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Existence of concave symmetric positive solutions for a three-point boundary value problems
Advances in Difference Equations volume 2014, Article number: 313 (2014)
In this paper, we investigate the existence of triple concave symmetric positive solutions for the nonlinear boundary value problems with integral boundary conditions. The proof is based upon the Avery and Peterson fixed point theorem. An example which supports our theoretical result is also indicated.
MSC:34B10, 39B18, 39A10.
Consider the boundary value problem (BVP)
where is continuous and . A function is said to be symmetric on if
By a symmetric positive solution of BVP (1.1) we mean a symmetric function such that for and satisfies BVP (1.1).
Recently, many authors have focused on the existence of symmetric positive solutions for ordinary differential equation boundary value problems; for example, see [1–5] and the references therein. However, multi-point boundary value problems included the most recent works [1–4, 6–9] and boundary value problems with integral boundary conditions for ordinary differential equations have been studied by many authors; one may refer to [5, 10–12]. Motivated by the works mentioned above, we aim to investigate existence results for concave symmetric positive solutions of BVP (1.1) by applying the fixed point theorem of Avery and Peterson.
The organization of this paper is as follows. Section 2 of this paper contains some preliminary lemmas. In Section 3, by applying the Avery and Peterson fixed point theorem, we obtain concave symmetric positive solutions for BVP (1.1). In Section 4, an example will be presented to illustrate the applicability of our result.
Throughout this paper, we always assume that the following assumption is satisfied:
(H1) , for , and for all .
In this section, we present several lemmas that will be used in the proof of our result.
Lemma 2.1 Let and , then the BVP
has a solution
Proof Suppose that is a solution of problem (2.1) and (2.2). Then we have
For , by integration from 0 to 1, we have
For , by integration again from 0 to 1, we have
From condition (2.2), we have
Integrating (2.6) from 0 to η, where , we have
and from , we have
therefore, (2.1) and (2.2) have a unique solution
From (2.4) and (2.5), we obtain
The proof is complete. □
The functions H and G have the following properties.
Lemma 2.2 If , then we have
Proof From the definition of , , and , we have . □
Lemma 2.3 , for .
Proof From the definition of , we get and for . □
Lemma 2.4 If and we let , then the unique solution u of BVP (1.1) satisfies for .
Proof From the definition of , Lemma 2.2, Lemma 2.3, and , we have . □
Let . Then E is Banach space with the norm , where .
We define the cone by
Define the operator as follows:
where and are given by (2.4) and (2.5). Clearly, u is the solution of BVP (1.1) if and only if u is a fixed point of the operator T.
Lemma 2.5 Let and . For , there exists a real number such that
Proof For any , we have
the concavity of u implies that
Now we divide the proof into two cases.
Case 1. If , then the concavity of u implies that
where is the area of rectangle, then
then from (2.8) and (2.9), we have
Case 2. If , then the concavity and symmetry of u imply that
then from (2.8) and (2.10), we have
where , these equations complete the proof. □
Lemma 2.6 Let and . For , there exists a real number such that
Proof For any , we have
and because the graph of u is concave, we have
Now we divide the proof into two cases.
Case 1. If , then using concavity and positivity of u, we get
then from (2.11) and (2.12), we have
Case 2. If , then using concavity and positivity of u, we get
where , and for , one can easily see that . These equations complete the proof. □
3 Existence of triple concave symmetric positive solutions for BVP (1.1)
In this section, we will apply the Avery-Peterson fixed point theorem to establish the existence of at least three concave symmetric positive solutions of BVP (1.1).
Let α, γ, θ, ψ be maps on P with α a nonnegative continuous concave functional; γ, θ nonnegative continuous convex functionals, and ψ a nonnegative continuous functional. Then for positive numbers a, b, c, d we define the following subsets of P:
Now we state the Avery-Peterson fixed point theorem.
Let P be cone in Banach space E. Let γ and θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P leading to
Suppose is completely continuous and there exist positive numbers a, b, c with such that
Then T has at least three fixed points such that
Lemma 3.1 Assume that (H1) is satisfied and let . Then the operator T is completely continuous.
Proof For any , from the expression of Tu, we know
Clearly, Tu is concave. From the definition of Tu, Lemma 2.2, and Lemma 2.3 we see that Tu is nonnegative on . We now show that Tu is symmetric about . From Lemma 2.3 and (H1), for , we have
The continuity of T with respect to is clear. We now show that T is compact. Suppose that is a bounded set. Then there exists r such that
For any , we have
So, we have from (2.7)
These equations imply that the operator T is uniformly bounded. Now we show that Tu is equi-continuous.
We separate these three conditions:
Case (i). ;
Case (ii). ;
Case (iii). .
We solely need to deal with Case (i) since the proofs of the other two are analogous. For , we have
So, we see that Tu is equi-continuous. By applying the Arzela-Ascoli theorem, we can guarantee that is relatively compact, which means T is compact. Then we find that T is completely continuous. This completes the proof. □
For convenience, we denote
Theorem 3.2 Suppose (H1) holds and let . Moreover, there exist nonnegative numbers such that
then BVP (1.1) has at least three concave symmetric positive solutions , , such that
Proof BVP (1.1) has a solution if and only if u solves the operator equation . So we need to verify that operator T satisfies the Avery-Peterson fixed point theorem, which will prove the existence of at least three fixed points of T.
Complete continuity of T is clear from Lemma 3.1. Define the nonnegative functionals α, θ, γ, and ψ by
Then in the cone P, θ and γ are convex as α is concave. It is well known that for all and . Moreover, from Lemma 2.5, . Now, we will prove the main theorem in four steps.
Step 1. We will show that .
If , then . Lemma 2.5 yields , then the condition (B1) implies that . On the other hand, for any , there is , then is concave, symmetric, and positive on and , and we have
Then . Therefore .
Step 2. To check if condition (S1) of Theorem 3.1 is satisfied, we choose . Clearly,
Thus, and .
If , then we have , . From condition (B2), we have , and it follows that
This shows that condition (S1) of Theorem 3.1 is satisfied.
Step 3. We will show that condition (S2) of Theorem 3.1 is satisfied. Take with . Then from Lemma 2.6, we have
so condition (S2) holds.
Step 4. We will show that condition (S3) of Theorem 3.1 is also satisfied. Obviously, , and we have . Assume that with . Then, from condition (B3), we have
It proves that condition (S3) holds. All conditions of Theorem 3.1 are satisfied and we assert that BVP (1.1) has at least three concave symmetric positive solutions such that
Therefore, our proof is complete. □
Example 4.1 We consider the following three-point second-order BVP with integral boundary conditions:
We can see from (4.1) that , . Then , , , . We choose , , . Clearly . Moreover, f satisfies (H1).
So, by Theorem 3.2 we find that BVP (4.1) has at least three concave symmetric positive solutions , , such that
Sun Y, Zhang X: Existence of symmetric positive solutions for an m -point boundary value problem. Bound. Value Probl. 2007., 2007: Article ID 79090
Kosmatov N: Symmetric solutions of a multi-point boundary value problem. J. Math. Anal. Appl. 2005, 309: 25-36. 10.1016/j.jmaa.2004.11.008
Zhao J, Miao C, Ge W, Zhang J: Multiple symmetric positive solutions to a new kind of four point boundary value problem. Nonlinear Anal. 2009, 71: 9-18. 10.1016/j.na.2008.10.026
Hao F: Existence of symmetric positive solutions for m -point boundary value problems for second-order dynamic equations on time scales. Math. Theory Appl. 2008, 28: 65-68.
Pang H, Tong Y: Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions. Bound. Value Probl. 2013., 2013: Article ID 150 10.1186/1687-2770-2013-150
Il’in VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differ. Equ. 1987, 23: 979-987.
Gupta CP: A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 1998, 89: 133-146. 10.1016/S0096-3003(97)81653-0
Feng W: On an m -point boundary value problem. Nonlinear Anal. 1997, 30: 5369-5374. 10.1016/S0362-546X(97)00360-X
Sun B, Ge W, Zhao D: Three positive solutions for multipoint one-dimensional p -Laplacian boundary value problems with dependence on the first order derivative. Math. Comput. Model. 2007, 45: 1170-1178. 10.1016/j.mcm.2006.10.002
Jankowski T: Differential equations with integral boundary conditions. J. Comput. Appl. Math. 2002, 147: 1-8. 10.1016/S0377-0427(02)00371-0
Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 2009, 70: 364-371. 10.1016/j.na.2007.12.007
Tariboon J, Sitthiwirattham T: Positive solutions of a nonlinear three-point integral boundary value problem. Bound. Value Probl. 2010., 2010: Article ID 519210 10.1155/2010/519210
Avery RI, Peterson AC: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 2001, 42: 313-322. 10.1016/S0898-1221(01)00156-0
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.