Multiple positive solutions for a fourth-order integral boundary value problem on time scales

In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications of our results, an example is also given in the article.

Boundary value problem (BVP) for ordinary differential equations arise in different areas of applied mathematics and physics and so on, the existence and multiplicity of positive solutions for such problems have become an important area of investigation in recent years, lots of significant results have been established by using upper and lower solution arguments, fixed point indexes, fixed point theorems and so on (see [1-8] and the references therein). Especially, the existence of positive solutions of nonlinear BVP with integral boundary conditions has been extensively studied by many authors (see [9-18] and the references therein).

However, the corresponding results for BVP with integral boundary conditions on time scales are still very few [19-21]. In this article, we discuss the multiple positive solutions for the following fourth-order system of integral BVP with a parameter on time scales

where a_i, b_i, c_i, d_i ≥ 0, and ρ_i = a_ic_iσ(T) + a_id_i + b_ic_i > 0(i = 1, 2), 0 < λ, μ < +∞, f, , ℝ⁺ = [0, +∞), A_i and B_i are nonnegative and rd-continuous on .

The main purpose of this article is to establish some sufficient conditions for the existence of at least two positive solutions for system (1.1) by using the fixed point theorem of cone expansion and compression type. This article is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using the fixed point theorem of cone expansion and compression type, we establish sufficient conditions for the existence of at least two positive solutions for system (1.1). An illustrative example is given in Section 4.

In this section, we will provide several foundational definitions and results from the calculus on time scales and give some lemmas which are used in the proof of our main results.

A time scale is a nonempty closed subset of the real numbers ℝ.

Definition 2.1. [22]For , we define the forward jump operator by , while the backward jump operator by .

In this definition, we put and , where ∅, denotes the empty set. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if and σ(t) = t, then t is called right-dense, and if and ρ(t) = t, then t is called left-dense. We also need, below, the set , which is derived from the time scale as follows: if has a left-scattered maximum m, then . Otherwise, .

Definition 2.2. [22]Assume that is a function and let . Then x is called differentiable at if there exists a θ ∈ ℝ such that for any given ε > 0, there is an open neighborhood U of t such that

In this case, x^Δ(t) is called the delta derivative of x at t. The second derivative of x(t) is defined by x^ΔΔ(t) = (x^Δ)^Δ(t).

In a similar way, we can obtain the fourth-order derivative of x(t) is defined by x^(4Δ)(t) = (((x^Δ)^Δ)^Δ)^Δ(t).

Definition 2.3. [22]A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist at left-dense points in . The set of rd-continuous functions will be denoted by .

Definition 2.4. [22]A function is called a delta-antiderivative of provide F^Δ(t) = f(t) holds for all . In this case we define the integral of f by

For convenience, we denote , and for i = 1, 2, we set

where

To establish the existence of multiple positive solutions of system (1.1), let us list the following assumptions:

In order to overcome the difficulty due to the dependence of f, g on derivatives, we first consider the following second-order nonlinear system

where A₀ is the identity operator, and

For the proof of our main results, we will make use of the following lemmas.

Lemma 2.1. The fourth-order system (1.1) has a solution (x, y) if and only if the nonlinear system (2.1) has a solution (u, v).

Proof. If (x, y) is a solution of the fourth-order system (1.1), let u(t) = x^ΔΔ(t), v(t) = y^ΔΔ(t), then it follows from the boundary conditions of system (1.1) that

Thus (u, v) = (x^ΔΔ(t), y^ΔΔ(t)) is a solution of the nonlinear system (2.1).

Conversely, if (u, v) is a solution of the nonlinear system (2.1), let x(t) = A₂u(t), y(t) = A₂v(t), then we have

which imply that

Consequently, (x, y) = (A₂u(t), A₂v(t)) is a solution of the fourth-order system (1.1). This completes the proof.

Lemma 2.2. Assume that D₁₁D₂₁ ≠ 1 holds. Then for any h₁ ∈ C(I', ℝ⁺), the following BVP

has a solution

where

Proof. First suppose that u is a solution of system (2.3). It is easy to see by integration of BVP(2.3) that

Integrating again, we can obtain

Let t = σ(T) in (2.4) and (2.5), we obtain

Substituting (2.6) and (2.7) into the second boundary value condition of system (2.3), we obtain

From (2.8) and the first boundary value condition of system (2.3), we have

Substituting (2.9) and (2.10) into (2.5), we have

By (2.11), we get

By (2.12) and (2.13), we get

Substituting (2.14) and (2.15) into (2.11), we have

Conversely, suppose , then

Direct differentiation of (2.17) implies

and

and it is easy to verify that

This completes the proof.

Lemma 2.3. Assume that D₁₂D₂₂ ≠ 1 holds. Then for any h₂ ∈ C(I', ℝ⁺), the following BVP

has a solution

where

Proof. The proof is similar to that of Lemma 2.2 and will omit it here.

Lemma 2.4. Suppose that (H₁) is satisfied, for all t, s ∈ I and i = 1, 2, we have

(i) G_i(t, s) > 0, H_i(t, s) > 0,

(ii) Lim_iG_i(σ(s), s) ≤ H_i(t, s) ≤ M_iG_i(σ(s), s),

(iii) mG_i(σ(s), s) ≤ H_i(t, s) ≤ MG_i(σ(s), s),

where

Proof. It is easy to verify that G_i(t, s) > 0, H_i(t, s) > 0 and G_i(t, s) ≤ G_i(σ(s), s), for all t, s ∈ I. Since

Thus G_i(t, s)/G_i(σ(s), s) ≥ L_i and we have

On the one hand, from the definition of L_i and m_i, for all t, s ∈ I, we have

and on the other hand, we obtain easily that from the definition of M_i, for all t, s ∈ I,

Finally, it is easy to verify that mG_i(σ(s), s) ≤ H_i(t, s) ≤ MG_i(σ(s), s). This completes the proof.

Lemma 2.5. [23]Let E be a Banach space and P be a cone in E. Assume that Ω₁ and Ω₂ are bounded open subsets of E, such that 0 ∈ Ω₁, , and let be a completely continuous operator such that either

(i) ||Tu|| ≤ ||u||, ∀u ∈ P ∩ ∂Ω₁ and ||Tu|| ≥ ||u||, ∀u ∈ P ∩ ∂Ω₂, or

(ii) ||Tu|| ≥ ||u||, ∀u ∈ P ∩ ∂Ω₁ and ||Tu|| ≤ ||u||, ∀u ∈ P ∩ ∂Ω₂

holds. Then T has a fixed point in .

To obtain the existence of positive solutions for system (2.1), we construct a cone P in the Banach space Q = C(I, ℝ⁺) × C(I, ℝ⁺) equipped with the norm by

It is easy to see that P is a cone in Q.

Define two operators T_λ, T_μ : P → C(I, ℝ⁺) by

Then we can define an operator T : P → C(I, ℝ⁺) by

Lemma 2.6. Let (H₁) hold. Then T : P → P is completely continuous.

Proof. Firstly, we prove that T : P → P. In fact, for all (u, v) ∈ P and t ∈ I, by Lemma 2.4(i) and (H₁), it is obvious that T_λ(u, v)(t) > 0, T_μ(u, v)(t) > 0. In addition, we have

which implies . And we have

In a similar way,

Therefore,

This shows that T : P → P.

Secondly, we prove that T is continuous and compact, respectively. Let {(u_k, v_k)} ∈ P be any sequence of functions with ,

from the continuity of f, we know that ||T_λ(u_k, v_k) - T_λ(u, v)|| → 0 as k → ∞. Hence T_λ is continuous.

T_λ is compact provided that it maps bounded sets into relatively compact sets. Let , and let Ω be any bounded subset of P, then there exists r > 0 such that ||(u, v)|| ≤ r for all (u, v) ∈ Ω. Obviously, from (2.16), we know that

so, T_λΩ is bounded for all (u, v) ∈ Ω. Moreover, let

We have

Thus, for any (u, v) ∈ Ω and ∀ε > 0, let , then for t₁, t₂ ∈ I, |t₁ - t₂| < δ, we have

So, for all (u, v) ∈ Ω, T_λΩ is equicontinuous. By Ascoli-Arzela theorem, we obtain that T_λ : P → P is completely continuous. In a similar way, we can prove that T_μ : P → P is completely continuous. Therefore, T : P → P is completely continuous. This completes the proof.

In this section, we will give our main results on multiplicity of positive solutions of system (1.1). In the following, for convenience, we set

where q_i(t), q_j(t) ∈ C_rd(I', ℝ⁺) satisfy

Theorem 3.1. Assume that (H₁) holds. Assume further that

(H₂) there exist a constant R > 0, and two functions p_i(t) ∈ C_rd(I, R₊) satisfying such that

and one of the folloeing conditions is satisfied

(E₁) , ,

(E₂) , ,

(E₃) , μ ∈ (0, N₄),

(E₄) λ ∈ (0, M₄), ,

where

O₁, O₂ satisfy . Then system (1.1) has at least two positive solutions.

Proof. We only prove the case in which (H₂) and (E₁) hold, the other case can be proved similarly. Firstly, from (2.2), we have

Take , and let Ω₁ = {(u, v) ∈ Q; ||(u, v)|| < R₁}. For any t ∈ I, (u, v) ∈ ∂Ω₁ ∩ P, it follows from λ < M₄, μ < N₄ and (H₂) that

and

Consequently, for any (u, v) ∈ ∂Ω₁ ∩ P, we have

Second, from , we can choose ε₁ > 0 such that λf₀ > M₃ + ε₁, then there exists 0 < l₁ < NR₁ such that for any and t ∈ I,

And since

Take

For all (u, v) ∈ Ω₂ ∩ P, where Ω₂ = {(u, v) ∈ Q; ||(u, v)|| < R₂}, we have

Thus, for all (u, v) ∈ Ω₂ ∩ P, we have

Consequently, for all (u, v) ∈ Ω₂ ∩ P, we have

Finally, from μ > N₃/g_∞, we can choose ε₂ > 0 such that μg_∞ > N₃ + ε₂. then, there exists such that for any and t ∈ I,

Take

For all (u, v) ∈ Ω₃ ∩ P, where Ω₃ = {(u, v) ∈ Q; ||(u, v)|| < R₃}, from (3.3), we have

Thus, for all (u, v) ∈ Ω₃ ∩ P, we have

Consequently, for all (u, v) ∈ Ω₃ ∩ P, we have

From (3.2), (3.4), and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u₁, v₁) ∈ P with R₂ ≤ ||(u₁, v₁)|| ≤ R₁. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x₁, y₁). In the same way, from (3.2), (3.6), and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u₂, v₂) ∈ P with R₁ ≤ ||(u₂, v₂)|| ≤ R₃. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x₂, y₂). Above all, system (1.1) has at least two positive solutions. This completes the proof.

Theorem 3.2. Assume that (H₁) holds. Suppose further that

(H₃) there exist a constant R₀ > 0, and two functions w_i(t) ∈ C_rd(I, R₊) (i = 1, 2) satisfying such that

or

Then system (1.1) has at least two positive solutions for each and , where

Proof. We only prove the case in which (3.7) holds. The other case in which (3.8) holds can be proved similarly.

Take

and let . For any t ∈ I, (u, v) ∈ ∂Ω₄ ∩ P, it follows from λ > M₅ and (H₃) that

Consequently, for any (u, v) ∈ ∂Ω₄ ∩ P, we have

From , , we know that , , we can choose ε₃ > 0 such that M₆ - ε₃ > 0, N₆ - ε₃ > 0 and λf⁰ < M₆ - ε₃, μg⁰ < N₆ - ε₃. Then there exists such that for any and t ∈ I,

Take and . Then, for any (u, v) ∈ Ω₅ ∩ P, from(3.1), we have

and

Consequently, for any (u, v) ∈ ∂Ω₅ ∩ P, we have

From , we know that , , we can choose ε₄ > 0 such that M₆ - ε₄ > 0, N₆ - ε₄ > 0 and λf^∞< M₆ - ε₄, μg^∞ < N₆ - ε₄. Then there exists such that for any and t ∈ I,

Take and let . Then, for any (u, v) ∈ Ω₆ ∩ P, we have

and

Consequently, for any (u, v) ∈ ∂Ω₆ ∩ P, we have

From (3.9), (3.10) and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u₁, v₁) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x₁, y₁). In the same way, from (3.9), (3.11) and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u₂, v₂) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x₂, y₂). Above all, system (1.1) has at least two positive solutions. This completes the proof.

Consider the following BVP with integral boundary conditions:

where A₁(t) = B₁(t) = t, A₂(t) = B₂(t) = t/2 and

we choose O₁ = 2, O₂ = 4, R = 1, p₁(t) = 2t, , q₁(t) = q₃(t) = 1. It is easy to check that f₀ = g_∞ = ∞, (H₁), (H₂) and (E₁) are satisfied. Therefore, by Theorem 3.1, system (4.1) has at least two positive solutions for each λ ∈ (0, M₄), μ ∈ (0, N₄).

The authors declare that they have no competing interests.

All authors contributed equally to the manuscript and typed, read and approved the final manuscript.

This study was supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

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