Positive solutions for nonlocal fourth-order boundary value problems with all order derivatives

In this article, by the fixed point theorem in a cone and the nonlocal fourth-order BVP's Green function, the existence of at least one positive solution for the nonlocal fourth-order boundary value problem with all order derivatives

is considered, where f is a nonnegative continuous function, λ > 0, 0 < A < π², p, q ∈ L[0, 1], p(s) ≥ 0, q(s) ≥ 0. The emphasis here is that f depends on all order derivatives.

The deformation of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by a fourth-order ordinary equation boundary value problem. Owing to its significance in physics, the existence of positive solutions for the fourth-order boundary value problem has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point index theory, the Krasnosel'skii's fixed point theorem and the method of upper and lower solutions, in reference [1-10].

In recent years, there has been much attention on the question of positive solutions of the fourth-order differential equations with one or two parameters. By the Krasnosel'skii's fixed point theorem in cone [11], Bai [5] investigated the following fourth-order boundary value problem with one parameter

where λ > 0, 0 < β < π², 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 fourth-order 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 fourth-order boundary value problem

Throughout, we assume

(H₁) λ > 0, 0 < A < π²;

(H₂) f: [0, 1] × R⁴ → 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 Leray-Schauder 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 fourth-order BVP's Green function, we give some new criteria for the existence of positive solutions to (1.1).

Let Y = C[0, 1] be the Banach space equipped with the norm

Set λ₁, λ₂ be the roots of the polynomial P(λ) = λ² + Aλ, namely λ₁ = 0, λ₂ = -A. By (H₁), it is obviously that -π² < λ₂ < 0.

Let Q₁(t, s), Q₂(t, s) be, respectively the Green's functions of the following problems

Then, carefully calculation yield

Denote

Lemma 2.1. [5] Suppose that (H₁) and (H₂) 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 (H₁) and (H₂) 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_iG_i(t, t)G_i(s, s), ∀t, s ∈ [0, 1];

(iii) G_i(t, s) ≤ c_iG_i(s, s), ∀t, s ∈ [0, 1].

where b₁ = 1, ; c₁ = 1, .

Let

Lemma 2.3. [5] Suppose that (H₁) and (H₂) hold and w₂, d_i, ξ_i are given as above. Then

one has

(i)

(ii)

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) = |λ|α(u), β(λu) = |λ|β(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 r₂ > r₁ > 0, L > 0 be constants and

two bounded open sets in X. Set

Assume T: K → K is a completely continuous operator satisfying

(A₁) α(Tu) < r₁, u ∈ D₁ ∩ K; α(Tu) > r₂, u ∈ D₂ ∩ K;

(A₂) β(Tu) < L, u ∈ K;

(A₃) there is a p ∈ (Ω₂ ∩ K) \ {0} such that α(p) ≠ 0 and α(u + λp) ≥ α(u), for all u ∈ K and λ ≥ 0.

Then T has at least one fixed point in .

Let X = C⁴[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) = |λ|α(u), β(λu) = |λ|β(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, u₀, v₀) satisfies the following growth conditions:

Let , where

We denote

Lemma 3.1. Suppose that (H₁) and (H₂) 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 c₁ = 1, c₂ > 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 f₁, Q₁(t, s), Q₂(t, s) is continuous, we show that T(B) is equicontinuous. By the Arzela-Ascoli theorem, a standard proof yields T: K → K is completely continuous.

Theorem 3.1. Suppose that (H₁)-(H₅) 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 ∈ D₁, we have

Hence, for u ∈ D₁ ∩ K, α(u) = c, we get

Whereas for u ∈ D₂ ∩ K, α(u) = b, there is or , By Lemma 2.4, we get

Therefore, using (H₄) and (3.3), we have

Hence,

By (3.2), (3.4), and (H₅), 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.

Example 4.1. Consider the following boundary value problem

where

In this problem, we know that , then we can get Further more, we obtain

then , , , , θb ≈ 3.06 > 3.

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

The authors declare that they have no competing interests.

The authors declare that the work was realized in collaboration with same responsibility.

All authors read and approved the final manuscript.

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.

1. 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

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

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

4. Zhao, JF, Ge, WG: Positive solutions for a higher-order four-point boundary value problem with a p-Laplacian. Comput Math Appl. 58, 1103–1112 (2009). Publisher Full Text

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

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

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

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

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

10. 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

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

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