Abstract
In this article, we study the following mpoint boundary value problem on time scales,
where is a time scale such that , ϕ_{p}(s) = s^{p2}s,p > 1,h ∈ C_{ld}((0, T), (0, +∞)), and f ∈ C([0,+∞), (0,+∞)), . By using several wellknown fixed point theorems in a cone, the existence of at least one, two, or three positive solutions are obtained. Examples are also given in this article.
AMS Subject Classification: 34B10; 34B18; 39A10.
Keywords:
positive solutions; cone; multipoint; boundary value problem; time scale1 Introduction
The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Further, the study of time scales has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, epidemic models, etc. [2].
Multipoint boundary value problems of ordinary differential equations (BVPs for short) arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross section and composed of N parts of different densities can be set up as a multipoint boundary value problem [3]. Many problems in the theory of elastic stability can be handled by the method of multipoint problems [4]. Small size bridges are often designed with two supported points, which leads into a standard twopoint boundary value condition and large size bridges are sometimes contrived with multipoint supports, which corresponds to a multipoint boundary value condition [5]. The study of multipoint BVPs for linear secondorder ordinary differential equations was initiated by Il'in and Moiseev [6]. Since then many authors have studied more general nonlinear multipoint BVPs, and the multipoint BVP on time scales can be seen as a generalization of that in ordinary differential equations.
Recently, the existence and multiplicity of positive solutions for nonlinear differential equations on time scales have been studied by some authors [711], and there has been some merging of existence of positive solutions to BVPs with pLaplacian on time scales [1219].
He [20] studied
subject to one of the following boundary conditions
where . By using a double fixedpoint theorem, the authors get the existence of at least two positive solutions to BVP (1.1) and (1.2).
Anderson [21] studied
subject to one of the following boundary conditions
by using a functionaltype cone expansioncompression fixedpoint theorem, the author gets the existence of at least one positive solution to BVP (1.3), (1.4) and BVP (1.3), (1.5).
However, to the best of the authors' knowledge, up to now, there are few articles concerned with the existence of mpoint boundary value problem with pLaplacian on time scales. So, in this article, we try to fill this gap. Motivated by the article mentioned above, in this article, we consider the following mpoint BVP with onedimensional pLaplacian,
where ϕ_{p}(s) = s^{p2}s,p > 1,h ∈ C_{ld}((0,T), (0, +∞)), . δ, β_{i }> 0, i = 1,..., m  2.
We will assume throughout
(S1) h ∈ C_{ld }((0, T), [0, ∞)) such that ;
(S2) f ∈ C([0, ∞), (0, ∞)), f ≢ 0 on ;
2 Preliminaries
In this section, we will give some background materials on time scales.
Definition 2.1. [7,22] For and , define the forward jump operator σ and the backward jump operator ρ, respectively,
for all . If σ(t) > t, t is said to be right scattered, and if ρ(r) < r, r is said to be left scattered. If σ(t) = t, t is said to be right dense, and if ρ(r) = r, r is said to be left dense. If has a right scattered minimum m, define ; Otherwise set . The backward graininess is defined by
If has a left scattered maximum M, define ; Otherwise set . The forward graininess is defined by
Definition 2.2. [7,22] For and , we define the "Δ" derivative of x(t), x^{Δ}(t), to be the number (when it exists), with the property that, for any ε > 0, there is neighborhood U of t such that
for all s ∈ U. For and , we define the "∇" derivative of x(t),x^{Δ }(t), to be the number(when it exists), with the property that, for any ε > 0, there is a neighborhood V of t such that
for all s ∈ V.
Definition 2.3. [22] If F^{Δ }(t) = f(t), then we define the "Δ" integral by
If F^{∇ }(t) = f(t), then we define the "∇" integral by
Lemma 2.1. [23]The following formulas hold:
Lemma 2.2. [7, Theorem 1.75 in p. 28] If f ∈ C_{rd }and , then
According to [23, Theorem 1.30 in p. 9], we have the following lemma, which can be proved easily. Here, we omit it.
Lemma 2.3. Let and f ∈ C_{ld}.
where the integral on the right is the usual Riemann integral from calculus.
(ii) If [a, b] consists of only isolated points, then
In what follows, we list the fixed point theorems that will be used in this article.
Theorem 2.4. [24]Let E be a Banach space and P ⊂ E be a cone. Suppose Ω_{1}, Ω_{2 }⊂ E open and bounded, . Assume is completely continuous. If one of the following conditions holds
(i) ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω_{1 }∩ P, ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω_{2 }∩ P;
(ii) ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω_{1 }∩ P, ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω_{2 }∩ P.
Then, A has a fixed point in .
Theorem 2.5. [25]Let P be a cone in the real Banach space E. Set
If α and γ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P with θ(0) = 0 such that for some positive constants r, M,
for all . Further, suppose there exists positive numbers a < b < r such that
If is completely continuous operator satisfying
(i) γ(Au) > r for all u ∈ ∂P(γ, r);
(ii) θ(Au) < b for all u ∈ ∂P(θ, r);
(iii) and α(Au) > a for all u ∈ ∂P(α, a).
Then, A has at least two fixed points u_{1 }and u_{2 }such that
Let a, b, c be constants, P_{r }= {u ∈ P : ∥u∥ < r}, P(ψ, b, d) = {u ∈ P : a ≤ ψ(u), ∥u∥ ≤ b}.
Theorem 2.6. [26]Let be a completely continuous map and ψ be a nonnegative continuous concave functional on P such that for , there holds ψ(u) ≤ ∥u∥. Suppose there exist a, b, d with 0 < a < b < d ≤ c such that
(i) and ψ(Au) > b for all u ∈ P(ψ, b, d);
(iii) ψ(Au) > b for all u ∈ P(ψ, b, d) with ∥Au∥ > d.
Then, A has at least three fixed points u_{1},u_{2}, and u_{3 }satisfying
Let the Banach space E = C_{ld}[0, T] be endowed with the norm ∥u∥ = sup_{t ∈ [0,T] }u(t), and cone P ⊂ E is defined as
It is obvious that ∥u∥ = u(T) for u ∈ P. Define A : P → E as
for t ∈ [0, T].
In what follows, we give the main lemmas which are important for getting the main results.
Lemma 2.7. A : P → P is completely continuous.
Proof. First, we try to prove that A : P → P.
Thus, (Au)^{Δ }(T) = 0 and by Lemma 2.1 we have (Au)^{Δ∇ }(t) = h(t)f(t, u(t)) ≤ 0 for t ∈ (0, T). Consequently, A : P → P.
By standard argument we can prove that A is completely continuous. For more details, see [27]. The proof is complete.
Lemma 2.8. For u ∈ P, there holds for t ∈ [0,T].
Proof. For u ∈ P, we have u^{Δ∇ }(t) ≤ 0, it follows that u^{Δ }(t) is nonincreasing. Therefore, for 0 < t < T,
and
thus
Combining (2.1) and (2.3) we have
as u(0) ≥ 0, it is immediate that
The proof is complete.
3 Existence of at least one positive solution
First, we give some notations. Set
Theorem 3.1. Assume in addition to (S1) and (S2), the following conditions are satisfied, there exists such that
Then, BVP (1.6) has at least one positive solution.
Proof. Cone P is defined as above. By Lemma 2.7 we know that A : P → P is completely continuous. Set Ω_{r }= {u ∈ E, ∥u∥ < r}. In view of (H1), for u ∈ ∂ Ω_{r }∩ P,
which means that for u ∈ ∂Ω_{r }∩ P, Au ≤ u.
On the other hand, for u ∈ P, in view of Lemma 2.8, there holds , for t ∈ [ξ_{1}, T]. Denote Ω_{ρ }= {u ∈ E, ∥u∥ < ρ}. Then for u ∈ ∂Ω_{ρ }∩ P, considering (H2), we have
which implies that for u ∈ ∂Ω_{ρ}∩P, ∥Au∥ ≥ ∥u∥ Therefore, the immediate result of Theorem 2.4 is that A has at least one fixed point u ∈ (Ω_{ρ}\Ω_{r}) ∩ P. Also, it is obvious that the fixed point of A in cone P is equivalent to the positive solution of BVP (1.6), this yields that BVP (1.6) has at least one positive solution u satisfies r ≤ ∥u∥ ≤ ρ. The proof is complete.
Here is an example.
Example 3.2. Let . Consider the following four point BVP on time scale .
where
and h(t) = 1, T = 4, ξ_{1 }= 2, ξ_{2 }= 3, δ = 2, β_{1 }= β_{2 }= 1,p = q = 2. In what follows, we try to calculate Λ, B. By Lemmas 2.2 and 2.3, we have
where
Thus, if all the conditions in Theorem 3.1 satisfied, then BVP (3.1) has at least one positive solution lies between 100 and 1000.
4 Existence of at least two positive solutions
In this section, we will apply fixed point Theorem 2.5 to prove the existence of at least two positive solutions to the nonlinear BVP (1.6).
and define the increasing, nonnegative, continuous functionals γ, θ,α on P by
We can see that, for u ∈ P, there holds
In addition, Lemma 2.8 implies that which means that
We also see that
For convenience, we give some notations,
Theorem 4.1. Assume in addition to (S1), (S2) there exist positive constants such that the following conditions hold
(H3) f(t, u) > ϕ_{p}(c/M) for t ∈ [ξ_{1},T] u ∈ [c,Tc/ξ_{1}];
(H4) f(t, u) < ϕ_{p}(b/K) for t ∈ [0,ξ_{m2}], u ∈ [b,Tb/ξ_{m2}];
(H5) f(t, u) > ϕ_{p}(a/L) for t ∈ [η,T], u ∈ [a,Ta/η].
Then BVP (1.6) has at least two positive solutions u_{1 }and u_{2 }such that
Proof. From Lemma 2.7 we know that A : P(γ, c) → P is completely continuous. In what follows, we will prove the result step by step.
Step one: To verify (i) of theorem 2.5 holds.
We choose u ∈ ∂P(γ,c), then . This implies that u(t) ≥ c for t ∈ [ξ_{1},T], considering that , we have
As a consequence of (H3),
Since Au ∈ P, we have
Thus, (i) of Theorem 2.5 is satisfied.
Step two: To verify (ii) of Theorem 2.5 holds.
Let u ∈ ∂P(θ,b), then , this implies that 0 ≤ u(t) ≤ b, t ∈ [0,ξ_{m2}] and since u ∈ P, we have ∥u∥ = u(T), note that . So,
From (H4) we know that for t ∈ [0, ξ_{m2}] and so
Thus, (ii) of Theorem 2.5 holds.
Step three: To verify (iii) of Theorem 2.5 holds.
Choose , obviously, u_{0}(t) ∈ P(α, a) and , thus .
Now, let u ∈ ∂P(α, a), then, α(u) = min_{t∈[η,T] }u(t) = u(η) = a. Recalling that . Thus, we have
From assumption (H5) we know that
and so
Therefore, all the conditions of Theorem 2.5 are satisfied, thus A has at least two fixed points in P(γ,c), which implies that BVP (1.6) has at least two positive solutions u_{1},u_{2 }which satisfies (4.1). The proof is complete.
Example 4.2. Let . Consider the following four point boundary value problem on time scale .
where
and h(t) = t, T = 8, ξ_{1 }= 1, ξ_{2 }= 2, δ = 1, β_{1 }= 1, β_{2 }= 2,p = 3/2, q = 3. In what follows, we try to calculate K, M, L. By Lemmas 2.2 and 2.3, we have
Let a = 10^{6}, b = 10^{8}, c = 10^{9}, then we have
(i) , for t ∈ [1, 8], u ∈ [10^{9}, 8 × 10^{9}];
(ii) , for t ∈ [0, 2], u ∈ [10^{8}, 4 × 10^{8}];
(iii) , for t ∈ [4, 8], u ∈ [10^{6}, 2 × 10^{6}].
Thus, if all the conditions in Theorem 4.1 are satisfied, then BVP (4.2) has at least two positive solutions satisfying (4.1).
5 Existence of at least three positive solutions
Let , then 0 < ψ(u) ≤ ∥u∥. Denote
In this section, we will use fixed point Theorem 2.6 to get the existence of at least three positive solutions.
Theorem 5.1. Assume that there exists positive number d, ν, g satisfying , such that the following conditions hold.
(H6) f(t, u) < ϕ_{p}(d/R), t ∈ [0,T],u ∈ [0,d];
(H7) f(t, u) > ϕ_{p}(ν/D), t ∈ [ξ_{1}, T], u ∈ [ν, Tυ/ξ_{1}];
(H8) f(t, u) ≤ ϕ_{p}(g/R), t ∈ [0,T],u ∈ [0,g],
then BVP (1.6) has at least three positive solutions u_{1}, u_{2}, u_{3 }satisfying
Proof. From Lemma 2.8 we know that A : P → P is completely continuous. Now we only need to show that all the conditions in Theorem 2.6 are satisfied.
Thus, . Similarly, by (H6), we can prove (ii) of Theorem 2.6 is satisfied.
In what follows, we try to prove that (i) of theorem 2.6 holds. Choose , obviously, ψ(u_{1}) > ν, thus . For u ∈ P(ψ,ν,Tν/ξ_{1}),
It remains to prove (iii) of Theorem 2.6 holds. For u ∈ P(ψ, ν, Tυ/ξ_{1}), with ∥Au∥ > Tν/ξ_{1}, in view of Lemma 2.8, there holds , which implies that (iii) of Theorem 2.6 holds.
Therefore, all the conditions in Theorem 2.6 are satisfied. Thus, BVP (1.6) has at least three positive solutions satisfying (5.1). The proof is complete.
Example 5.2. Let . Consider the following four point boundary value problem on time scale .
where
and h(t) = e^{t}, T = 2, ξ_{1 }= 1/2, ξ_{2 }= 1, δ = 3, β_{1 }= 2, β_{2 }= 3, p = 4, q = 4/3. In what follows, we try to calculate D, R. By Lemmas 2.2 and 2.3, we have
Let d = 40, ν = 50, g = 400, then we have
(i) f(t, u) < 7.027 = (40/20.8832)^{3 }= ϕ_{p}(d/R), for t ∈ [0, 2], u ∈ [0, 40];
(ii) f(t, u) > 23.2375 = (50/17.5216)^{3 }= ϕ_{p}(ν/D), for t ∈ [1/2, 2], u ∈ [50, 200];
(iii) f(t, u) < 7027.305 = (400/20.8832)^{3 }= ϕ_{p}(g/R), for t ∈ [0, 2], u ∈ [0, 400].
Thus, if all the conditions in Theorem 5.1 are satisfied, then BVP (5.2) has at least three positive solutions satisfying (5.1).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
WG and HL conceived of the study, and participated in its coordination. JZ drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors were very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The study was supported by Preresearch project and Excellent Teachers project of the Fundamental Research Funds for the Central Universities (2011YYL079, 2011YXL047).
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