Abstract
In this article, by using the fixed point theorem of strictsetcontractions operator, we discuss the existence of positive solution for boundary value problems with pLaplacian
in Banach spaces E, where^{: }θ is the zero element of E. Although the fixed point theorem of strictsetcontractions operator is used extensively in yielding positive solutions for boundary value problems in Banach spaces, this method has not been used to study those boundary value problems with pLaplacian in Banach spaces. So this article may be regarded as an illustration of fixed point theorem of strictsetcontractions operator in a new area.
MSC: 34B18.
Keywords:
boundary value problems; pLaplacian; positive solution; strictsetcontractions1 Introduction
In the last ten years, the theory of ordinary differential equations in Banach spaces has become an important new branch, so boundary value problems in Banach Space has been studied by some researchers, we refer the readers to [19] and the references therein.
For abstract space, it is here worth mentioning that Guo and Lakshmikantham [10] discussed the multiple solutions of the following twopoint boundary value problems (BVP for short) of ordinary differential equations in Banach space
Very recently, by using the fixedpoint principle in cone and the fixedpoint index theory for strictsetcontraction operator, Zhang et al. [11] investigated the existence, nonexistence, and multiplicity of positive solutions for the following nonlinear threepoint boundary value problems of nthorder differential equations in ordered Banach spaces
On the other hand, boundary value problems with pLaplacian have received a lot of attention in recent years. They often occur in the study of the ndimensional pLaplacian equation, nonNewtonian fluid theory, and the turbulent flow of gas in porous medium [1219]. Many studies have been carried out to discuss the existence of solutions or positive solutions and multiple solutions for the local or nonlocal boundary value problems.
However, to the authors' knowledge, this is the first article can be found in the literature on the existence of positive solutions for boundary value problems with pLaplacian in Banach spaces. As is well known, the main difficulty that appears when passing from p = 2 to p ≠ 2 is that, when p = 2, we can change the differential equation into a equivalent integral equation easily and therefore a Green's function exists, so we can easily prove the equivalent integral operator is a strictsetcontractions operator, which is a very important result for discussing positive solution for boundary value problems in Banach space. However, for p ≠ 2, it is impossible for us to find a Green's function in the equivalent integral operator since the differential operator (ϕ_{p}(u'))' is nonlinear. To authors' knowledge, this is the first article to use the fixed point theorem of strictsetcontractions to deal with boundary value problems with pLaplacian in Banach spaces. Such investigations will provide an important platform for gaining a deeper understanding of our environment.
Basic facts about an ordered Banach space E can be found in [1,4]. Here we just recall a few of them. Let the real Banach spaces E with norm  · be partially ordered by a cone P of E, i.e., x ≤ y if and only if y  x ∈ P , and P* denotes the dual cone of P. P is said to be normal if there exists a positive constant N such that θ ≤ x ≤ y implies x ≤ Ny, where θ denotes the zero element of E, and the smallest N is called the normal constant of P (it is clear, N ≥ 1). Set I = 0 [1], (C[I, E], · _{C}) is a Banach space with x_{C }= max_{t∈I }x(t). Clearly, Q = {x ∈ C[I, E]x(t) ≥ θ for t ∈ I} is a cone of the Banach space C[I, E].
For a bounded set S in a Banach space, we denote by α(S) the Kuratowski measure of noncompactness. In this article, we denote by α(·) the Kuratowski measure of noncompactness of a bounded set in E and in C[I, E].
The operator T : D → E(D ⊂ E) is said to be a kset contraction if T : D → E is continuous and bounded and there is a constant k ≥ 0 such that α(T (S)) ≤ kα(S) for any bounded S ⊂ D; a kset contraction with k < 1 is called a strict set contraction.
In this article, we will consider the boundary value problems with pLaplacian
in Banach spaces E, where ϕ_{p}(s) = s^{p1}, p > 1, (ϕ_{p})^{1 }= ϕ_{q}, , θ is the zero element of E, f ∈ C(P, P).
A function u is called a positive solution of BVP (1) and (2) if it satisfies (1) and (2) and u ∈ Q, u(t) ≢ Q.
The main tool of this article is the following fixed point Theorems.
Theorem 1. [5] Let K be a cone in a Banach space E and K_{r, R }= {x ∈ K, r ≤ x ≤ R}, R > r > 0. Suppose that A : K_{r, R }→ K is a strictset contraction such that one of the following two conditions is satisfied:
Then, A has a fixed point x ∈ K_{r, R }such that r ≤ x ≤ R.
2 Preliminaries
Lemma 2.1. If y ∈ C[I, E], then the unique solution of
is
Lemma 2.2. If y ∈ Q, then the unique solution u of the problem (3) and (4) satisfies u(t) ≥ θ, t ∈ I, that is u ∈ Q.
Lemma 2.3. Let , J_{δ }= [δ, 1δ], then for any y ∈ Q, the unique solution u of the problem (3) and (4) satisfies u(t) ≥ δu(s), t ∈ J_{δ}, s ∈ I.
Lemma 2.4. We define an operator T by
Then u is a solution of problem (1) and (2) if and only if u is a fixed point of T.
In the following, the closed balls in spaces E and C[I, E] are denoted by T_{r }= {x ∈ Ex ≤ r} (r > 0) and B_{r }= {x ∈ C[I, E]x_{c }≤ r}, M = sup {f(u): u ∈ Q ⋂ B_{r}}.
Lemma 2.5. Suppose that, for any r > 0, f is uniformly continuous and bounded on P ⋂ T_{r }and there exists a constant L_{r }with
such that
Then, for any r > 0, operator T is a strictsetcontraction on D ⊂ P ⋂ T_{r}.
Proof. Since f is uniformly continuous and bounded on P ⋂ T_{r}, we see from Lemma 2.4 that T is continuous and bounded on Q ⋂ B_{r}. Now, let S ⊂ Q ⋂ B_{r }be given arbitrary, there exists a partition We set α{y : y ∈ S} = α(S)·
By virtue of Lemma 2.4, it is easy to show that the functions {Tyy ∈ S} are uniformly bounded and equicontinuous, and so by [11],
where T (S(t)) = {Tu(t)u ∈ S, t is fixed}⊂ P ⋂ T_{r }for any t ∈ I.
Let u_{1},u_{2 }∈ S_{i},
So, we have
where B = {y(s) s ∈ I, y ∈ S}⊂ P ⋂ T_{r}. Similarly, to the proof of [10], we have α(B) ≤ 2α(S)·It follows from (6), (7), and (8), that
and consequently T is a strictsetcontraction on S ⊂ Q ⋂ B_{r }because of (q1)M^{q2 }L_{r }< 1. □
3 Existence of positive solution to BVP (1) and (2)
In the following, for convenience, we set
where β = 0 or ∞, ψ ∈ P* and ψ = 1.
Furthermore, we list some condition:
(H_{1}): For any r > 0, f is uniformly continuous and bounded on P ⋂ T_{r }and there exists a constant L_{r }with (q  1)M^{q2}L_{r }< 1 such that
Theorem 3.1. Let (H_{1}) hold, cone P be normal. If , then BVP (1) and (2) has at least one positive solution.
Proof. Set
It is clear that K is a cone of the Banach space C[I, E] and K ⊂ Q. By Lemma 2.4, we know T (Q) ⊂ K, and so
We first assume that ϕ_{q}(f^{0}) < 1 Then, there exists a constant such that, for any u ∈ K, we have f(u) ≤ (f^{0}+ε_{1})ϕ_{p}(u), where ε_{1 }> 0 satisfies ϕ_{q}(f^{0 }+ ε_{1}) ≤ 1. Let then for any t ∈ I, u ∈ K, u_{C }= r_{1}, we have
i.e., u ∈ K, u_{C }= r_{1 }implies Tu_{C }≤ u_{C}·
On the other hand, since , there exists such that
Choose , then, for any t ∈ J_{δ}, u ∈ K, u_{C }= r_{2}, we have
then,
i.e., for any u ∈ K, u_{C }= r_{2}, we have
On the other hand, by Lemma 2.5, T is a strict set contraction from into . Consequently, Theorem 1 implies that T has a fixed point in , and the proof is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
This study was sponsored by the National Natural Science Foundation of China (No. (11071014))and the Tianjin City High School Science and Technology Fund Planning Project (No. (20091008)) and Tianyuan Fund of Mathematics in China (No. (11026176)) and Natural Science Foundation of Shandong Province of China (No. (ZR2010AM035)). The authors thank the referee for his/her careful reading of the article and useful suggestions.
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