Abstract
In this article, by the fixed point theorem in a cone and the nonlocal fourthorder BVP's Green function, the existence of at least one positive solution for the nonlocal fourthorder boundary value problem with all order derivatives
is considered, where f is a nonnegative continuous function, λ > 0, 0 < A < π^{2}, p, q ∈ L[0, 1], p(s) ≥ 0, q(s) ≥ 0. The emphasis here is that f depends on all order derivatives.
Keywords:
fourthorder boundary value problem; fixed point theorem; Green's function; positive solution1 Introduction
The deformation of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by a fourthorder ordinary equation boundary value problem. Owing to its significance in physics, the existence of positive solutions for the fourthorder boundary value problem has been studied by many authors using nonlinear alternatives of LeraySchauder, the fixed point index theory, the Krasnosel'skii's fixed point theorem and the method of upper and lower solutions, in reference [110].
In recent years, there has been much attention on the question of positive solutions of the fourthorder differential equations with one or two parameters. By the Krasnosel'skii's fixed point theorem in cone [11], Bai [5] investigated the following fourthorder boundary value problem with one parameter
where λ > 0, 0 < β < π^{2}, f: C([0, 1] × [0, ∞) × (∞, 0], [0, ∞)) is continuous, p, q ∈ L[0, 1], p(s) ≥ 0, q(s) ≥ 0, , .
By the fixed point index in cone, Ma [7] proved the existence of symmetric positive solutions for the nonlocal fourthorder boundary value problem
All the above works were done under the assumption that all order derivatives u', u″, u‴ are not involved explicitly in the nonlinear term f. In this article, we are concerned with the existence of positive solutions for the nonlocal fourthorder boundary value problem
Throughout, we assume
(H_{1}) λ > 0, 0 < A < π^{2};
(H_{2}) f: [0, 1] × R^{4 }→ R^{+ }is continuous, p, q ∈ L[0, 1], p(s) ≥ 0, q(s) ≥ 0, , .
We will impose all order derivatives in f and make use of two continuous convex functionals which will ensure the existence of at least one positive solution to (1.1). Bai [5] applied Krasnoselskii's fixed point theorem. Ma [8] used fixed point index in cone and LeraySchauder degree. In this article, to show the existence of positive solutions to (1.1), we define two positive continuous convex functionals. Then, using the new fixed point theorem [12] in a cone and the nonlocal fourthorder BVP's Green function, we give some new criteria for the existence of positive solutions to (1.1).
2 The preliminary lemmas
Let Y = C[0, 1] be the Banach space equipped with the norm
Set λ_{1}, λ_{2 }be the roots of the polynomial P(λ) = λ^{2 }+ Aλ, namely λ_{1 }= 0, λ_{2 }= A. By (H_{1}), it is obviously that π^{2 }< λ_{2 }< 0.
Let Q_{1}(t, s), Q_{2}(t, s) be, respectively the Green's functions of the following problems
Then, carefully calculation yield
Denote
Lemma 2.1. [5] Suppose that (H_{1}) and (H_{2}) hold. Then for any y(t) ∈ C[0, 1], the problem
has a unique solution
where
By (2.2), we get
Lemma 2.2. [5] Assume that (H_{1}) and (H_{2}) hold. Then one has
(i) Q_{i}(t, s) ≥ 0, ∀t, s ∈ [0, 1]; Q_{i}(t, s) > 0, ∀t, s ∈ (0, 1);
(ii) G_{i}(t, s) ≥ b_{i}G_{i}(t, t)G_{i}(s, s), ∀t, s ∈ [0, 1];
(iii) G_{i}(t, s) ≤ c_{i}G_{i}(s, s), ∀t, s ∈ [0, 1].
where b_{1 }= 1, ; c_{1 }= 1, .
Let
Lemma 2.3. [5] Suppose that (H_{1}) and (H_{2}) hold and w_{2}, d_{i}, ξ_{i }are given as above. Then
one has
Lemma 2.4. If y(t) ∈ C[0, 1] and y(t) ≥ 0, then the unique solution u(t) of problem (2.1)
satisfies
Proof. By (2.2) and (iii) of Lemma 2.2, we get
So,
Using (ii) of Lemma 2.2, we have
By (2:4) and (iii) of Lemma 2.2, we get
So,
Using (ii) of Lemma 2.2, we have
The proof is completed.
Let X be a Banach space and K ⊂ X a cone. Suppose α, β: × → R^{+ }are two continuous convex functionals satisfying α(λu) = λα(u), β(λu) = λβ(u), for u ∈ X, λ ∈ R, and u ≤ M max{α(u), β(u)}, for u ∈ X and α(u) ≤ α(v) for u, v ∈ K, u ≤ v, where M > 0 is a constant.
Theorem 2.1. [12] Let r_{2 }> r_{1 }> 0, L > 0 be constants and
two bounded open sets in X. Set
Assume T: K → K is a completely continuous operator satisfying
(A_{1}) α(Tu) < r_{1}, u ∈ D_{1 }∩ K; α(Tu) > r_{2}, u ∈ D_{2 }∩ K;
(A_{2}) β(Tu) < L, u ∈ K;
(A_{3}) there is a p ∈ (Ω_{2 }∩ K) \ {0} such that α(p) ≠ 0 and α(u + λp) ≥ α(u), for all u ∈ K and λ ≥ 0.
3 The main results
Let X = C^{4}[0, 1] be the Banach space equipped with the norm and is a cone in X.
Define two continuous convex functionals and , for each u ∈ X, then u ≤ 2 max{α(u), β(u)} and α(λu) = λα(u), β(λu) = λβ(u), for u ∈ X, λ ∈ R; α(u) ≤ α(v) for u, v ∈ K, u ≤ v.
In the following, we denote
We will suppose that there are L > b > θb > c > 0 such that f(t, u, v, u_{0}, v_{0}) satisfies the following growth conditions:
We denote
Lemma 3.1. Suppose that (H_{1}) and (H_{2}) hold. Then T: K → K is completely continuous.
Proof. For u ∈ K, by (3.1), (3.3) and Lemma 2.2, it is obviously that Tu ≥ 0, (Tu)″ ≤ 0. In view of c_{1 }= 1, c_{2 }> 1, so
By Lemma 2.3, (3.1) and (3.3), we have
So we can get T(K) ⊂ K: Let B ⊂ K is bounded, it is clear that T(B) is bounded. Using f_{1}, Q_{1}(t, s), Q_{2}(t, s) is continuous, we show that T(B) is equicontinuous. By the ArzelaAscoli theorem, a standard proof yields T: K → K is completely continuous.
Theorem 3.1. Suppose that (H_{1})(H_{5}) hold. Then BVP (1.1) has at least one positive solution u(t) satisfying
Proof. Take
two bounded open sets in X, and
By Lemma 3.1, T: K → K is completely continuous. Let . It is easy to see that α(u + λp) ≥ α(u), for all u ∈ K and λ ≥ 0.
Let u ∈ D_{1}, we have
Hence, for u ∈ D_{1 }∩ K, α(u) = c, we get
Whereas for u ∈ D_{2 }∩ K, α(u) = b, there is or , By Lemma 2.4, we get
Therefore, using (H_{4}) and (3.3), we have
Hence,
By (3.2), (3.4), and (H_{5}), for u ∈ K, we have
So,
Theorem 2.1 implies there is such that u = Tu. So, u(t) is a positive solution for BVP (1.1) satisfying
Thus, Theorem 3.1 is completed.
4 Example
Example 4.1. Consider the following boundary value problem
where
In this problem, we know that , then we can get Further more, we obtain
If we take c = 2, b = 40, L = 16000, then we get
Then all the conditions of Theorem 3.1 are satisfied. Therefore, by Theorem 3.1 we know that boundary value problem (4.1) has at least one positive solution u(t) satisfying
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with same responsibility.
All authors read and approved the final manuscript.
Acknowledgements
The project is supported by the Natural Science Foundation of China (10971045) and the Natural Science Foundation of Hebei Province (A2009000664, A2011208012). The research item financed by the talent training project funds of Hebei Province. The authors would like to thank the referee for helpful comments and suggestions.
References

Bai, ZB: The method of lower and upper solution for a bending of an elastic beam equation. J Math Anal Appl. 248, 195–202 (2000). Publisher Full Text

Bai, ZB: The method of lower and upper solutions for some fourthorder boundary value problems. Nonlinear Anal. 67, 1704–1709 (2007). Publisher Full Text

Chai, GQ: Existence of positive solutions for fourthorder boundary value problem with variable parameters. Nonlinear Anal. 66, 870–880 (2007). Publisher Full Text

Zhao, JF, Ge, WG: Positive solutions for a higherorder fourpoint boundary value problem with a pLaplacian. Comput Math Appl. 58, 1103–1112 (2009). Publisher Full Text

Bai, ZB: Positive solutions of some nonlocal fourthorder boundary value problem. Appl Math Comput. 215, 4191–4197 (2010). Publisher Full Text

Li, YX: Positive solutions of fourthorder boundary value problems with two parameters. J Math Anal Appl. 281, 477–484 (2003). Publisher Full Text

Ma, HL: Symmetric positive solutions for nonlocal boundary value problems of fourthorder. Nonlinear Anal. 68, 645–651 (2008). Publisher Full Text

Ma, RY: Existence of positive solutions of a fourthorder boundary value problem. Appl Math Comput. 168, 1219–1231 (2005). Publisher Full Text

Ma, R, Wang, H: On the existence of positive solutions of fourthorder ordinary differential equations. Appl Anal. 59, 225–231 (1995). Publisher Full Text

Yao, QL: Local existence of multiple positive solutions to a singular cantilever beam equation. J Math Anal Appl. 363, 138–154 (2010). Publisher Full Text

Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)

Guo, YP, Ge, WG: Positive solutions for threepoint boundary value problems with dependence on the first order derivatives. J Math Anal Appl. 290, 291–301 (2004). Publisher Full Text