Monotone and convex positive solutions for fourth-order multi-point boundary value problems

The existence results of multiple monotone and convex positive solutions for some fourth-order multi-point boundary value problems are established. The nonlinearities in the problems studied depend on all order derivatives. The analysis relies on a fixed point theorem in a cone. The explicit expressions and properties of associated Green's functions are also given.

MSC: 34B10; 34B15.

Boundary value problems for second and higher order nonlinear differential equations play a very important role in both theory and applications. For example, the deformations of an elastic beam in the equilibrium state can be described as a boundary value problem of some fourth-order differential equations. Owing to its importance in application, the existence of positive solutions for nonlinear second and higher order boundary value problems has been studied by many authors. We refer to recent contributions of Ma [1–3], He and Ge [4], Guo and Ge [5], Avery et al. [6, 7], Henderson [8], Eloe and Henderson [9], Yang et al. [10], Webb and Infante [11, 12], and Agarwal and O'Regan [13]. For survey of known results and additional references, we refer the reader to the monographs by Agarwal [14] and Agarwal et al. [15].

When it comes to positive solutions for nonlinear fourth-order ordinary differential equations, two point boundary value problems are studied extensively, see [16–24]. Few papers deal with the multi-point cases. Furthermore, for nonlinear fourth-order equations, only the situation that the nonlinear term does not depend on the first, second and third order derivatives are considered, see [16–23]. Few paper deals with the situation that lower order derivatives are involved in the nonlinear term explicitly. In fact, the derivatives are of great importance in the problem in some cases. For example, in the linear elastic beam equation (Euler-Bernoulli equation)

where u(t) is the deformation function, L is the length of the beam, f(t) is the load density, E is the Young's modulus of elasticity and I is the moment of inertia of the cross-section of the beam. In this problem, the physical meaning of the derivatives of the function u(t) is as follows: u⁽⁴⁾(t) is the load density stiffness, u'''(t) is the shear force stiffness, u''(t) is the bending moment stiffness and the u'(t) is the slope. If the payload depends on the shear force stiffness, bending moment stiffness or the slope, the derivatives of the unknown function are involved in the nonlinear term explicitly.

In this paper, we are interested in the positive solution for fourth-order nonlinear differential equation

subject to multi-point boundary condition

or

where 0 < ξ₁ < ξ₂ < ⋯ < ξ_m-2< 1, β _i > 0, 1 = 1, 2, ..., m - 2, {\sum }_{i=1}^{m-2}{\beta }_i>1, and f ∈ C([0, 1] × R⁴, [0, +∞)).

One can see that all lower order derivatives are involved in the nonlinear term explicitly and the BCs are the m-point cases. In this sense, the problems studied in this paper are more general than before. In the paper, multiple monotone and convex positive solutions for problems (1.1), (1.2) and (1.1), (1.3) are established. The results presented extend the study for fourth-order boundary value problems of nonlinear ordinary differential equations.

This paper is organized as follows. In Section 2, we present some preliminaries and lemmas. Section 3 is devoted to the existence of at least three convex and increasing positive solutions for problem (1.1), (1.2). In Section 4, we prove that there exist at least three convex and decreasing positive solutions for problem (1.1), (1.3).

In this section, some preliminaries and lemmas used later are presented.

Definition 2.1 The map α is said to be a nonnegative continuous convex functional on cone P of a real Banach space E provided that α : P → [0, +∞) is continuous and

Definition 2.2 The map β is said to be a nonnegative continuous concave functional on cone P of a real Banach space E provided that β : P → [0, +∞) is continuous and

Let γ, 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. Then for positive numbers a, b, c and d, we define the following convex sets:

and a closed set

Lemma 2.1 [25] Let P be a cone in Banach space E. Let γ, θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional and ψ be a nonnegative continuous functional on P satisfying

such that for some positive numbers l and d,

for all x\in \overline{P\left(\gamma ,d\right)}. Suppose T:\overline{P\left(\gamma ,d\right)}\to \overline{P\left(\gamma ,d\right)} is completely continuous and there exist positive numbers a, b, c with a < b such that

(S₁) {x ∈ P (γ, θ, α, b, c, d)|α(x) > b} ≠ ∅ and α(Tx) > b for x ∈ P (γ, θ, α, b, c, d);

(S₂) α(Tx) > b for x ∈ P (γ, α, b, d) with θ(Tx) > c;

(S₃) 0 ∉ R(γ, ψ, a, d) and ψ(Tx) < a for x ∈ R(γ, ψ, a, d) with ψ(x) = a.

Then T has at least three fixed points x₁, x₂, x_3\in \overline{P\left(\gamma ,d\right)} such that:

We begin with the fourth-order m-point boundary value problem

where 0 < ξ₁ < ξ₂ < ⋯ < ξ_m-2< 1, β _i > 0, i = 1, 2, ..., m - 2.

The following assumption will stand throughout this section:

Lemma 3.1 Denote ξ₀ = 0, ξ_{m -1}= 1, β₀ = β_{m -1}= 0, and y(t) ∈ C[0, 1]. Problem (3.1), (3.2) has the unique solution

where

for ξ_i-1≤ s ≤ ξ _i , i = 1, 2, ..., m -1.

Proof Let G(t, s) be the Green's function of problem x⁽⁴⁾(t) = 0 with boundary condition (3.2). We can suppose

Considering the definition and properties of Green's function together with the boundary condition (3.2), we have

A straightforward calculation shows that

These give the explicit expression of the Green's function and the proof of Lemma 3.1 is completed.

Lemma 3.2 One can see that G(t, s) ≥ 0, t, s ∈ [0, 1].

Proof For ξ_i-1≤ s ≤ ξ _i , i = 1, 2, ..., m - 1,

Then \frac{\partial G\left(t,s\right)}{\partial t}\ge 0, 0 ≤ t, s ≤ 1. Thus G(t, s) is increasing on t. By a simple computation, we see

These ensures that G(t, s) ≥ 0, t, s ∈ [0, 1].

Lemma 3.3 Suppose x(t) ∈ C³[0, 1] and

Furthermore x⁽⁴⁾(t) ≥ 0 and there exist t₀ such that x⁽⁴⁾(t₀) > 0. Then x(t) has the following properties:

1. (1)

   \underset{0\le t\le 1}{min}\left|x\left(t\right)\right|\ge \delta \underset{0\le t\le 1}{max}\left|x\left(t\right)\right|,

2. (2)

   \underset{0\le t\le 1}{max}\left|x\left(t\right)\right|\le \gamma \underset{0\le t\le 1}{max}\left|x^{\prime }\left(t\right)\right|,

3. (3)

   \underset{0\le t\le 1}{max}\left|x^{\prime }\left(t\right)\right|\le \underset{0\le t\le 1}{max}\left|x^{″}\left(t\right)\right|\underset{0\le t\le 1}{max}\left|x^{″}\left(t\right)\right|\le \underset{0\le t\le 1}{max}\left|x^{‴}\left(t\right)\right|,

where \delta =\left(1-{\sum }_{i=1}^{m-2}{\beta }_i{\xi }_i\right)∕{\sum }_{i=1}^{m-2}{\beta }_i\left(1-{\xi }_i\right),\gamma ={\sum }_{i=1}^{m-2}{\beta }_i\left(1-{\xi }_i\right)∕\left({\sum }_{i=1}^{m-2}{\beta }_i-1\right) are positive constants.

Proof Since x⁽⁴⁾(t) ≥ 0, t ∈ [0, 1], then x'''(t) is increasing on [0, 1]. Considering x'''(1) = 0, we have x'''(t) ≤ 0, t ∈ [0, 1]. Thus x''(t) is decreasing on [0, 1]. Considering this together with the boundary condition x''(1) = 0, we conclude that x''(t) ≥ 0. Then x(t) is convex on [0, 1]. Taking into account that x'(0) = 0, we get that

1. (1)

   From the concavity of x(t), we have

   {\xi }_i\left(x\left(1\right)-x\left(0\right)\right)\ge x\left({\xi }_i\right)-x\left(0\right).

Multiplying both sides with β _i and considering the boundary condition, we have

Thus

1. (2)

   Considering the mean-value theorem together with the concavity of x(t), we have

   x\left(1\right)-x\left({\xi }_i\right)\le \left(1-{\xi }_i\right)x^{\prime }\left(1\right).

   (3.4)

Multiplying both sides with β _i and considering the boundary condition, we have

which yields that x\left(1\right)\le {\sum }_{i=1}^{m-2}{\beta }_i\left(1-{\xi }_i\right)∕\left({\sum }_{i=1}^{m-2}{\beta }_i-1\right)\left|x^{\prime }\left(1\right)\right|=\gamma \underset{0\le t\le 1}{max}\left|x^{\prime }\left(t\right)\right|.

1. (3)

   For x^{\prime }\left(t\right)=x^{\prime }\left(0\right)+{\int }_0^t{x}^{″}\left(s\right)ds and x'(0) = 0, we get

   \left|x^{\prime }\left(t\right)\right|=\left|{\int }_0^t{x}^{″}\left(s\right)ds\right|\le {\int }_0^1\left|x^{″}\left(s\right)\right|ds.

For x^{″}\left(t\right)=x^{″}\left(1\right)-{\int }_t^1{x}^{‴}\left(s\right)ds and x"(1) = 0, we get

Consequently

These give the proof of Lemma 3.3.

Remark Lemma 3.3 ensures that

Let Banach space E = C³[0, 1] be endowed with the norm

Define the cone P ⊂ E by

Let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals γ, θ and the nonnegative continuous functional ψ be defined on the cone by

By Lemma 3.3, the functionals defined above satisfy

Denote

Assume that there exist constants 0 < a, b, d with a < b < λd such that

Theorem 3.1 Under assumptions (A₁)-(A₃), problem (1.1), (1.2) has at least three positive solutions x₁, x₂, x₃ satisfying

Proof Problem (1.1, 1.2) has a solution x = x(t) if and only if x solves the operator equation

Then

For x\in \overline{P\left(\gamma ,d\right)}, considering Lemma 3.3 and assumption (A₁), we have f(t, x(t), x'(t), x''(t), x'''(t)) ≤ d. Thus

Hence T:\overline{P\left(\gamma ,d\right)}\to \overline{P\left(\gamma ,d\right)}. An application of the Arzela-Ascoli theorem yields that T is a completely continuous operator. The fact that the constant function x(t) = b/δ ∈ P(γ, θ, α, b, c, d) and α(b/δ) > b implies that

For x ∈ P(γ, θ, α, b, c, d), we have b ≤ x(t) ≤ b/δ and |x'''(t)| < d. From assumption (A₂), we see

Hence, by definition of α and the cone P, we can get

which means α(Tx) > b, ∀x ∈ P(γ, θ, α, b, b/δ, d). This ensures that condition (S1) of Lemma 2.1 is fulfilled.

Second, with (3.4) and b < λd, we have

for all x ∈ P(γ, α, b, d) with \theta \left(Tx\right)>\frac{b}{\delta }.

Thus, condition (S₂) of Lemma 2.1 holds. Finally we show that (S₃) also holds. We see ψ(0) = 0 < a and 0 ∉ R(γ, ψ, a, d). Suppose that x ∈ R(γ, ψ, a, d) with ψ(x) = a, then by the assumption of (A₃),

which ensures that condition (S₃) of Lemma 2.1 is fulfilled. Thus, an application of Lemma 2.1 implies that the fourth-order m-point boundary value problem (1.1, 1.2) has at least three positive convex increasing solutions x₁, x₂, x₃ with the properties that

The following assumption will stand throughout this section:

Lemma 4.1 Denote ξ₀ = 0, ξ_m-1= 1, β₀ = β_m-1= 0, the Green's function of problem

is

for i = 1, 2, ..., m - 1.

Proof Suppose that

Considering the definition and properties of Green's function together with the boundary condition (4.2), we have

Consequently

The proof of Lemma 4.1 is completed.

Lemma 4.2 One can see that H(t, s) ≥ 0, t, s ∈ [0, 1].

Proof For ξ_i-1≤ s ≤ ξ _i , i = 1, 2, ..., m - 1,

Then \frac{\partial H\left(t,s\right)}{\partial t}\le 0, 0 ≤ t, s ≤ 1, which implies that H(t, s) is decreasing on t. The fact that

ensures that H(t, s) ≥ 0, t, s ∈ [0, 1].

Lemma 4.3 If x(t) ∈ C³[0, 1],

and x⁽⁴⁾(t) ≥ 0, there exists t₀ such that x⁽⁴⁾(t₀) > 0, then

1. (1)

   \underset{0\le t\le 1}{min}\left|x\left(t\right)\right|\ge {\delta }_1\underset{0\le t\le 1}{max}\left|x\left(t\right)\right|,

2. (2)

   \underset{0\le t\le 1}{max}\left|x\left(t\right)\right|\le {\gamma }_1\underset{0\le t\le 1}{max}\left|x^{\prime }\left(t\right)\right|,

3. (3)

   \underset{0\le t\le 1}{max}\left|x^{\prime }\left(t\right)\right|\le \underset{0\le t\le 1}{max}\left|x^{″}\left(t\right)\right|,\underset{0\le t\le 1}{max}\left|x^{″}\left(t\right)\right|\le \underset{0\le t\le 1}{max}\left|x^{‴}\left(t\right)\right|

where {\delta }_1=\left(1-{\sum }_{i=1}^{m-2}{\beta }_i\left(1-{\xi }_i\right)\right)∕{\sum }_{i=1}^{m-2}{\beta }_i{\xi }_i,{\gamma }_1={\sum }_{i=1}^{m-2}{\beta }_i{\xi }_i∕\left({\sum }_{i=1}^{m-2}{\beta }_i-1\right) are positive constants.

Proof It follows from the same methods as Lemma 3.3 that x(t) is convex on [0, 1]. Taking into account that x'(1) = 0, one can see that x(t) is decreasing on [0, 1] and