Research Article Multiple Positive Symmetric Solutions to p-laplacian Dynamic Equations on Time Scales

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1 Discrete Dynamics in Nature and Society Volume 29, Article ID , 27 pages doi:1.1155/29/ Research Article Multiple Positive Symmetric Solutions to p-laplacian Dynamic Equations on Time Scales You-Hui Su 1 and Can-Yun Huang 2 1 School of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Xuzhou, Jiangsu 2218, China 2 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 735, China Correspondence should be addressed to Can-Yun Huang, canyun h@sina.com Received 1 July 29; Accepted 18 November 29 Recommended by Leonid Shaikhet This paper makes a study on the existence of positive solution to p-laplacian dynamic equations on time scales T. Somenewsufficient conditions are obtained for the existence of at least single or twin positive solutions by using Krasnosel skii s fixed point theorem and new sufficient conditions are also obtained for the existence of at least triple or arbitrary odd number positive solutions by using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem. As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations, as well as in the general time-scale setting. Copyright q 29 Y.-H. Su and C.-Y. Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Initiated by Hilger in his Ph.D. thesis 1 in 1988, the theory of time scales has been improved ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of difference equations in the discrete case. For the time being, it remains a field of vitality and attracts attention of many distinguished scholars worldwide. In particular, the theory is also widely applied to biology, heat transfer, stock market, wound healing, epidemic models 2 5, and so forth. Recent research results indicate that considerable achievement has been made in the existence problems of positive solutions to dynamic equations on time scales. For details, please see 6 13 and the references therein. Symmetry and pseudosymmetry have been widely used in science and engineering 14. The reason is that symmetry and pseudosymmetry are not only of its theoretical value in studying the metric manifolds 15 and symmetric graph 16, 17, and so forth, but also of its practical value, for example, we can

2 2 Discrete Dynamics in Nature and Society apply this characteristic to study graph structure 18, 19 and chemistry structure 2. Yet, few literature resource 21, 22 is available concerning the characteristics of positive solutions to p-laplacian dynamic equations on time scales. Throughout this paper, we denote the p-laplacian operator by ϕ p u, thatis,ϕ p u u p 2 u for p>1with ϕ p 1 ϕ q and 1/p 1/q 1. For convenience, we think of the blanket as an assumption that a, b are points in T, for an interval a, b T we always mean a, b T. Other type of intervals is defined similarly. We would like to mention the results of Sun and Li 11, 12. In 12, Sun and Li considered the two-point BVP ( Δ (ϕ p u t Δ h t f u σ t, t a, b T, u a B (u Δ a, u Δ σ b, 1.1 and established the existence theory for positive solutions of the above problem. They 11 also considered the m-point boundary value problem with p-laplacian (ϕ p ( u Δ t h t f t, u t, t,t T, u Δ, m 2 u T c i u ξ i, i and gave the existence of single or multiple positive solutions to the above problem. The main tools used in these two papers are some fixed-point theorems It is also noted that the researchers mentioned above 11, 12 only considered the existence of positive solutions. As a results, they failed to further provide characteristics of solutions, such as, symmetry. Naturally, it is quite necessary to consider the characteristics of solutions to p-laplacian dynamic equations on time scales. Let T be a symmetric time scale such that,t T. we consider the following p- Laplacian boundary value problem on time scales T of the form: (ϕ p ( u Δ t h t f u t, t,t T, u u T, u Δ u Δ T. 1.3 By using symmetric technique, the Krasnosel skii s fixed point theorem 24, the generalized Avery-Henderson fixed point theorem 26, and Avery-Peterson fixed point theorem 27, we obtain the existence of at least single, twin, triple, or arbitrary odd positive symmetric solutions of problem 1.3. As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations as well as in the general time-scale setting. The rest of the paper is organized as follows. In Section 2, we present several fixed point results. In Section 3, by using Krasnosel skii s fixed point theorem, we obtain the existence of at least single or twin positive symmetric solutions to problem 1.3. InSection 4, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to

3 Discrete Dynamics in Nature and Society 3 problem 1.3 are established. In Section 5, we present two simple examples to illustrate our results. For convenience, we now give some symmetric definitions. Definition 1.1. The interval,t T is said to be symmetric if any given t,t T, we have T t,t T. We note that such a symmetric time scale T exists. For example, let T {,.5,.1,.15}.22,.44 {.5,.85,.9,.95, 1}.56, It is obvious that T is a symmetric time scale. Definition 1.2. A function u :,T T R is said to be symmetric if u is symmetric over the interval,t T. That is, u t u T t, for any given t,t T. Definition 1.3. We say u is a symmetric solution to problem 1.3 on,t T provided that u is a solution to boundary value problem 1.3 and is symmetric over the interval,t T. Basic definitions on time scale can be found in 6, 7, 28. Another excellent sources on dynamical systems on measure chains are the book in 29. Throughout this paper, it is assumed that H1 f :,, is continuous, and does not vanish identically; H2 h C ld,t T,, is symmetric over the interval,t T and does not vanish identically on any closed subinterval of,t T, where C ld,t T,, denotes the set of all left dense continuous functions from,t T to,. 2. Preliminaries Let E C ld,t T, R and equip norm u sup t,t T u t, 2.1 then E is a Banach space. Define a cone P E by P { u E u u T,u is symmetric, nonnegative, and concave on the interval,t T }. 2.2 Assume that r,,t/2 T with < r. By using the symmetric and concave characters of u P and u u T, it is easy to obtain the following results. Lemma 2.1. Assume that r,,t/2 T with <r.ifu P, then i u /r u r ; ii T/2 u r ru T/2.

4 4 Discrete Dynamics in Nature and Society From the previous lemma we know that u u T/2 for u P. The operator A : P E is defined by Au t t T ϕ q h r f u r r Δs, s t ϕ q( s T/2 h r f u r r Δs, t t [, T ], 2 T [ ] T 2,T. T 2.3 It is obvious that A is completely continuous operator and all the fixed points of A are the solutions to the boundary value problem 1.3. In addition, it is easy to see that the operator A is symmetric. In fact, for t,t/2 T, we have T t T/2,T T, by using the integral transform, we have Au T t T s q( T tϕ T/2 t ( T s s T s ϕ q T/2 r T r t t h r f u r r Δs h r f u r r Δs ϕ q h T r f u T r r Δs s ϕ q h r f u r r Δs Au t. s 2.4 Hence, A is symmetric. Now, we provide some background material from the theory of cones in Banach spaces 24, 26, 27, 3, and then state several fixed point theorems needed later. Firstly, we list the Krasnosel skii s fixed point theorem 24. Lemma 2.2 see 24. Let P be a cone in a Banach space E. Assume that Ω 1, Ω 2 are open subsets of E with Ω 1, Ω 1 Ω 2. If A : P Ω 2 \ Ω 1 P is a completely continuous operator such that either i Ax x, for all x P Ω 1 and Ax x, for all x P Ω 2 or ii Ax x, for all x P Ω 1 and Ax x, for all x P Ω 2, then A has a fixed point in P Ω 2 \ Ω 1. Given a nonnegative continuous functional γ on a cone P of a real Banach space E, we define, for each d>, the set P γ,d {x P : γ x <d}. Secondly, we state the generalized Avery-Henderson fixed point theorem 26.

5 Discrete Dynamics in Nature and Society 5 Lemma 2.3 see 26. Let P be a cone in a real Banach space E. Let α, β, and γ be increasing, nonnegative continuous functional on P such that for some c> and H>,γ x β x α x and x Hγ x for all x P γ,c. Suppose that there exist positive numbers a and b with a<b<c and A : P γ,c P is a completely continuous operator such that i γ Ax <cfor all x P γ,c ; ii β Ax >bfor all x P β, b ; iii P α, a / and α Ax <afor x P α, a, then A has at least three fixed points x 1, x 2, and x 3 belonging to P γ,c such that α x 1 <a<α x 2 with β x 2 <b<β x 3,γ x 3 <c. 2.5 The following lemma can be found in 21. Lemma 2.4 see 21. Let P be a cone in a real Banach space E. Let α, β, and γ be increasing, nonnegative continuous functional on P such that for some c> and H>,γ x β x α x and x Hγ x for all x P γ,c. Suppose that there exist positive numbers a and b with a<b<c and A : P γ,c P is a completely continuous operator such that: i γ Ax >cfor all x P γ,c ; ii β Ax <bfor all x P β, b ; iii P α, a / and α Ax >afor x P α, a, then A has at least three fixed points x 1,x 2, and x 3 belonging to P γ,c such that α x 1 <a<α x 2 with β x 2 <b<β x 3,γ x 3 <c. 2.6 Let β and φ be nonnegative continuous convex functionals on P, λ is a nonnegative continuous concave functional on P, and ϕ is a nonnegative continuous functional, respectively on P. We define the following convex sets: P ( φ, λ, b, d { x P : b λ x,φ x d }, P ( φ, β, λ, b, c, d { x P : b λ x,β x c, φ x d }, 2.7 and a closed set R φ, ϕ, a, d {x P : a ϕ x,φ x d}. Finally, we list the fixed point theorem due to Avery-Peterson 27.

6 6 Discrete Dynamics in Nature and Society Lemma 2.5 see 27. Let P be a cone in a real Banach space E and β, φ, λ, ϕ defined as above, moreover, ϕ satisfies ϕ λ x λ ϕ x for λ 1 such that, for some positive numbers h and d, λ x ϕ x, x hφ x 2.8 for all x P φ, d. Suppose that A : P φ, d P φ, d is completely continuous and there exist positive real numbers a, b, c,witha<bsuch that i {x P φ, β, λ, b, c, d : λ x >b} / and λ A x >bfor x P φ, β, λ, b, c, d ; ii λ A x >bfor x P φ, λ, b, d with β A x >c; iii / R φ, ϕ, a, d and λ A x <afor all x R φ, ϕ, a, d with ϕ x a, then A has at least three fixed points x 1,x 2,x 3 P φ, d such that φ x i d for i 1, 2, 3, b < λ x 1, a < ϕ x 2, λ x 2 <b with ϕ x 3 <a Single or Twin Solutions Let f u f lim u ϕ p u, f f u lim u ϕ p u. 3.1 We define i number of zeros in the set {f,f } and i number of infinities in the set {f,f }. Clearly, i, i, 1, or 2 and there exist six possible cases: i i 1andi 1; ii i andi ; iii i andi 1; iv i andi 2; v i 1andi ; vi i 2andi. In the following, by using Krasnosel skii s fixed point theorem in a cone, we study the existence of positive symmetric solutions to problem 1.3 under the above six possible cases For the Case i 1 and i 1 In this subsection, we discuss the existence of single positive symmetric solution of the problem 1.3 under i 1andi 1. Theorem 3.1. Problem 1.3 has at least one positive symmetric solution in the case i i 1. 1 and Proof. We divide the proof into two cases. Case 1 f andf. Inviewoff, there exists an H 1 > such that f u ϕ p ε ϕ p u ϕ p εu for u,h 1, where ε arbitrary small and satisfies < εt/2 ϕ q T/2 h r r 1.

7 Discrete Dynamics in Nature and Society 7 If u P with u H 1, then Au sup t,t T Au ( T Au 2 T/2 ϕ q h r f u r r Δs s ε u T/2 ϕ q h r r Δs 3.2 u εt 2 ϕ q h r r u. We let Ω H1 {u E : u <H 1 }, then Au u for u P Ω H1. From f, there exists an H 2 > such that f u ϕ p k ϕ p u ϕ p ku for u H 2,, where k>, and satisfies the following inequality: 2k 2 T ϕ q h r r Set { H 2 max 2H 1,H 2 T 2 }, Ω H2 {u E : u <H 2 }. 3.4 If u P with u H 2, then, by Lemma 2.1, one has min u t 2 ( T t,t/2 T T u H

8 8 Discrete Dynamics in Nature and Society For u P Ω H2, in terms of 3.3 and 3.5, weget Au sup t,t T Au Au ( ϕ q h r f u r r Δs s ϕ q h r f u r r Δs ϕ q h r ϕ p ku r r Δs 2k2 T u ϕ q h r r 3.6 u. Thus, by i of Lemma 2.2, problem 1.3 has at least single positive symmetric solution u in P Ω H2 \ Ω H1 with H 1 u H 2. Case 2 f and f. Since f, there exists an H 3 > such that f u ϕ p m ϕ p u ϕ p mu for u,h 3, where m is such that 2m 2 T ϕ q h r r If u P with u H 3, then, by 3.7, one has Au sup Au t,t T ϕ q h r f u r r Δs ϕ q h r ϕ p mu r r m 2 T u ϕ q h r r 3.8 u. If we let Ω H3 {u E : u <H 3 }, then Au u for u P Ω H3.

9 Discrete Dynamics in Nature and Society 9 Now, we consider f. By definition, there exists H 4 > such that f u ϕ p δ ϕ p u ϕ p δu for u [ H 4,, 3.9 where δ>satisfies ( δt T/2 2 ϕ q h r r Suppose that f is bounded, then f u ϕ p K for all u, and some constant K>. Pick { H 4 max 2H 3, KT 2 ϕ q } h s Δs If u P with u H 4, then ( T Au Au 2 T/2 ϕ q h r f u r r Δs K T 2 ϕ q h s Δs 3.12 H 4 u. Suppose that f is unbounded. From f C,,,, we have f u C 3 for arbitrary u,c 4, here C 3 and C 4 are arbitrary positive constants. This implies that f u if u. Hence, it is easy to know that there exists H 4 max{2h 3, T/2 H 4 } such

10 1 Discrete Dynamics in Nature and Society that f u f H 4 for u,h 4. If u P with u H 4, then by using 3.9 and 3.1, we have ( T Au Au 2 T/2 ϕ q h r f u r r Δs s T/2 ϕ q h r f H 4 r Δs 3.13 ( T T/2 δh 4 2 ϕ q h r r u. Consequently, in either case, if we take Ω H4 {u E : u <H 4 }, then, for u P Ω H4, we have Au u. Thus, condition ii of Lemma 2.2 is satisfied. Consequently, problem 1.3 has at least single positive symmetric solution u in P Ω H4 \ Ω H3 with H 3 u H 4. The proof is complete For the Case i and i In this subsection, we discuss the existence of positive symmetric solutions to problems 1.3 under i andi. First, we will state and prove the following main result of problem 1.3. Theorem 3.2. Suppose that the following conditions hold: i there exists constant p > such that f u ϕ p p Λ 1 for u,p, where Λ 1 { T/2 ϕ q T/2 h r r } 1 ; ii there exists constant q > such that f u ϕ p q Λ 2 for u 2/T q,q, where Λ 2 {ϕ q T/2 h r r } 1, furthermore, p / q, then problem 1.3 has at least one positive symmetric solution u such that u lies between p and q.

11 Discrete Dynamics in Nature and Society 11 Proof. Without loss of generality, we may assume that p <q. Let Ω p {u E : u <p }. For any u P Ω p, in view of condition i, we have ( T Au Au 2 T/2 ϕ q h r f u r r Δs s 3.14 ( p T T/2 Λ 1 2 ϕ q h r r p, which yields Au u for u P Ω p Now, set Ω q {u E : u <q }. For u P Ω q, Lemma 2.1 implies that 2 T q u t q for t [, T ] T Hence, by condition ii we get ( T Au Au 2 ϕ q h r f u r r Δs q Λ 2 ϕ q h r r 3.17 q. So, if we take Ω q {u E : u <q }, then Au u, u P Ω q Consequently, in view of p <q, 3.15 and 3.18, it follows from Lemma 2.2 that problem 1.3 has a positive symmetric solution u in P Ω q \ Ω p. The proof is complete For the Case i 1 and i or i and i 1 In this subsection, under the conditions i 1andi ori andi 1, we discuss the existence of positive symmetric solutions to problem 1.3.

12 12 Discrete Dynamics in Nature and Society Theorem 3.3. Suppose that f,ϕ p Λ 1 and f ϕ p T/2 Λ 2, hold. Then problem 1.3 has at least one positive symmetric solution. Proof. It is easy to see that under the assumptions, conditions i and ii in Theorem 3.2 are satisfied. So the proof is easy and we omit it here. Theorem 3.4. Suppose that f ϕ p T/2 Λ 2, and f,ϕ p Λ 1 hold, then problem 1.3 has at least one positive symmetric solution. Proof. Firstly, let ε 1 f ϕ p T/2 Λ 2 >, there exists a sufficiently small q > thatsatisfies ( f u T ϕ p u f ε 1 ϕ p 2 Λ 2 for u (,q ] Thus, u 2/T q,q, we have f u ϕ p ( T 2 Λ 2 ϕ p u ϕ p ( Λ2 q, 3.2 which implies that condition ii in Theorem 3.2 holds. Nextly, for ε 2 ϕ p Λ 1 f >, there exists a sufficiently large p >q such that f u ϕ p u f ε 2 ϕ p Λ 1 for u [ p, We consider two cases. Case 1. Assume that f is bounded, that is, f u ϕ p K 1 for u,, 3.22 here K 1 > some constant. If we take sufficiently large p such that p max{k 1 /Λ 1,p }, then f u ϕ p K 1 ϕ p ( Λ1 p for u [,p ] Consequently, from the above inequality, condition i of Theorem 3.2 is true. Case 2. Assume that f is unbounded. From f C,,,, there exists p >p such that f u f ( p for u [,p ]. 3.24

13 Discrete Dynamics in Nature and Society 13 Since p >p,by 3.21, wegetf p ϕ p Λ 1 p, hence f u f ( p ϕ p ( Λ1 p for u [,p ] Thus, condition i of Theorem 3.2 is fulfilled. Consequently, Theorem 3.2 implies that the conclusion of this theorem holds. From the proof of Theorems 3.1 and 3.2, respectively, we have the following two results. Corollary 3.5. Suppose that f and condition (ii in Theorem 3.2 hold, then problem 1.3 has at least one positive symmetric solution. Corollary 3.6. Suppose that f and condition (ii in Theorem 3.2 hold, then problem 1.3 has at least one positive symmetric solution. Theorem 3.7. Suppose that f,ϕ p Λ 1 and f hold, then problem 1.3 has at least one positive symmetric solution. Proof. First, in view of f, then by inequality 3.7, we have Au u for u P Ω H2. Next, by f,ϕ p Λ 1, for ε 3 ϕ p Λ 1 f >, there exists a sufficiently small p,h 2 such that f u ( f ε 3 ϕp u ϕ p Λ 1 u ϕ p ( Λ1 p for u [,p ], 3.26 which implies that i of Theorem 3.2 holds, that is, 3.14 is true. Hence, we obtain Au u for u P Ω p. The result is obtained and the proof is complete. Theorem 3.8. Suppose that f and f,ϕ p Λ 1 hold, then problem 1.3 has at least one positive symmetric solution. Proof. On one hand, since f, by inequality 3.9,onegets Au u, u P Ω H3. On the other hand, since f,ϕ p Λ 1, from the technique similar to the second part proof in Theorem 3.4, one obtains that condition i of Theorem 3.2 is satisfied, that is, inequality 3.14 holds, one has Au u, u P Ω p, where p >H 3. Hence, problem 1.3 has at least one positive symmetric solution. The proof is complete. From Theorems 3.7 and 3.8, respectively, it is easy to obtain the following two corollaries. Corollary 3.9. Assume that f and condition (i in Theorem 3.2 hold, then problem 1.3 has at least one positive symmetric solution. Corollary 3.1. Assume that f and condition (i in Theorem 3.2 hold, then problem 1.3 has at least one positive symmetric solution.

14 14 Discrete Dynamics in Nature and Society 3.4. For the Case i and i 2 or i 2 and i In this subsection, under i andi 2ori 2andi, we study the existence of multiple positive solutions to problems 1.3. Combining the proofs of Theorems 3.1 and 3.2, it is easy to prove the following two theorems. Theorem Suppose that i and i 2 and condition (i of Theorem 3.2 hold, then problem 1.3 has at least two positive solutions u 1,u 2 P such that < u 1 <p < u 2. Theorem Suppose that i 2 and i and condition (ii of Theorem 3.2 hold, then problem 1.3 has at least two positive solutions u 1,u 2 P such that < u 1 <q < u Triple Solutions In the previous section, we have obtained some results on the existence of at least single or twin positive symmetric solutions to problem 1.3. In this section, we will further discuss the existence of positive symmetric solutions to problem 1.3 by using two different methods. And the conclusions we will arrive at are different with their own distinctive advantages. Based on the obtained symmetric solution position and local properties, we can only get some local properties of solutions by using method one; however, the position of solutions is not determined. In contrast, by means of method two, we cannot only get some local properties of solutions but also give the position of all solutions, with regard to some subsets of the cone, which has to meet some conditions which are stronger than those of method one. Obviously, the local properties of obtained solutions are different by using the two different methods. Hence, it is convenient for us to comprehensively comprehend the solutions of the models by using the two different techniques. In Section 5, two examples are given to illustrate the differences of the results obtained by the two different methods. For the notational convenience, we denote M ξ ϕ q h r r, N ξ ϕ q h r r, L ξ rϕ q h r r, L θ rϕ q h r r, W ξ T 2 ϕ q r 4.1 h r r Result 1 In this subsection, in view of the generalized Avery-Henderson fixed-point theorem 26,the existence criteria for at least triple and arbitrary odd positive symmetric solutions to problems 1.3 are established. For u P, we define the nonnegative, increasing, continuous functionals γ,β,and α by γ u max t, T u t u (, β u min t,t/2 T u t u (, α u max t,r T u t u r. 4.2

15 Discrete Dynamics in Nature and Society 15 It is obvious that γ u β u α u for each u P. By Lemma 2.1, oneobtains u C γ u for all u P, here C T/2. We now present the results in this subsection. Theorem 4.1. If there are positive numbers a, b, c such that a < 2r/T b < 2r/T c N ξ /M ξ. In addition, f u satisfies the following conditions: i f u <ϕ p c /M ξ for u, T/2 c ; ii f u >ϕ p b /N ξ for u b, T/2 b ; iii f u <ϕ p a /L ξ for u, T/2r a. Then problem 1.3 has at least three positive symmetric solutions u 1, u 2, and u 3 such that min u 2 t <b < t,t/2 T < max t,r T u 1 t <a < max min t,t/2 T u 3 t, t,r T u 2 t, max u 3 t <c. t, T 4.3 Proof. By the definition of completely continuous operator A and its properties, it has to be demonstrated that all the conditions of Lemma 2.3 hold with respect to A. It is easy to obtain that A : P γ,c P. Firstly, we verify that if u P γ,c, then γ Au <c. If u P γ,c, then γ u max t, T u t u ( c. 4.4 Lemma 2.1 implies that u T 2 u( T 2 c, 4.5 we have u t T 2 c, t [, T ] T

16 16 Discrete Dynamics in Nature and Society Thus, by condition i, one has γ Au max Au t t, T Au ( ϕ q h r f u r r Δs s ϕ q h r f u r r Δs ( c <ϕ q h r ϕ p M ξ c ϕ q h r r M ξ r 4.7 c. have Secondly, we show that β Au >b for u P β, b. If we choose u P β, b, then β u min t,t/2 T u t b. In view of Lemma 2.1, we u T 2 u( T 2 b. 4.8 So b u t T 2 b, t [, T ] T Using condition ii,weget β Au Au ( ϕ q h r f u r r Δs ( b >ϕ q h r ϕ p N ξ > b ϕ q h r r N ξ r 4.1 b.

17 Discrete Dynamics in Nature and Society 17 Finally, we prove that P α, a / and α Au <a for all u P α, a. In fact, the constant function a /2 P α, a. Moreover, for u P α, a, we have α u max t,r T u t a, which implies u t a for t,r T. Hence, u T/2r u r. Therefore By using assumption iii, one has u t T 2r a, t [, T ] T α Au Au r r ϕ q h r f u r rδs r r s ϕ q h r f u r r Δs ( a ϕ q h r ϕ p L ξ a rϕ q h r r L ξ r Δs 4.12 a. Thus, all the conditions in Lemma 2.3 are satisfied. From H1 and H2, we have that the solutions to problem 1.3 do not vanish identically on any closed subinterval of,t T. Consequently, problem 1.3 has at least three positive symmetric solutions u 1, u 2,andu 3 belonging to P γ,c, and satisfying 4.3. The proof is complete. From Theorem 4.1, we see that, when assumptions as i, ii, and iii are imposed appropriately on f, we can establish the existence of an arbitrary odd number of positive symmetric solutions to problem 1.3. Theorem 4.2. Let l 1, 2,...,n.Suppose that there exist positive numbers a s l, b s l, c s l such that a s 1 < 2r T b s 1 < 2r N ξ c s T M 1 <a s 2 < 2T ξ r b s 2 < 2r N ξ c s T M 2 <a s 3 < <a s l ξ In addition, f u satisfies the following conditions: i f u <ϕ p c s l /M ξ for u, T/2 c s l ; ii f u >ϕ p b s l /N ξ for u b s l, T/2 b s l ; iii f u <ϕ p a s l /L ξ for u, T/2r a s l. Then problem 1.3 has at least 2l 1 positive symmetric solutions.

18 18 Discrete Dynamics in Nature and Society Proof. When l 1, it is clear that Theorem 4.1 holds. Then we can obtain at least three positive symmetric solutions u 1,u 2,andu 3 satisfying < max u 1 t <a s1 < max u 2 t, t,r T t,r T min u 2 t <b s1 < min u 3 t, t i,t/2 T t i,t/2 T max u 3 t <c s1. t, T 4.14 Following this way, we finish the proof by induction. The proof is complete. Using Lemma 2.4, it is easy to have the following results. Theorem 4.3. Suppose that there are positive numbers a, b, c such that a < L θ /M ξ b < 2/T L θ /M ξ c. In addition, f u satisfies the following conditions: i f u >ϕ p c /N ξ for u c, T/2 c ; ii f u <ϕ p b /M ξ for u, T/2 b ; iii f u >ϕ p a /L θ for u a, T/2r a. Then problem 1.3 has at least three positive symmetric solutions u 1, u 2, and u 3 such that min u 2 t <b < t,t/2 T < max t,r T u 1 t <a < max min t,t/2 T u 3 t, t,r T u 2 t, max u 3 t <c. t, T 4.15 From Theorem 4.3, we can obtain Theorem 4.4 and Corollary 4.5. Theorem 4.4. Let l 1, 2,...,n.Suppose that there existence positive numbers a λ l, b λ l, c λ l such that a λ 1 < L θ b M λ 1 < 2 L θ c λ 1 <a ξ T N λ 2 < L θ b ξ M λ 2 < 2 L θ c λ 2 <a ξ T N λ 3 < <a λ l ξ In addition, f u satisfies the following conditions: i f u >ϕ p c λ l /N ξ for u c λ l, T/2 c λ l ; ii f u <ϕ p b λ l /M ξ for u, T/2 b λ l ; iii f u >ϕ p a λ l /L θ for u a λ l, T/2r a λ l. Then problem 1.3 has at least 2l 1 positive symmetric solutions. Corollary 4.5. Assume that f satisfies the following conditions: i f,f, ii there exists c > such that f u <ϕ p 2/T c /M ξ for u,c, then problem 1.3 has at least three positive symmetric solutions.

19 Discrete Dynamics in Nature and Society 19 Proof. First, by condition ii,letb 2/T c, one gets f u <ϕ p ( b M ξ for u [ ] T, 2 b, 4.17 which implies that ii of Theorem 4.3 holds. Second, choose K 3 sufficiently large to satisfy K 3 L θ K 3 rϕ q h r r > r Since f, there exists r 1 > sufficiently small such that f u ϕ p K 3 ϕ p u ϕ p K 3 u for u [,r 1] Without loss of generality, suppose r 1 L θt/2rm ξ b. Choose a > such that a < 2r/T r 1. For a u T/2r a, we have u r 1 and a < L θ /M ξ b. Thus, by 4.18 and 4.19, we have ( a f u ϕ p K 3 u ϕ p K 3 a >ϕ p L θ for u [a, T2r a ], 4.2 this implies that iii of Theorem 4.3 is true. Third, choose K 2 sufficiently large such that K 2 N ξ K 2 ϕ q h r r > Since f, there exists r 2 > sufficiently large such that f u ϕ p K 2 ϕ p u ϕ p K 2 u for u r Without loss of generality, suppose r 2 > T/2 b. Choose c r 2. Then f u ϕ p K 2 u ϕ p K 2 c >ϕ p ( c N ξ for u [c, T2 c ], 4.23 which means that i of Theorem 4.3 holds. From above analysis, we get <a < L θ b < 2 L θ c, M ξ T M ξ 4.24 then, all conditions in Theorem 4.3 are satisfied. Hence, problem 1.3 has at least three positive symmetric solutions.

20 2 Discrete Dynamics in Nature and Society In terms of Theorem 4.1, we also have the following corollary. Corollary 4.6. Assume that f satisfies conditions i f,f ; ii there exists c > such that f u >ϕ p 2/T c /N ξ for u 2/T c,c, then problem 1.3 has at least three positive symmetric solutions Result 2 In this subsection, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to problems 1.3 are established by using the Avery-Peterson fixed point theorem 27. Define the nonnegative continuous convex functionals φ and β, nonnegative continuous concave functional λ, and nonnegative continuous functional ϕ, respectively, on P by φ u max u t u t,t/2 T ( T 2 λ u ϕ u, β u max t r,t/2 T κ min t,t/2 T u t u (. u Δ t u Δ r, 4.25 Now, we list and prove the results in this subsection. Theorem 4.7. Suppose that there exist constants a, b, d such that < a < 2/T b < 2/T N ξ d /W ξ. In addition, suppose that W ξ >ϕ q T/2 h s s holds, f satisfies the following conditions: i f u ϕ p d /W ξ for u,d ; ii f u >ϕ p b /N ξ for u b,d ; iii f u <ϕ p a /M ξ for u, T/2 a, then problem 1.3 has at least three positive symmetric solutions u 1,u 2,u 3 such that u i d, i 1, 2, 3, b <u 1 (, u2 ( <b, u 3 ( <a Proof. By the definition of completely continuous operator A and its properties, it suffices to show that all the conditions of Lemma 2.5 hold with respect to A. For all u P, λ u ϕ u and u u T/2 φ u. Hence, condition 2.8 is satisfied.

21 Discrete Dynamics in Nature and Society 21 Firstly, we show that A : P φ, d P φ, d. For any u P φ, d, in view of φ u u d and assumption i, one has ( T Au Au 2 T/2 ϕ q h r f u r r Δs T/2 d W ξ T 2 ϕ q s ϕ q h r f u r r Δs h r r 4.27 d. From the above analysis, it remains to show that i iii of Lemma 2.5 hold. Secondly, we verify that condition i of Lemma 2.5 holds, let u t kb with k W ξ /N ξ > 1. From the definitions of N ξ,w ξ,andβ u, respectively, it is easy to see that u t kb >b and β u <kb. In addition, in view of b < N ξ /W ξ d, we have φ u kb <d. Thus { u P ( φ, β, λ, b,kb,d : λ x >b } / For any u P φ, β, λ, b,kb,d, then we get b u t d for all t, T/2 T. Hence, by assumption ii, we have λ Au Au ( ϕ q h r f u r r Δs s ϕ q h r f u r r Δs > b ϕ q h r r N ξ 4.29 b. Thirdly, we prove that condition ii of Lemma 2.5 holds. For any u P φ, λ, b,d with β Au >kb, that is, β Au Au Δ r ϕ q h s f u s s >kb. r 4.3

22 22 Discrete Dynamics in Nature and Society So, in view of k W ξ /N ξ, W ξ >ϕ q T/2 h s s and 4.3, one has λ Au Au ( > ϕ q h r f u r r Δs s ϕ q h r f u r r Δs ϕ q h r f u r r Δs r 4.31 >kb >b. Finally, we check condition iii of Lemma 2.5. Clearly, since ϕ <a, we have / R φ, ϕ, a,d. If u R φ, ϕ, a,d with ϕ u min t,t/2 T u t a, then Lemma 2.1 implies that u T 2 u( T 2 a This yields u t T/2 a for all t,t/2 T. Hence, by assumption iii, we have λ Au Au ( ϕ q h r f u r r Δs < a ϕ q M ξ h r r a Consequently, all conditions of Lemma 2.5 are satisfied. The proof is completed. i : We remark that condition i in Theorem 4.7 can be replaced by the following condition ( f u 1 lim ϕ p u ϕ p u W ξ, 4.34 which is a special case of i. Corollary 4.8. If condition (i in Theorem 4.7 is replaced by (i, then the conclusion of Theorem 4.7 also holds. Proof. By Theorem 4.7, we only need to prove that i implies that i holds, that is, if i holds, then there is a number d max{ T/2 a, W ξ /N ξ b } such that f u ϕ p d /W ξ for u,d.

23 Discrete Dynamics in Nature and Society 23 Suppose on the contrary that for any d max{ T/2 a, W ξ /N ξ b }, there exists u c,d such that f u c >ϕ p d /W ξ. Hence, if we choose { T c n > max 2 a, W } ξ b N ξ n 1, 2, with c n, then there exist u n,c n such that ( c f u n >ϕ n p, 4.36 W ξ and so lim f u n. n 4.37 Since condition i holds, then there exists τ> such that f u ϕ p ( u W ξ for u τ, Hence, we have u n τ. Otherwise, if u n >τ,then it follows from 4.38 that ( ( un c f u n ϕ p ϕ n p, 4.39 W ξ W ξ which contradicts Let W max u,τ f u, then f u n W n 1, 2,..., which also contradicts The proof is complete. Theorem 4.9. Let l 1, 2,...,n.Suppose that there exist constants a l, b l, d l such that <a 1 < 2 T b 1 < 2 d 1 N ξ T W ξ <a 2 < 2 i T b 2 < 2 d 2 N ξ <a 3 T W < <a l. 4.4 ξ In addition, suppose that W ξ >ϕ q T/2 h s s holds, then f satisfies the following conditions: i f u <ϕ p d l /W ξ for u,d l ; ii f u >ϕ p b l /N ξ for u b l,d l ; iii f u <ϕ p a l /M ξ for u, T/2 a l, then problem 1.3 has at least 2l 1 positive symmetric solutions. Proof. Similar to the proof of Theorem 4.2, we omit it here.

24 24 Discrete Dynamics in Nature and Society 5. Examples In this section, we give two simple examples to illustrate that the conclusions we will arrive at are different with their own distinctive advantages. Example 5.1. Let T {,.5,.1,.15,.45,.48,.5,.55,.52,.85,.9,.95, 1}.22,.44.56, Consider the following boundary value problem with p 3: (ϕ p ( u Δ t h t f u t, t, 1 T, u u 1, u Δ u Δ 1, 5.2 where t ρ t, t,.5 T, h t 1 t ρ 1 t, t.5, 1 T,.16, u,.6, 149.6u 629.6, u.6, 1, f u 42, u 1, 5, u 45.82, u 5, 6, 576, u 6,. Note that T 1. If we choose.1, r.25, then direct calculation shows that ( 1/2 M ξ ϕ q h r r.5, 5.4 and N ξ.489,l ξ.125, W ξ.25. If we take a.3, b 1, c 12, then a.3 < 2r T b.9714 < 2r c N ξ T M ξ

25 Discrete Dynamics in Nature and Society 25 Furthermore, f u.16 < 5.76 ϕ p ( a L ξ f u 42 > ϕ p ( b N ξ f u 576 < 576 ϕ p ( c M ξ, u,.6,, u 1, 5,, u, By Theorem 4.1 we see that the boundary value problem 5.2 has at least three positive symmetric solutions u 1,u 2,andu 3 such that <u 1.25 <.3 <u 2.25, u 2.1 < 1 <u 3.1, u 3.1 < Yet, ϕ q 1/2.1 h s s.489 > W ξ.25, hence, the existence of positive solutions of boundary value problem 5.2 is not obtained by using Theorem 4.7. Example 5.2. Let T {,.5,.1,.15,.45,.46,.48,.5,.52,.54,.55,.85,.9,.95, 1}.22,.44.56, Consider the following boundary value problem: (ϕ p ( u Δ t h t f u t, t, 1 T, u u 1, u Δ u Δ 1, 5.9 where t ρ t, t,.5 T, h t 1 t ρ 1 t, t.5, 1 T. 5.1 Note that T 1. If we choose.45, r.48, then a direct calculation shows that ( 1/2 M ξ ϕ q h r r.225, (.5 N ξ.981, L ξ.24, W ξ.25, W ξ.25 >.218 ϕ q h r r

26 26 Discrete Dynamics in Nature and Society Let ε be an arbitrary small positive number, a,b,andd are arbitrary positive numbers with <a < 2b < 2 N ξd W ξ That is <a <.9b <.3924d, ( a ϕ p ε, u a,.225 f u l u, a u b, ( b ϕ p ε, b u d, where l u satisfy l a ϕ p a /.225 ε, l b ϕ p b /.981 ε, l Δ u Δ, u a,b. It is obvious that i, ii,and iii in Theorem 4.7 are satisfied. By Theorem 4.7,wesee that the boundary value problem 5.9 has at least three positive solutions u 1,u 2,andu 3 such that u i d, i 1, 2, 3, b <u 1.45, a <u 2.45, u 3.45 <a However, for arbitrary positive numbers a, b, d, condition iii of Theorem 4.1 is not satisfied. Therefore, Theorem 4.1 is not fit to study the boundary value problem 5.9. Acknowledgments This work was supported by the Grant of Department of Education Jiangsu Province 9KJD116 and Science Foundation of Lanzhou University of Technology BS1293. References 1 S. Hilger, Ein maßkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, Würzburg, Germany, F. M. Atici, D. C. Biles, and A. Lebedinsky, An application of time scales to economics, Mathematical and Computer Modelling, vol. 43, no. 7-8, pp , V. Jamieson and V. Spedding, Taming nature s numbers, New Scientist: The Global Science and Technology Weekly, vol. 244, pp , S. Hilger, Analysis on measure chains a unified approach to continuous and discrete calculus, Results in Mathematics, vol. 18, no. 1-2, pp , D. M. Thomas, L. Vandemuelebroeke, and K. Yamaguchi, A mathematical evolution model for phytoremediation of metals, Discrete and Continuous Dynamical Systems B, vol. 5, no. 2, pp , M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston, Mass, USA, 21.

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