we have
It follows from equation (4.17) that .
-
(2)
Fixing , we get . From equation (4.10) in the proof of Lemma 4.2, we easily get
(4.18)
Since and , the Lebesgue Dominated Convergent theorem guarantees that
Since H is continuous, we have
Letting in equation (4.18) and noticing and equation (4.19), we have
It follows from equation (4.20) that . This complete the proof. □
Example 4.1 Consider
with boundary conditions
where
Then the BVP (4.21)-(4.22) has at least one positive solution.
Let , , , , , . Let and . We have
where and
Then (H1)-(H5) hold. Now Theorem 4.1 guarantees that the BVP (4.21)-(4.22) has at least one positive solution.
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The author thanks the referees for their suggestions and this research is supported by Young Award of Shandong Province (ZR2013AQ008).
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Yan, B. Positive solutions for the singular nonlocal boundary value problems involving nonlinear integral conditions. Bound Value Probl 2014, 38 (2014). https://doi.org/10.1186/1687-2770-2014-38
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DOI: https://doi.org/10.1186/1687-2770-2014-38