Abstract
In this article, we use the Krasnosel’skii fixed point theorem, the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem to obtain some results for the existence of at least one, two or three positive solutions of m-point integral boundary value problems for nonlinear second-order p-Laplacian dynamic equations on time scales. Two examples are presented to illustrate the applications of the results.
MSC:34B15, 34N05.
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1 Introduction
Analysis on measure chains was initiated by Stefan Hilger [1] as a bridge between continuous and discrete calculus. Dynamic equations on time scales have been a component of applied analysis on measure chains to describe the processes that feature both continuous and discrete elements [2–6]. This subject not only gives a unified approach to the study of differential and difference equations, but also gives an extended approach to the study of dynamic equations with nonuniform step size or a combination of real and discrete domains. Further, the study of time scale equations has led to several important applications, e.g., in the study of economics, insect population models, heat transfer, stock market and epidemic models (see [7–10]), etc. Integral boundary value problems occur in the study of nonlocal phenomena in many different areas of applied mathematics, physics and engineering, e.g., in heat conduction, chemical engineering, underground water flow, thermo-elasticity, plasma physics, etc. (see [11–15] and the references therein).
Throughout this paper, we denote the one-dimensional p-Laplacian operator by , i.e., for with , where . For convenience, we make the blanket assumption that 0, T are points in a time scale ; for an interval , we always mean . Other types of an interval are defined similarly.
In 2007, Sun and Li [16] discussed the existence of at least one, two or three positive solutions of the following boundary value problem:
They used the Krasnosel’skii fixed point theorem, the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem to prove the existence of multiple positive solutions to problem (1.1)-(1.2).
In 2009, Zhang and Qiao [17] studied the existence criteria for the m-point boundary value problem:
They obtained some results for the existence of multiple positive solutions of problem (1.3)-(1.4) by using the Krasnosel’skii fixed point theorem, the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem.
In 2011, Li and Zhang [18] considered the existence of at least three positive solutions for the boundary value problem with integral boundary conditions:
They established some sufficient conditions for the existence of positive solutions to problem (1.5)-(1.6) by using the Legget-Williams fixed point theorem. For some recent results on the existence of positive solutions for p-Laplacian dynamic equations on time scales, see [19–27]. However, to the best of the authors’ knowledge, existence results for positive solutions of m-point integral boundary value problems for nonlinear p-Laplacian dynamic equations on time scales have not been studied.
In this article, we are concerned with the existence of multiple positive solutions to the m-point integral boundary value problem for a second-order p-Laplacian dynamic equation on time scale :
where is a time scale, and
(H1) such that for , ;
(H2) ;
(H3) and there exists such that .
The rest of the paper is organized as follows. In Section 2, we state and prove some lemmas which are used later. In Section 3, we use the Krasnosel’skii [28] fixed point theorem to obtain the existence of at least one positive solution of problem (1.7)-(1.8). In Section 4, by using the Avery-Henderson [29] fixed point theorem, we establish sufficient conditions for the existence of at least two positive solutions of problem (1.7)-(1.8). In Section 5, the existence of at least three positive solutions of problem (1.7)-(1.8) are proved by using the Leggett-Williams [30] fixed point theorem. Two illustrative examples are given in Section 6.
For convenience, we list the following well-known definitions which can be found in [4] and the references therein.
Definition 1.1 A time scale is an arbitrary nonempty closed subset of the real set ℝ with topology and ordering inherited from ℝ.
The forward and backward jump operators and the graininess are defined, respectively, by
for all . If , t is said to be right scattered, and if , t is said to be left scattered; if , t is said to be right dense, and if , t is said to be left dense. If has a left-scattered maximum M, define ; otherwise set .
Definition 1.2 A function is rd-continuous (rd-continuous is short for right-dense continuous) provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by .
Definition 1.3 For and , the delta derivative of f at the point t is defined to be the number (provided it exists), with the property that for each , there is a neighborhood U of t such that
for all .
Definition 1.4 For a function , the delta derivative is defined at the point t by
if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined by
provided this limit exists.
Definition 1.5 If , then we define the delta integral by
2 Preliminaries
In this section, we first prove and recall some lemmas which are used in what follows.
Lemma 2.1 Let . Then, for , the problem
has a unique solution
Proof Integrating (2.1) from 0 to t and using the first condition of (2.2), one gets
Integrating (2.4) from 0 to t, we obtain
In particular, for , we have
Using the second condition of (2.2), we get that
Hence,
Substituting the value of in (2.5), we obtain the solution (2.3). □
Lemma 2.2 Let . If , then the unique solution u of problem (2.1)-(2.2) satisfies
Proof From (2.4), we have for . In fact, is a monotone increasing continuously differentiable function and
Then, by the chain rule [4], we get for . □
Lemma 2.3 Let . If , then the unique solution u of problem (2.1)-(2.2) satisfies
Proof From Lemma 2.2, for , we know that u is nonincreasing on . Consequently, for each and , it holds that .
Therefore,
If , then the second condition of (2.2) together with (2.6) implies that
This contradicts the fact that .
If , it follows that since u is nonincreasing. Hence, we get a contradiction. Indeed, if and , we again obtain a contradiction. □
Lemma 2.4 Let . If , then problem (2.1)-(2.2) has no positive solutions.
Proof Suppose that problem (2.1)-(2.2) has a positive solution u satisfying for . Then for all . By the second condition of (2.2) and (2.6), we have
getting a contradiction. □
Let E denote the Banach space with the norm . Define the cone , by
Define the operator by
where a positive constant . In view of Lemma 2.1, the solutions of problem (1.7)-(1.8) are given by the operator equation, .
From (2.8), we claim that for each , and satisfies (1.8). In fact, for , we get
This implies that for . As in Lemma 2.2, we can prove that , for . In addition, we find that and . So, . It is also easy to check that is completely continuous.
Lemma 2.5 Let (H1) hold. If , then
where
which .
Proof Since for , we have , .
Thus,
From for and (2.11), we get
This implies that
Note that (H1) yields
Thus we have . The proof of Lemma 2.5 is complete. □
In the following, for the sake of convenience, we set constants
3 Existence of at least one positive solution
Now we are in a position to establish the main result. Our first result is based on the Krasnosel’skii fixed point theorem.
Theorem 3.1 (see [28])
Let E be a Banach space, and let be a cone. Assume that and are bounded open subsets of E with , , and let be a completely continuous operator such that either
-
(i)
for , for ; or
-
(ii)
for , for hold.
Then A has a fixed point in .
Theorem 3.2 Assume that (H1)-(H3) hold. In addition, suppose that there exist numbers such that
(A1) for and ;
(A2) for and ,
where constants L, M are defined by (2.12) and (2.13), respectively.
Then problem (1.7)-(1.8) has at least one positive solution.
Proof Firstly, we define a cone P and a completely continuous operator as in (2.7) and (2.8), respectively.
Let . For any with , from condition (A1), we obtain
This implies that for .
Set . Since , it follows that . Hence from condition (A2), for any , we have
Therefore, for .
Thus, from Theorem 3.1, it follows that A has a fixed point u in such that . Therefore, problem (1.7)-(1.8) has at least one positive solution. □
4 Existence of at least two positive solutions
In this section, we obtain the existence of at least two positive solutions of problem (1.7)-(1.8) by using the Avery-Henderson fixed point theorem which is as follows.
Theorem 4.1 (see [29])
Let P be a cone in a real Banach space E. Set
Let ν and Φ be increasing nonnegative continuous functionals on P, and let θ be a nonnegative continuous functional on P with such that, for some and ,
for all . Suppose there exist a completely continuous operator and such that
and
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
and for all .
Then A has at least two fixed points and belonging to satisfying
Define a constant such that . Let Φ, θ and ν be increasing, nonnegative and continuous functionals on P, defined by
Obviously, for each . Moreover, Lemma 2.5 implies for each . It is easy to see that and for all and .
We can now prove the following theorem.
Theorem 4.2 Assume that (H1)-(H3) hold, and suppose that there exist positive numbers such that the function f satisfies the following conditions:
(B1) for and ;
(B2) for and ;
(B3) for and ,
where constants L, M, N are defined by (2.12), (2.13) and (2.14), respectively.
Then problem (1.7)-(1.8) has at least two positive solutions and such that with and with .
Proof We now wish to prove that all of the conditions of Theorem 4.1 are satisfied. For this purpose, we define the cone P as (2.7) and a completely continuous operator by (2.8).
To check condition (i) of Theorem 4.1, we choose , then . This implies that . For , we have . From condition (B3), we get that for . Since , we obtain
Hence, condition (i) of Theorem 4.1 holds.
We now prove that condition (ii) in Theorem 4.1 holds. In fact, for , we have . This implies that for . From condition (B2), we have
This shows that condition (ii) of Theorem 4.1 is satisfied.
Now, we assert that condition (iii) of Theorem 4.1 also holds. If for , then . Thus . Let , then . So that . From condition (B1), for any , we have
Therefore, condition (iii) of Theorem 4.1 is satisfied.
Thus, by Theorem 4.1, problem (1.7)-(1.8) has at least two positive solutions and such that with and with . □
5 Existence of at least three positive solutions
In this section, we use the Leggett-Williams fixed point theorem to prove the existence of at least three positive solutions to problem (1.7)-(1.8). The Leggett-Williams fixed point theorem is as follows.
Theorem 5.1 (see [30])
Let P be a cone in the real Banach space E. Set
Let be a completely continuous operator and let Ψ be a nonnegative continuous concave functional on P with for all . Suppose that there exists such that the following conditions hold:
-
(i)
and for all ;
-
(ii)
for ;
-
(iii)
for with .
Then A has at least three fixed points , and in satisfying , , with .
We now prove the following result.
Theorem 5.2 Assume that (H1)-(H3) hold. Suppose that there exist constants such that
(C1) for and ;
(C2) for and ;
(C3) for and ,
where constants L, M are defined by (2.12) and (2.13), respectively.
Then problem (1.7)-(1.8) has at least three positive solutions , and such that , , with .
Proof We will show that all the conditions of Leggett-Williams Theorem 5.1 hold with respect to the operator A defined in (2.8).
At first, we define a nonnegative continuous concave functional by , where the cone P is defined by (2.7). In fact, for , we get . If , then . From condition (C1), we obtain
This implies that . Therefore, we have . Since and , then . For , we get . By using condition (C2), we obtain
Hence, condition (i) of Theorem 5.1 is satisfied.
Indeed, if , then condition (C3) implies that
Thus . Therefore, condition (ii) of Theorem 5.1 holds.
We finally show that condition (iii) of Theorem 5.1 also holds. Assume that , with . Then we obtain
So, condition (iii) of Theorem 5.1 is satisfied. Therefore, an application of Theorem 5.1 implies that problem (1.7)-(1.8) has at least three positive solutions , and such that , and with . □
6 Numerical examples
In this section, we present some examples to illustrate our results.
Example 6.1 Consider the following six-point integral boundary value problem with and :
where
Set , , , , , , , , , and . We can show that
Through a simple calculation we can get
Choose and , then satisfies
and
By Theorem 3.2, we have that boundary value problem (6.1)-(6.2) has at least one positive solution.
Example 6.2 Consider the following six-point integral boundary value problem with and (ℕ stands for the natural number set).
where
Set , , , , , , , , , and . We can show that
Through a simple calculation we can get
Choose , and , then satisfies
and
and
By Theorem 5.2, we get that problem (6.3)-(6.4) has at least three positive solutions , and such that , and with .
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Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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Thiramanus, P., Tariboon, J. Positive solutions of m-point integral boundary value problems for second-order p-Laplacian dynamic equations on time scales. Adv Differ Equ 2013, 206 (2013). https://doi.org/10.1186/1687-1847-2013-206
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DOI: https://doi.org/10.1186/1687-1847-2013-206