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Multi-point boundary value problems for an increasing homeomorphism and positive homomorphism on time scales
Advances in Difference Equations volume 2012, Article number: 13 (2012)
Abstract
Investigated here are interesting aspects of the positive solutions for two kinds of m-point boundary value problems for an increasing homeomorphism and positive homo-morphism on time scales. By using the Avery-Peterson fixed point theorem, we obtain the existence of at least three positive solutions for these problems. The interesting point is that the nonlinear term depends on the first-order delta-derivative explicitly.
1 Introduction
With the development of boundary value problems for differential equations [1–5], difference equations [6, 7], and the theory of time scales [8–12], the existence of solutions for boundary value problems on time scales have attracted many author's attention. Recently in [13], the authors considered positive solutions for boundary value problem of the following second-order dynamic equations on time scales
where ϕ: R → R is an increasing homeomorphism and positive homomorphism and ϕ(0) = 0. Here a projection ϕ: R → R is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:
-
(i)
if x ≤ y, then ϕ(x) ≤ ϕ(y), ∀x, y ∈ R;
-
(ii)
ϕ is a continuous bijection and its inverse mapping is also continuous;
-
(iii)
ϕ (xy) = ϕ (x)ϕ (y), ∀x, y ∈ R.
By using a fixed point theorem, they obtained an existence theorem for positive solutions for this problem. In [14], Han and Jin established existence results of positive solutions for problem (1.1, 1.2) by using fixed point index theory. Sang et al. [15] considered the problem
By using a fixed point index theorem, the existence results of positive solutions for this problem were established.
However, the nonlinear terms f in [13–15] does not depend on the first order delta derivative. It is well-known that many difficulties occur when the nonlinear term f depends on the first order delta derivative explicitly. To the author's best knowledge, positive solutions are not available for the case when the boundary value problem for an increasing homeomorphism and positive homomorphism on a time scale in which the nonlinear term depends on the first order delta derivative. This article will fill this gap in the literature. In this article, we consider the existence of positive solutions for the second-order nonlinear m-point dynamic equation on a time scale with an increasing homeomorphism and positive homomorphism,
where for i ∈ {1,2,...,m -2}, T is a time scale.
We will assume that the following conditions are satisfied throughout this:
(H1) α i , β i ∈ [0, +∞) satisfy .
(H2) f ∈ [0, T] × [0, ∞) × R → [0, ∞) is continuous.
Our main results will depend on an application of a fixed point theorem due to Avery and Peterson which deals with fixed points of a cone-preserving operator defined on an ordered Banach space. By using analysis techniques and the Avery-Peterson fixed point theorem, we obtain sufficient conditions for existence of at least three positive solutions of the problems (1.5, 1.6) and (1.5, 1.7).
2 Preliminaries
First we present some basic definitions on time scales which can be found in Atici and Guseinov [8].
A time scale T is a closed nonempty subset of R. For t < sup T and r > inf T, we define the forward jump operator σ and the backward jump operator ρ respectively by
for all t ∈ T. If σ(t) > t, t is said to be right scattered, and if σ(t) = t, t is said to be right dense. If ρ(t) < t, t is said to be left scattered, and if ρ(t) = t, t is said to be left dense. A function f is left-dense continuous, if f is continuous at each left dense point in T and its right-sided limits exists at each right dense points.
For u : T → R and t ∈ T, we define the delta derivative of u(t), uΔ(t), to be the number (when it exists), with the property that for each ε > 0, there is a neighborhood U of t such that
for all s ∈ U.
For u: T → R and t ∈ T, we define the nabla derivative of u(t), u∇(t), to be the number (when it exists), with the property that for each ε > 0, there is a neighborhood U of t such that
for all s ∈ U.
We present here the necessary definitions of the theory of cones in Banach spaces and the Avery-Peterson fixed point theorem.
Definition 2.1. Let E be a real Banach space over R. A nonempty convex closed set P ⊂ E is said to be a cone provided that:
-
(1)
au ∈ P, for all u ∈ P, a ≥ 0;
-
(2)
u, -u ∈ P implies u = 0.
Definition 2.2. An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.
Definition 2.3. The map α is said to be a nonnegative continuous convex functional on a cone P of a real Banach space E provided that α : P → [0, + ∞) is continuous and
Definition 2.4. The map β is said to be a nonnegative continuous concave functional on a 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. [16] 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 on P, and ψ be a nonnegative continuous functional on P satisfying
such that for some positive numbers l and d,
for all . Suppose is completely continuous and there exist positive numbers a, b, c with a < b such that
(S 1) 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) and ψ(Tx) < a for x ∈ R(γ, ψ, a, d) with ψ(x) = a.
Then T has at least three fixed points such that:
3 Positive solutions for problem (1.5, 1.6)
Lemma 3.1. [13] Suppose that condition (H1) holds, then the boundary value problem
has the unique solution
where
Lemma 3.2. Suppose that condition (H1) holds, for h ∈ C ld [0, T] and h(t) ≥ 0, the unique solution of problem (3.1, 3.2) satisfies
(1)u(t) ≥ 0, t ∈ [0, T].
-
(2)
inft ∈ [0,T]u(t) ≥ δ maxt ∈ [0,T]|u(t)|, where
-
(3)
, where
Proof. Parts (1) and (2) were established in [13]. We give the proof of (3). It is easy to check that
For the concavity of u and the boundary condition, we get
This together with conclusion (2) ensures that conclusion (3) is satisfied.
Let E be the real Banach space E = CΔ[0, σ(T)] to be the set of all Δ- differential functions with continuous Δ-derivative on [0, σ(T)] with the norm
where
We 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 P by
By Lemmas 3.3 and 3.4, the functionals defined above satisfy
Therefore condition (2.2) of Lemma 2.1 is satisfied.
Define an operator F : P → E by
To present our main results, we assume there exist constants 0 < a, b, c, d with a < b < d such that
(A1) ;
(A2) ;
(A3) .
Theorem 3.1. Under the assumptions (A1)-(A3), the boundary value problem (1.5)-(1.6) has at least three positive solutions u1, u2, u3 satisfying
Proof.
It is easy to check that problem (1.5), (1.6) has a solution u(t) if and only if u is a fixed
point of operator F.
If , then . Thus
Then,
Hence .
To check condition (S1) of Lemma 2.1, we choose . It's easy to see and . So .
If u ∈ P(γ, θ, α, b, c, d), we have . From assumption (A2), we have
Thus,
so α(Fu) > b, ∀u ∈ P(γ, θ, α, b, b/δ, d).
Second, with (4.1), we have α(Fu) ≥ δθ(Fu) > δb/δ = b for all u ∈ P(γ, α, b, d) with θ(Fu) > b/δ. Thus, condition (S2) of Lemma 2.1 is satisfied.
Finally we show that (S3) also holds. Clearly, as ψ(0) = 0 < a, we see 0 ∉ R(γ, ψ, a, d). Suppose that x ∈ R(γ, ψ, a, d) with ψ(u) = a, then assumption (A3) holds. then
So we verify that condition (S3) of Lemma 2.1 is satisfied. Thus, an application of Lemma 2.1 implies that the boundary value problem (1.5)-(1.6) has at least three positive solutions u1, u2, u3 satisfying (3.3).
4 Positive solutions for problem (1.5, 1.7)
In this section, we present the existence of positive solutions for problem (1.5, 1.7).
Lemma 4.1. [15] Suppose that condition (H1) holds, then boundary value problem
has the unique solution
where
Lemma 4.2. Suppose that condition (H1) holds, for h ∈ C ld [0,T] and h(t) ≥ 0, the unique solution of problem (4.1, 4.2) satisfies
-
(1)
u(t) ≥ 0, t ∈ [0,T]
-
(2)
inft ∈ [0,T]u(t) ≥ δ1 maxt ∈ [0,T]|u(t)|, where is a constant.
-
(3)
, where . is a constant.
Proof. Parts (1) and (2) are established in [15]. It is easy to check that
For the concavity of u and the boundary condition, we get
This together with conclusion (2) ensures that conclusion (3) is satisfied. We define the cone P1 ⊂ E by
Define an operator G : P → E by
To present our main results, we assume there exist constants 0 < a1, b1, c 1, d with a1 < b1 < d1 such that
A4) ;
A5) ;
A6) .
Theorem 4.1. Under the assumptions (A4)-(A6), the boundary value problem (1.5), (1.7) has at least three positive solutions u1, u2, u3 satisfying
The proof of Theorem 4.1 is similar with the Theorem 3.1 and is omitted here.
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Acknowledgements
This study was supported by the Anhui Provincial Natural Science Foundation (10040606Q50), National Natural Science Foundation of China (No. 11071164), Shanghai Natural Science Foundation (No. 10ZR1420800) and Shanghai Leading Academic Discipline Project (No. S30501).
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Yang, L., Zhang, W. Multi-point boundary value problems for an increasing homeomorphism and positive homomorphism on time scales. Adv Differ Equ 2012, 13 (2012). https://doi.org/10.1186/1687-1847-2012-13
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DOI: https://doi.org/10.1186/1687-1847-2012-13
Keywords
- boundary value problem
- time scale
- fixed point
- cone
- increasing homeo-morphism and positive homomorphism