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Multiple positive solutions for a fourth-order integral boundary value problem on time scales
Boundary Value Problems volume 2011, Article number: 59 (2011)
Abstract
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.
1 Introduction
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 i c i σ(T) + a i d i + b i c 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.
2 Preliminaries
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, xΔ(t) is called the delta derivative of x at t. The second derivative of x(t) is defined by xΔΔ(t) = (xΔ)Δ(t).
In a similar way, we can obtain the fourth-order derivative of x(t) is defined by x(4Δ)(t) = (((xΔ)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 FΔ(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) = xΔΔ(t), v(t) = yΔΔ(t), then it follows from the boundary conditions of system (1.1) that
Thus (u, v) = (xΔΔ(t), yΔΔ(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 i G i (σ(s), s) ≤ H i (t, s) ≤ M i G 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 λ Ω 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 λ Ω 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.
3 Main results
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.
4 An example
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).
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Acknowledgements
This study was supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
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Li, Y., Dong, Y. Multiple positive solutions for a fourth-order integral boundary value problem on time scales. Bound Value Probl 2011, 59 (2011). https://doi.org/10.1186/1687-2770-2011-59
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DOI: https://doi.org/10.1186/1687-2770-2011-59