Research Article | Open Access
Existence of Positive Solutions for -Point Boundary Value Problems on Time Scales
We study the one-dimensional -Laplacian -point boundary value problem , , , , where is a time scale, , , some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by using fixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional -Laplacian -point boundary value problem on time scales has been studied.
With the development of p-Laplacian dynamic equations and theory of time scales, a few authors focused their interest on the study of boundary value problems for p-Laplacian dynamic equations on time scales. The readers are referred to the paper [1–7].
In 2005, He  considered the following boundary value problems:where is a time scales, The author showed the existence of at least two positive solutions by way of a new double fixed point theorem.
In 2004, Anderson et al.  used the virtue of the fixed point theorem of cone and obtained the existence of at least one solution of the boundary value problem:
In 2007, Geng and Zhu  used the Avery-Peterson and another fixed theorem of cone and obtained the existence of three positive solutions of the boundary value problem:Also, in 2007, Sun and Li  discussed the existence of at least one, two or three positive solutions of the following boundary value problem:
In this paper, we are concerned with the existence of multiple positive solutions to the m-point boundary value problem for the one dimension p-Laplcaian dynamic equation on time scale where is a time scale, and
(H1)(H2)(H3) and there exists such that
In this paper, we have organized the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krassnoselskiifs  fixed point theorem to prove the existence of at least one positive solution to the MBVP(1.5). In Section 4, conditions for the existence of at least two positive solutions to the MBVP (1.5) are discussed by using Avery and Henderson  fixed point theorem. In Section 5, to prove the existence of at least three positive solutions to the MBVP (1.5) are discussed by using Leggett and Williams  fixed point theorem.
For completeness, we introduce the following concepts and properties on time scales.
A time scale is a nonempty closed subset of , assume that has the topology that it inherits from the standards topology on
Definition 1.1. Let be a time scale, for , one defines the forward jump operator by , and the backward jump operator by while the graininess function is defined by . If one says that is right-scattered, while if one says that tis left-scattered. Also, if and then is called right-dense, and if and then is called left-dense. One also needs below the set as follows: if has a left-scattered maximum then otherwise For instance, if then
Definition 1.2. Assume is a function and let Then , one defines to be the number (provided it exists) with the property that any given there is a neighborhood of such thatfor all One says that is delta differentiable (or in short: differentiable) on provided exist for all
If then if then
A function .
(i)If is continuous , then is rd-continuous.(ii)The jump operator is rd-continuous.(iii)If is rd-continuous, then so is
A function is called an antidervative of , provided holds for all . One defines the definite integral byFor all . If then is nondecreasing.
2. The Preliminary Lemmas
Lemma 2.2. Assume that conditions (H1)–(H3) are satisfied, then the solution of the MBVP (1.5) on satisfies
Lemma 2.3 (see ). If the conditions (H1)–(H3) are satisfied, thenwhere
Lemma 2.4 (see ).
Let denote the Banach space with the norm
Define the cone by
The solutions of MBVP (1.5) are the points of the operator defined bySo, It is easy to check that is completely continuous.
3. Existence of at least One Positive Solutions
Theorem 3.1 (see ). Let be a Banach space, and let be a cone. Assume and are open boundary subsets of with and let be a completely continuous operator such that
(i) for for or(ii) for for hold. Then A has a fixed point in
Theorem 3.2. Assume conditions (H1)–(H3) are satisfied. In addition, suppose there exist numbers such that if and if where
Then the MBVP (1.5) has at least one positive solution.
the cone as in (2.5), define a completely continuous
integral operator byFrom (H1)–(H3), Lemmas 2.1 and 2.2, .
If with then we get This implies that So, if we set then for .
Let us now set
Then for with by Lemma 2.4 we have Therefore, we haveHence, for Thus by the Theorem 3.1, has a fixed point in Therefore, the MBVP (1.5) has at least one positive solution.
4. Existence of at least Two Positive Solutions
Theorem 4.1 (see Avery and Henderson ). Let be a cone in a real Banach space Set
If and are increasing, nonnegative continuous functionals on let be a nonnegative continuous functional on with such that, for some positive constants and and for all . Suppose that there exist positive numbers such that for all and
If is a completely continuous operator satisfying
(i) for all ;(ii) for all ;(iii) and for all then has at least two fixed points and such that with and with
Let and Define the increasing, nonnegative and continuous functionals and on by and
From Lemma 2.4, for each
In addition, for each Lemma 2.3 implies
We also see that and for all and .
Theorem 4.2. Assume (H1)–(H3) hold, suppose there exist positive numbers such that the function satisfies the following conditions:
(B1) for and ;(B2) for and (B3) for and
Then the MBVP (1.5) has at least two positive solutions and such that with and with
now verify that all of the conditions of Theorem 4.1 are satisfied.
Define the cone as (2.5), define a completely continuous integral operator by
and as in (3.1). To verify that condition (i) of Theorem 4.1 holds, we choose , then This implies . Note that We have for As a consequence of (B3), for Since we have from Lemma 2.2, Then condition (i) of Theorem 4.1 holds.
Let . Then This implies for From (B2), we haveHence condition (ii) of Theorem 4.1 holds.
If we first define for , then So
Now, let then This mean that From (B1) and Lemma 2.4, we getThen condition (iii) of Theorem 4.1 holds.
Since all conditions of Theorem 4.1 are satisfied, the MBVP (1.5) has at least two positive solutions and such that with and with
5. Existence of at least Three Positive Solutions
Theorem 5.1 (see Leggett and Williams ). Let be a cone in the real Banach space Set
Suppose be a completely continuous operator and be a nonnegative continuous concave functional on with for all . If there exists such that the following condition hold:
(i) and for all (ii) for (iii) for with then has at least three fixed points and in satisfying with
Theorem 5.2. Assume (H1)–(H3) hold . Suppose that there exist constants such that
(C1) for and (C2) for and (C3) for and
Then the MBVP (1.5) has at least three positive solutions and such that with
conditions of Theorem 5.1 will be shown to be satisfied. Define the nonnegative
continuous concave functional to be the cone as in (2.5), and as in (3.1). We have for all If ,
then and from assumption (C1) , then we have
This implies that Thus, we have . Since and For we have Using assumption (C2), we obtainHence, condition (i) of Theorem 5.1 holds.
If from assumption (C3), we obtainThis implies that
Consequently, condition (ii) of Theorem 5.1 holds.
We suppose that with Then we getHence, condition (iii) of Theorem 5.1 holds.
Because all of the hypotheses of the Leggett-Williams fixed point theorem are satisfied, the nonlinear MBVP (1.5) has at least three positive solutions and such that and with
This work is supported by the Research and Development Foundation of College of Shanxi Province (no. 200811043).
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Copyright © 2009 Ying Zhang and ShiDong Qiao. 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.