# Existence of positive solutions for nonlinear m-point boundary value problems on time scales

Junfang Zhao1*, Hairong Lian1 and Weigao Ge2

### Author affiliations

1 School of Science, China University of Geosciences, Beijing 100083, P.R. China

2 Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China

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Boundary Value Problems 2012, 2012:4  doi:10.1186/1687-2770-2012-4

 Received: 4 May 2011 Accepted: 17 January 2012 Published: 17 January 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we study the following m-point boundary value problem on time scales,

where is a time scale such that , ϕp(s) = |s|p-2s,p > 1,h Cld((0, T), (0, +∞)), and f C([0,+∞), (0,+∞)), . By using several well-known 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; multi-point; boundary value problem; time scale

### 1 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 multi-point boundary value problem [3]. Many problems in the theory of elastic stability can be handled by the method of multi-point problems [4]. Small size bridges are often designed with two supported points, which leads into a standard two-point boundary value condition and large size bridges are sometimes contrived with multi-point supports, which corresponds to a multi-point boundary value condition [5]. The study of multi-point BVPs for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [6]. Since then many authors have studied more general nonlinear multi-point BVPs, and the multi-point 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 [7-11], and there has been some merging of existence of positive solutions to BVPs with p-Laplacian on time scales [12-19].

He [20] studied

(1.1)

subject to one of the following boundary conditions

(1.2)

where . By using a double fixed-point theorem, the authors get the existence of at least two positive solutions to BVP (1.1) and (1.2).

Anderson [21] studied

(1.3)

subject to one of the following boundary conditions

(1.4)

(1.5)

by using a functional-type cone expansion-compression fixed-point 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 m-point boundary value problem with p-Laplacian 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 m-point BVP with one-dimensional p-Laplacian,

(1.6)

where ϕp(s) = |s|p-2s,p > 1,h Cld((0,T), (0, +∞)), . δ, βi > 0, i = 1,..., m - 2.

We will assume throughout

(S1) h ∈ Cld ((0, T), [0, ∞)) such that ;

(S2) f C([0, ∞), (0, ∞)), f ≢ 0 on ;

(S3) By ϕq we denote the inverse to ϕp, where ;

(S4) By t ∈ [a, b] we mean that , where 0 ≤ a b T.

### 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), xΔ(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),xΔ (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 FΔ (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:

(i) ,

(ii) ,

(iii) ,

(iv) .

Lemma 2.2. [7, Theorem 1.75 in p. 28] If f Crd 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 Cld.

(i) If , then

where the integral on the right is the usual Riemann integral from calculus.

(ii) If [a, b] consists of only isolated points, then

(iii) If , where h > 0, then

(iv) If , 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 1, Ω2 E open and bounded, . Assume is completely continuous. If one of the following conditions holds

(i) ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω1 P, ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω2 P;

(ii) ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω1 P, ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω2 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 u1 and u2 such that

Let a, b, c be constants, Pr = {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);

(ii) ∥Au∥ < a for all ;

(iii) ψ(Au) > b for all u P(ψ, b, d) with Au∥ > d.

Then, A has at least three fixed points u1,u2, and u3 satisfying

Let the Banach space E = Cld[0, T] be endowed with the norm ∥u∥ = supt ∈ [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 uΔ∇ (t) ≤ 0, it follows that uΔ (t) is non-increasing. Therefore, for 0 < t < T,

(2.1)

and

(2.2)

thus

(2.3)

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

(H1) ;

(H2) .

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 Ωr = {u E, ∥u∥ < r}. In view of (H1), for u r P,

which means that for u ∈ ∂Ωr P, ||Au|| ≤ ||u||.

On the other hand, for u P, in view of Lemma 2.8, there holds , for t ∈ [ξ1, T]. Denote Ωρ = {u E, ∥u∥ < ρ}. Then for u ∈ ∂Ωρ P, considering (H2), we have

which implies that for u ρP, ∥Au∥ ≥ ∥u∥ Therefore, the immediate result of Theorem 2.4 is that A has at least one fixed point u ∈ (Ωρ\Ωr) ∩ 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 .

(3.1)

where

and h(t) = 1, T = 4, ξ1 = 2, ξ2 = 3, δ = 2, β1 = β2 = 1,p = q = 2. In what follows, we try to calculate Λ, B. By Lemmas 2.2 and 2.3, we have

where

Thus, . Let . Then, we have

(i)

(ii)

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).

Fix such that

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) > ϕp(c/M) for t ∈ [ξ1,T] u ∈ [c,Tc/ξ1];

(H4) f(t, u) < ϕp(b/K) for t ∈ [0,ξm-2], u ∈ [b,Tb/ξm-2];

(H5) f(t, u) > ϕp(a/L) for t ∈ [η,T], u ∈ [a,Ta/η].

Then BVP (1.6) has at least two positive solutions u1 and u2 such that

(4.1)

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 ∈ [ξ1,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,ξm-2] and since u P, we have ∥u∥ = u(T), note that . So,

From (H4) we know that for t ∈ [0, ξm-2] and so

Thus, (ii) of Theorem 2.5 holds.

Step three: To verify (iii) of Theorem 2.5 holds.

Choose , obviously, u0(t) ∈ P(α, a) and , thus .

Now, let u ∂P(α, a), then, α(u) = mint∈[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 u1,u2 which satisfies (4.1). The proof is complete.

Example 4.2. Let . Consider the following four point boundary value problem on time scale .

(4.2)

where

and h(t) = t, T = 8, ξ1 = 1, ξ2 = 2, δ = 1, β1 = 1, β2 = 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 = 106, b = 108, c = 109, then we have

(i) , for t ∈ [1, 8], u ∈ [109, 8 × 109];

(ii) , for t ∈ [0, 2], u ∈ [108, 4 × 108];

(iii) , for t ∈ [4, 8], u ∈ [106, 2 × 106].

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) < ϕp(d/R), t ∈ [0,T],u ∈ [0,d];

(H7) f(t, u) > ϕp(ν/D), t ∈ [ξ1, T], u ∈ [ν, Tυ/ξ1];

(H8) f(t, u) ≤ ϕp(g/R), t ∈ [0,T],u ∈ [0,g],

then BVP (1.6) has at least three positive solutions u1, u2, u3 satisfying

(5.1)

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.

For . By (H8), one has

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(u1) > ν, thus . For u P(ψ,ν,Tν/ξ1),

It remains to prove (iii) of Theorem 2.6 holds. For u P(ψ, ν, Tυ/ξ1), with ∥Au∥ > /ξ1, 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 .

(5.2)

where

and h(t) = et, T = 2, ξ1 = 1/2, ξ2 = 1, δ = 3, β1 = 2, β2 = 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 = ϕp(d/R), for t ∈ [0, 2], u ∈ [0, 40];

(ii) f(t, u) > 23.2375 = (50/17.5216)3 = ϕp(ν/D), for t ∈ [1/2, 2], u ∈ [50, 200];

(iii) f(t, u) < 7027.305 = (400/20.8832)3 = ϕp(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 Pre-research project and Excellent Teachers project of the Fundamental Research Funds for the Central Universities (2011YYL079, 2011YXL047).

### References

1. Hilger, S: Analysis on measure chains--a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

2. Agarwal, RP, Bohner, M, Li, WT: Nonoscillation and Oscillation Theory for Functional Differential Equations. Pure and Applied Mathematics Series, Dekker, FL (2004)

3. Moshinsky, M: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Bol Soc Mat Mexicana. 7, 10–25 (1950)

4. Timoshenko, S: Theory of Elastic Stability. McGraw-Hill, New York (1961)

5. Zou, Y, Hu, Q, Zhang, R: On numerical studies of multi-point boundary value problem and its fold bifurcation. Appl Math Comput. 185, 527–537 (2007). Publisher Full Text

6. Il'in, VA, Moiseev, EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Diff Equ. 23, 979–987 (1987)

7. Agarwal, RP, Bohner, M: Basic calculus on time scales and some of its applications. Results Math. 35, 3–22 (1999)

8. Agarwal, RP, O'Regan, D: Nonlinear boundary value problems on time scales. Nonlinear Anal. 44, 527–535 (2001). Publisher Full Text

9. Anderson, D: Solutions to second-order three-point problems on time scales. J Diff Equ Appl. 8, 673–688 (2002). Publisher Full Text

10. Kaufmann, ER: Positive solutions of a three-point boundary value problem on a time scale. Electron J Diff Equ. 82, 1–11 (2003)

11. Chyan, CJ, Henderson, J: Twin solutions of boundary value problems for differential equations on measure chains. J Comput Appl Math. 141, 123–131 (2002). Publisher Full Text

12. Agarwal, RP, Lü, HSh, O'Regan, D: Eigenvalues and the one-dimensional p-Laplacian. J Math Anal Appl. 266, 383–400 (2002). Publisher Full Text

13. Sun, H, Tang, L, Wang, Y: Eigenvalue problem for p-Laplacian three-point boundary value problem on time scales. J Math Anal Appl. 331, 248–262 (2007). Publisher Full Text

14. Geng, F, Zhu, D: Multiple results of p-Laplacian dynamic equations on time scales. Appl Math Comput. 193, 311–320 (2007). Publisher Full Text

15. He, Z, Jiang, X: Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales. J Math Anal Appl. 321, 911–920 (2006). Publisher Full Text

16. Hong, S: Triple positive solutions of three-point boundary value problems for p-Laplacian dynamic equations. J Comput Appl Math. 206, 967–976 (2007). Publisher Full Text

17. Graef, J, Kong, L: First-order singular boundary value problems with p-Laplacian on time scales. J Diff Equ Appl. 17, 831–839 (2011). Publisher Full Text

18. Anderson, DR: Existence of solutions for a first-order p-Laplacian BVP on time scales. Nonlinear Anal. 69, 4521–4525 (2008). Publisher Full Text

19. Goodrich, CS: Existence of a positive solution to a first-order p-Laplacian BVP on a time scale. Nonlinear Anal. 74, 1926–1936 (2011). Publisher Full Text

20. He, ZM: Double positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales. Appl Anal. 84, 377–390 (2005). Publisher Full Text

21. Anderson, DR: Twin n-point boundary value problem. Appl Math Lett. 17, 1053–1059 (2004). Publisher Full Text

22. Bohner, M, Peterson, A: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)

23. Bohner, M, Peterson, A: Dynamic Equations on Time Scales. An Introduction with Applications, Birkhauser, Boston (2001)

24. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)

25. Avery, RI, Henderson, J: Two positive fixed points of nonlinear operators on ordered Banach spaces. Comm Appl Nonlinear Anal. 8, 27–36 (2001)

26. Leggett, RW, Williams, LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ Math J. 28, 673–688 (1979). Publisher Full Text

27. Zhao, J: Nonlocal boundary value problems of ordinary differential equations and dynamical equations on time scales. Doctoral thesis, Beijing Institute of Technology (2009)