Abstract
This paper is concerned with the existence of one and two positive solutions for the following SturmLiouville boundary value problem on time scales
Under a locally nonnegative assumption on the nonlinearity f and some other suitable hypotheses, positive solutions are sought by considering the corresponding truncated problem, constructing the variational framework and combining the subsupersolution method with the mountain pass lemma.
MSC: 34B10, 34B18.
Keywords:
positive solution; SturmLiouville; time scales; subsupersolution; variational methods1 Introduction
The theory of dynamic equations on time scales has become a new important mathematical branch [1,2] since it was initiated by Hilger in 1988 [3]. Since then, boundary value problems (BVPs) for dynamic equations on time scales have received considerable attention, various fixed point theorems, subsupersolution method and LeraySchauder degree theory have been applied to get many interesting results about the existence of solutions; see [1,2,412] and the references therein. Variational method is also an important method for dealing with the existence of BVPs. Recently, some authors have used the theory to study the existence of solutions of some BVPs on time scales [4,5,13,14].
Especially, in [4,5], Agarwal et al. studied the following dynamic equation on time scales:
and
with the Dirichlet boundary condition. They gave some sufficient conditions for the existence of single and multiple positive solutions by using the variational method and critical point theory.
In [14], we considered the problem
and obtained the existence of many solutions depending on the value of the parameter λ which lie in some different intervals under some suitable hypotheses. The main approaches are also the variational method and some known critical point theorems and a three critical point theorem established in [15]. Erbe et al.[8] also established some existence criteria of positive solutions by a fixed point theorem in a cone with the globally nonnegative hypothesis of f.
Motivated by the papers [4,5,8,14], in this paper, we continue to study the problem (1.1) in the case of , that is,
Here is a time scale and , , , , , , , and . The purpose of this paper is to discuss the existence and multiplicity of positive solutions to the problem (1.2) under the local nonnegativity assumption of f and some other hypotheses. The main tools are the truncated method, the variational method, the subsupersolution method and the mountain pass lemma. First, inspired by the method in [4], we convert the existence of a positive solution of (1.2) to the existence of a solution of an associated problem of (1.2). In contrast with the paper [4], the appearance of term , our problem is more complicated and the proof is also different from [4] (see Lemma 2.3 for details). Next, we construct a supersolution of (1.2) and give the existence of one positive solution. Finally, under our weaker assumption on f (see (H1) and (H2) for details), since we cannot verify that the corresponding functional for the associated problem satisfies the P.S. condition, we consider the corresponding truncated problem. To prove the existence of the second positive solution by the mountain pass lemma, we also give an estimate of a nonnegative solution of (1.2) and prove the solution of a truncated problem is also a solution of (1.2) for n large enough (see Theorem 3.3 for details). To the best of our knowledge, the results are new both in the continuous and in the discrete case.
The paper is organized as follows. In Section 2, we present some basic properties of some related Sobolev space on time scales, construct the variational framework, give some properties of this framework and some necessary lemmas. In Section 3, we firstly get the existence of a single positive solution of (1.2) by using the subsupersolution method; then applying the truncated method, analytic technique and mountain pass lemma, we establish the existence of two positive solutions.
2 Preliminaries and variational formulation
In this section, we list the definition and properties of the Sobolev space on time scales [16], give some lemmas which we need for the proof of the main result and construct a variational framework.
For convenience, for [16,17], we denote . We let [18] denote the class of absolutely continuous functions on and the Sobolev space is defined as
with the norm
We know the immersion is compact. Analogous to the proof in the real numbers situation, one can deduce the following result on time scales.
and
Using Lemma 2.1, by a similar proof of , one can derive the following.
Lemma 2.2If, thenand, Δa.e. in, where, .
For convenience, we denote
In order to discuss the existence of a positive solution of (1.2), we consider the following problem:
First, we give an important lemma.
Lemma 2.3Iffor, uis a solution of (2.1), thenuis nonnegative in. Furthermore, iffor, andis nondecreasing in, then, .
Proof Let u be a solution of (2.1). In view of Lemma 2.2, we know and , Δa.e. in . Multiplying (2.1) by , integrating over and employing the integration by parts formula for an absolutely continuous function on , we find that
Next, we show that if is nondecreasing in , and for , then , .
In fact, if the conclusion is false, we can suppose that there exists such that and one of the following two cases holds:
(ii) for and there exists such that for .
For the case (i), if , then , there exists such that for . According to the nonnegativity of u on and , it is easy to see that is impossible. Thus we have
If , then we know , and . If , then , . Hence, in this case, we always have
Therefore
However, since u is a solution of (1.2), (2.2) and , we know that
which contradicts (2.4).
For the case (ii), it can be divided into two cases to consider.
(1) If , then one can deduce that , . From this and , we have a contradiction.
(2) If , then we always have . If , then , . Similar to (i), we get (2.2) and (2.3). But this is impossible from (2.4) and (2.5).
If and , then , , . So, we get a contradiction to (2.4) and (2.5).
If and , we have , . Hence, we get (2.2) and (2.3). But this contradicts (2.4) and (2.5).
If and , then . By assumption, u is a solution of (1.2), so we have , which contradicts and for . □
Remark 2.4 From the proof of Lemma 2.3, we can easily find that if , then the monotonicity assumption of can be omitted.
By Lemma 2.3, under the hypothesis
(H1) is nondecreasing in , and for ,
in order to prove the existence of a positive solution of (1.2), it suffices to consider the existence of a solution of (2.1). Now we establish the corresponding variational formulations for (2.1). We set
then E is a Banach space with the norm , and we can find that can be taken as an equivalent norm on E. Define the functional I on E as
Note that the appearance of the term in the functional I guarantees that I is , see next lemma. By the definition of Fréchet derivative and the fact that the immersion is compact, we have the following results.
Lemma 2.5The following statements are valid.
(ii) We define
Then
Jis weakly continuous inEandis compact.
(iii) The solutions of (2.1) match up to the critical points ofIinE.
For the eigenvalue problem
we have the following lemma.
Lemma 2.6 [[14], Lemma 3.1]
The eigenvalues of (2.6) may be arranged as , and there exists a countable orthonormal basis ofEconsisting of eigenfunction associated eigenvalues of (2.6) and
Remark 2.7 By (2.7) and Lemma 2.2, we know the eigenfunction corresponding to the eigenvalue satisfies for . Furthermore, by the KreinRutman theorem [[19], Theorem 7.C], we know with
Lemma 2.8 [[1], Theorem 4.73], [8]
The problem (1.2) has the Green function
and
respectively, and satisfy, , , , , , , .
Lemma 2.9The function defined by
belongs to, whereis given in Remark 2.7.
Proof Clearly, is well defined in .
If , by and , we have . Hence .
If , then . By Remark 2.7, we know . If , then . If , then there exists such that , for , then by L’Hôspital rule [[1], Theorem 1.119] and Lemma 2.8, we know
Hence, is bounded for s close to 0.
Similarly, we can derive that is bounded for s close to . □
In order to derive the main result, we list the following wellknown mountain pass lemma.
Lemma 2.10 [[20], Theorem 6.1]
Supposesatisfies the P.S. condition. Supposeand
(i) there exist, such thatforwith;
Define
3 Main results
In this section, we establish some existence criteria of a positive solution of (1.1) by employing the subsupersolution method and critical point theory.
First, using a method analogous to that in [21], we construct a supersolution to employ the subsupersolution method.
Theorem 3.1Assume that (H1) holds and there are constantsandsuch that
whereis fixed, , represents the unique positive solution of
Then the problem (1.2) has at least one positive solution.
Proof For fixed , let v be the unique positive solution of
Then, from the definition of , we have . Then, by the assumptions, it is easy to see that v is a supersolution of (1.2). In addition, condition (H1) guarantees that the constant function 0 is a strict subsolution of (1.2). Therefore, the subsupersolution method implies (1.2) has a positive solution . □
Remark 3.2 Furthermore, by Lemma 2.8, we know
Theorem 3.3Under the hypothesis of Theorem 3.1 and suppose the condition
holds, then the problem (1.2) has at least two positive solutions.
In order to prove this theorem, we first present some necessary lemmas.
Lemma 3.4Letv, be given in the proof of Theorem 3.1, thenis a local minimizer ofIinE.
Proof Denote
By the assumptions and Lemma 2.8, we easily find , . Hence is a local minimizer of I in W.
Next, by a similar argument to that in [22], we assert that is also a local minimizer of I in E.
In fact, if is not a local minimizer of I in E, then for every there is such that , and . By the Lagrange multiplier rule, we know there exists a constant such that
Note that is a solution of (1.2), so
Thus, from (3.2), we have
It is easy to show that as in . But this contradicts the fact that is a local minimizer of I in W. □
Next, under hypothesis (H2), in order to show the existence of the second positive solution of (1.2) by employing the mountain pass lemma, we need to show that I satisfies the P.S. condition. However, by (H2), we cannot justify this; therefore, we consider the truncation function and the truncation functional defined as follows.
Let and be an increasing positive sequence with as . For , define
and
Lemma 3.5Assume that (H1) and (H2) hold, then there existssuch that the functionalsatisfies the P.S. condition inEfor.
Proof For given n, let be the P.S. sequence of , that is, is bounded and as . If is bounded, one can deduce that I satisfies the P.S. condition by a similar proof to Proposition B.35 in [23].
Suppose that is unbounded. Since
we have as . Denote , then . So, without loss of generality, we can assume that in E, in . Note that
by the definition of and passing to limit in (3.4), one can derive that .
In view of (H2) and the definition of , for small enough, there exist independent of n, such that
Passing to the limit in (3.6), we know . Hence , which contradicts . Therefore is bounded. So, satisfies the P.S. condition for . □
By Lemmas 3.4 and 3.5, we deduce that for n large enough, has a nontrivial critical point by using the mountain pass lemma and Theorem 1 in [24]. In order to obtain a solution of (1.2), we need to get an estimate of . Therefore, we first give an estimate of a nonnegative solution of (1.2) employing a method similar to that in [25].
Lemma 3.6Suppose (H2) holds, then there issuch that for any nonnegative solutionuof (1.2), we have.
Proof If u is a nonnegative solution of (1.2), then by the definition of , we have
Condition (H2) implies there exist , such that
Hence, from (3.7) and (3.8), we derive that
So, using (3.7), we know
Thus, by (H2), we know is uniformly bounded. Note that by Lemma 2.9, for ,
Hence, there exists such that . □
Remark 3.7 Note that we only need (H2) to derive (3.8). Hence, (3.5) implies Lemma 3.6 is also valid for the truncation problem.
Proof of Theorem 3.3 Since the positive solution derived from Theorem 3.1 is a local minimizer of I and , we can choose large enough such that for every . Hence, is also a local minimizer of for . Then, from the definition of and Lemma 3.5, we know the mountain pass lemma and Theorem 1 in [24] imply that has the second critical point . Furthermore, by Remark 3.7, we know is also a critical point of I. Thus the problem (1.2) has the second positive solution . □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors typed, read, and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The research of HR Sun has been supported by the program for New Century Excellent Talents in University (NECT120246) and FRFCU (lzujbky2013k02).
References

Bohner, M, Peterson, A: Dynamic Equations on Time Scales, Birkhäuser, Boston (2001)

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

Hilger, S: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universität Würzburg (1988) (in German)

Agarwal, RP, OteroEspinar, V, Perera, K, Vivero, DR: Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods. J. Math. Anal. Appl.. 331, 1263–1274 (2007). Publisher Full Text

Agarwal, RP, OteroEspinar, V, Perera, K, Vivero, DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Anal.. 67, 368–381 (2007). Publisher Full Text

Anderson, D: Solutions to secondorder threepoints problems on time scales. J. Differ. Equ. Appl.. 8, 673–688 (2002). Publisher Full Text

Atici, EM, Guseinov, GS: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math.. 141, 75–99 (2002). PubMed Abstract  Publisher Full Text

Erbe, L, Peterson, A, Mathsen, R: Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain. J. Comput. Appl. Math.. 113, 365–380 (2000). Publisher Full Text

Rynne, BP: spaces and boundary value problems on time scales. J. Math. Anal. Appl.. 328, 1217–1236 (2007). Publisher Full Text

Sun, HR: Triple positive solutions for pLaplacian mpoint boundary value problem on time scales. Comput. Math. Appl.. 58, 1736–1741 (2009). Publisher Full Text

Sun, HR, Li, WT: Positive solutions for nonlinear threepoint boundary value problems on time scales. J. Math. Anal. Appl.. 299, 508–524 (2004). Publisher Full Text

Sun, HR, Li, WT: Existence theory for positive solutions to onedimensional pLaplacian boundary value problems on time scales. J. Differ. Equ.. 240, 217–248 (2007). Publisher Full Text

Jiang, L, Zhou, Z: Existence of weak solutions of twopoint boundary value problems for secondorder dynamic equations on time scales. Nonlinear Anal.. 69, 1376–1388 (2008). Publisher Full Text

Zhang, QG, Sun, HR: Variational approach for SturmLiouville boundary value problems on time scales. J. Appl. Math. Comput.. 36, 219–232 (2011). Publisher Full Text

Bonanno, G, Candito, P: Nondifferentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ.. 244, 3031–3059 (2008). Publisher Full Text

Agarwal, RP, OteroEspinar, V, Perera, K, Vivero, DR: Basic properties of Sobolev’s spaces on time scales. Adv. Differ. Equ.. 2006, Article ID 38121 (2006)

Guseinov, GS: Integration on time scales. J. Math. Anal. Appl.. 285, 107–127 (2003). Publisher Full Text

Cabada, A, Vivero, DR: Criterions for absolute continuity on time scales. J. Differ. Equ. Appl.. 11, 1013–1028 (2005). Publisher Full Text

Zeidler, E: Nonlinear Functional Analysis and Its Application I: FixedPoint Theorems, Springer, New York (1985)

DE Figueiredo, DG, Lions, PL: On pairs of positive solutions for a class of semilinear elliptic problems. Indiana Univ. Math. J.. 34, 591–606 (1985). Publisher Full Text

Brezis, H, Nirenberg, L: versus local minimizers. C. R. Acad. Sci., Sér. 1 Math.. 317, 465–472 (1993)

Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)

Ghoussoub, N, Preiss, D: A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 6, 321–330 (1989)

Mawhin, J, Omana, W: A priori bounds and existence of positive solutions for some SturmLiouville superlinear boundary value problems. Funkc. Ekvacioj. 35, 333–342 (1992)