Abstract
In this article, we investigate the multiplicity of positive solutions for a fourthorder 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.
Keywords:
positive solutions; fixed points; integral boundary conditions; time scales1 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 [18] 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 [918] and the references therein).
However, the corresponding results for BVP with integral boundary conditions on time scales are still very few [1921]. In this article, we discuss the multiple positive solutions for the following fourthorder 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 rdcontinuous 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 rightscattered, while if ρ(t) < t, we say that t is leftscattered. Also, if and σ(t) = t, then t is called rightdense, and if and ρ(t) = t, then t is called leftdense. We also need, below, the set , which is derived from the time scale as follows: if has a leftscattered 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 fourthorder derivative of x(t) is defined by x^{(4Δ)}(t) = (((x^{Δ})^{Δ})^{Δ})^{Δ}(t).
Definition 2.3. [22]A function is called rdcontinuous provided it is continuous at rightdense points in and its leftsided limits exist at leftdense points in . The set of rdcontinuous functions will be denoted by .
Definition 2.4. [22]A function is called a deltaantiderivative 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 secondorder nonlinear system
where A_{0 }is the identity operator, and
For the proof of our main results, we will make use of the following lemmas.
Lemma 2.1. The fourthorder 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 fourthorder 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) = A_{2}u(t), y(t) = A_{2}v(t), then we have
which imply that
Consequently, (x, y) = (A_{2}u(t), A_{2}v(t)) is a solution of the fourthorder system (1.1). This completes the proof.
Lemma 2.2. Assume that D_{11}D_{21 }≠ 1 holds. Then for any h_{1 }∈ 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
Direct differentiation of (2.17) implies
and
and it is easy to verify that
This completes the proof.
Lemma 2.3. Assume that D_{12}D_{22 }≠ 1 holds. Then for any h_{2 }∈ 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 (H_{1}) 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 Ω_{1 }and Ω_{2 }are bounded open subsets of E, such that 0 ∈ Ω_{1}, , and let be a completely continuous operator such that either
(i) Tu ≤ u, ∀u ∈ P ∩ ∂Ω_{1 }and Tu ≥ u, ∀u ∈ P ∩ ∂Ω_{2}, or
(ii) Tu ≥ u, ∀u ∈ P ∩ ∂Ω_{1 }and Tu ≤ u, ∀u ∈ P ∩ ∂Ω_{2}
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 (H_{1}) 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 (H_{1}), it is obvious that T_{λ}(u, v)(t) > 0, T_{μ}(u, v)(t) > 0. In addition, 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 t_{1}, t_{2 }∈ I, t_{1 } t_{2} < δ, we have
So, for all (u, v) ∈ Ω, T_{λ}Ω is equicontinuous. By AscoliArzela 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 (H_{1}) holds. Assume further that
(H_{2}) 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
where
O_{1}, O_{2 }satisfy . Then system (1.1) has at least two positive solutions.
Proof. We only prove the case in which (H_{2}) and (E_{1}) hold, the other case can be proved similarly. Firstly, from (2.2), we have
Take , and let Ω_{1 }= {(u, v) ∈ Q; (u, v) < R_{1}}. For any t ∈ I, (u, v) ∈ ∂Ω_{1 }∩ P, it follows from λ < M_{4}, μ < N_{4 }and (H_{2}) that
and
Consequently, for any (u, v) ∈ ∂Ω_{1 }∩ P, we have
Second, from , we can choose ε_{1 }> 0 such that λf_{0 }> M_{3 }+ ε_{1}, then there exists 0 < l_{1 }< NR_{1 }such that for any and t ∈ I,
And since
Take
For all (u, v) ∈ Ω_{2 }∩ P, where Ω_{2 }= {(u, v) ∈ Q; (u, v) < R_{2}}, we have
Thus, for all (u, v) ∈ Ω_{2 }∩ P, we have
Consequently, for all (u, v) ∈ Ω_{2 }∩ P, we have
Finally, from μ > N_{3}/g_{∞}, we can choose ε_{2 }> 0 such that μg_{∞ }> N_{3 }+ ε_{2}. then, there exists such that for any and t ∈ I,
Take
For all (u, v) ∈ Ω_{3 }∩ P, where Ω_{3 }= {(u, v) ∈ Q; (u, v) < R_{3}}, from (3.3), we have
Thus, for all (u, v) ∈ Ω_{3 }∩ P, we have
Consequently, for all (u, v) ∈ Ω_{3 }∩ P, we have
From (3.2), (3.4), and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u_{1}, v_{1}) ∈ P with R_{2 }≤ (u_{1}, v_{1}) ≤ R_{1}. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x_{1}, y_{1}). 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 (u_{2}, v_{2}) ∈ P with R_{1 }≤ (u_{2}, v_{2}) ≤ R_{3}. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x_{2}, y_{2}). Above all, system (1.1) has at least two positive solutions. This completes the proof.
Theorem 3.2. Assume that (H_{1}) holds. Suppose further that
(H_{3}) there exist a constant R_{0 }> 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) ∈ ∂Ω_{4 }∩ P, it follows from λ > M_{5 }and (H_{3}) that
Consequently, for any (u, v) ∈ ∂Ω_{4 }∩ P, we have
From , , we know that , , we can choose ε_{3 }> 0 such that M_{6 } ε_{3 }> 0, N_{6 } ε_{3 }> 0 and λf^{0 }< M_{6 } ε_{3}, μg^{0 }< N_{6 } ε_{3}. Then there exists such that for any and t ∈ I,
Take and . Then, for any (u, v) ∈ Ω_{5 }∩ P, from(3.1), we have
and
Consequently, for any (u, v) ∈ ∂Ω_{5 }∩ P, we have
From , we know that , , we can choose ε_{4 }> 0 such that M_{6 } ε_{4 }> 0, N_{6 } ε_{4 }> 0 and λf^{∞}< M_{6 } ε_{4}, μg^{∞ }< N_{6 } ε_{4}. Then there exists such that for any and t ∈ I,
Take and let . Then, for any (u, v) ∈ Ω_{6 }∩ P, we have
and
Consequently, for any (u, v) ∈ ∂Ω_{6 }∩ P, we have
From (3.9), (3.10) and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u_{1}, v_{1}) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x_{1}, y_{1}). 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 (u_{2}, v_{2}) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x_{2}, y_{2}). 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 A_{1}(t) = B_{1}(t) = t, A_{2}(t) = B_{2}(t) = t/2 and
we choose O_{1 }= 2, O_{2 }= 4, R = 1, p_{1}(t) = 2t, , q_{1}(t) = q_{3}(t) = 1. It is easy to check that f_{0 }= g_{∞ }= ∞, (H_{1}), (H_{2}) and (E_{1}) are satisfied. Therefore, by Theorem 3.1, system (4.1) has at least two positive solutions for each λ ∈ (0, M_{4}), μ ∈ (0, N_{4}).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
Acknowledgements
This study was supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
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