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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 826504, 12 pages
Positive Solutions for Nonlinear First-Order m-Point Boundary Value Problem on Time Scales
Department of Mathematics, Pamukkale University, 20070 Denizli, Turkey
Received 7 August 2012; Accepted 24 October 2012
Academic Editor: Yongfu Su
Copyright © 2012 İsmail Yaslan. 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.
By means of fixed-point theorems, we investigate the existence of positive solutions for nonlinear first-order -point boundary value problem , , where is a time scale, , are given constants.
The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988 (see ). The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences; see the monographs of Aulbach and Hilger , Bohner and Peterson [3, 4], and Lakshmikantham et al.  and the references therein.
A time scale is an arbitrary nonempty closed subset of real numbers . A book on the subject of time scales by Bohner and Peterson  also summarizes and organizes much of the time scale calculus. The closed interval in is defined as where with .
In this study, we consider the nonlinear first-order -point boundary value problem where is a time scale, are given constants. is regressive and rd-continuous, and is continuous.
In , Cabada studied the following first-order periodic boundary value problem on time scales: He developed the monotone method in the presence of lower and upper solutions to obtain the existence of extremal solutions. When , and , BVP (1.2) is reduced to (1.3).
In , Sun studied the first-order boundary value problem where . Some existence results for at least two positive solutions were established, by using Avery-Henderson fixed-point theorem. When and , BVP (1.2) is reduced to (1.5).
In , Shu and Chunhua are concerned with the existence of three positive solutions for the following nonlinear first-order boundary value problem on time scale: where is fixed, , and is continuous. When and , BVP (1.2) is reduced to (1.5).
Sun and Li  studied the following first-order periodic boundary value problem on time scales: Conditions for the existence of at least one solution were obtained by using novel inequalities and the Schaefer fixed-point theorem. When and , BVP (1.2) is reduced to (1.6).
In , Tian and Ge studied the existence and uniqueness results for first-order three-point boundary value problem by using several well-known fixed-point theorems. When , BVP (1.2) is reduced to (1.7).
We have arranged the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krasnosel'skii fixed-point theorem, Avery-Henderson fixed-point theorem, and Leggett-Williams fixed-point theorem to prove the existence of at least one, two, and three positive solutions to BVP (1.2). In Section 4, as an application, the examples are included to illustrate our results.
Let denote the Banach space with the norm . For , we consider the following linear boundary value problem:
Lemma 2.1. For , BVP (2.1) has the unique solution where .
Proof. From , we have By using the boundary condition, we get Thus, satisfies (2.2).
Let be Green's function for the boundary value problem By Lemma 2.1, we obtain where for all .
Lemma 2.2. Green's function in (2.6) has the following properties:(i) for .(ii), where and .(iii) for .
Let denote the Banach space with the norm . Define the cone by Equation (1.2) is equivalent to the nonlinear integral equation
We can define the operator by Therefore solving (2.9) in is equivalent to finding fixed-points of the operator .
So, we have .
3. Main Results
To prove the existence of at least one positive solution for the BVP (1.2), we will need the following (Krasnosel'skii) fixed-point theorem.
Theorem 3.1 (Krasnosel'skii fixed-point theorem ). Let be a Banach space, and let be a cone. Assume and are open bounded subsets of 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. Let there exist numbers , satisfying such that for Then BVP (1.2) has at least one positive solution satisfying , .
Proof. It is easy to check by the Arzela-Ascoli theorem that the operator is completely continuous. Let us now define two bounded open sets as follows:
Then . For , we obtain
Hence for .
If , then and for . We have
Thus for . By the first part of Theorem 3.1, has a fixed-point in . Therefore, the BVP (1.2) has at least one positive solution satisfying , .
Now, we will apply the following (Avery-Henderson) fixed-point theorem to prove the existence of at least two positive solutions to BVP (1.2).
Theorem 3.3 (see ). 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 ,
for all . Suppose that there exist positive numbers such that
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
Theorem 3.4. Suppose there exist numbers , , and satisfying such that the function satisfies the following conditions:(i) for and ;(ii) for and ;(iii) for and .
Then the BVP (1.2) has at least two positive solutions and such that
Proof . Let the nonnegative increasing continuous functionals , , and be defined on the cone by
For each , we have and
Then . In addition, and for all , we obtain .
Now we will verify the remaining conditions of Theorem 3.3.
Claim 1. If , then . Since , we have for . Then, we get by hypothesis .
Claim 2. If , then . Since , for . Thus, by hypothesis we have
Claim 3. and for all . Since and , . If , we get for . Hence, we obtain by hypothesis . This completes the proof.
To prove the existence of at least three positive solutions for the BVP (1.2), we will apply the following (Leggett-Williams) fixed-point theorem.
Theorem 3.5 (see ). Let P be a cone in the real Banach space . Set
Suppose is a completely continuous operator and is 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
Theorem 3.6. Suppose that there exist numbers , , and satisfying such that for the function satisfies the following conditions:(i), ,(ii), ,(iii), . Then (1.2) has at least three positive solutions , and satisfying
Proof. Define the nonnegative continuous concave functional to be and the cone as in (2.8). For all , we have . If , then and from the hypothesis . Then we get
by Lemma 2.2. This proves that . Similarly, by the hypothesis , the condition of Theorem 3.5 is satisfied.
Since and , . For all , we have for . Using the hypothesis and Lemma 2.2, we find Hence, the condition of Theorem 3.5 holds.
For the condition of Theorem 3.5, we suppose that with . Then, from Lemma 2.2 we obtain This completes the proof.
Example 4.1. Let . We consider the first-order four-point BVP as follows: Taking , we have , and . If we take , and ; then all the assumptions in Theorem 3.4 are satisfied. Finally, BVP (4.1) has at least two positive solutions and such that
Example 4.2. Let . We consider the first-order four-point BVP as follows: where , and Hence, we obtain , and . If we take , and ; then all the assumptions in Theorem 3.6 are satisfied. Finally, BVP (4.3) has at least three positive solutions , and such that
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