Existence of Solutions for Nonlinear Four-Point -Laplacian Boundary Value Problems on Time Scales
© S. Gulsan Topal et al. 2009
Received: 16 March 2009
Accepted: 20 July 2009
Published: 19 August 2009
We are concerned with proving the existence of positive solutions of a nonlinear second-order four-point boundary value problem with a -Laplacian operator on time scales. The proofs are based on the fixed point theorems concerning cones in a Banach space. Existence result for -Laplacian boundary value problem is also given by the monotone method.
Let be any time scale such that be subset of . The concept of dynamic equations on time scales can build bridges between differential and difference equations. This concept not only gives us unified approach to study the boundary value problems on discrete intervals with uniform step size and real intervals but also gives an extended approach to study on discrete case with non uniform step size or combination of real and discrete intervals. Some basic definitions and theorems on time scales can be found in [1, 2].
the function ,
the function and does not vanish identically on any closed subinterval of and ,
is continuous and satisfies that there exist such that for .
In recent years, the existence of positive solutions for nonlinear boundary value problems with -Laplacians has received wide attention, since it has led to several important mathematical and physical applications [3, 4]. In particular, for or is linear, the existence of positive solutions for nonlinear singular boundary value problems has been obtained [5, 6]. -Laplacian problems with two-, three-, and m-point boundary conditions for ordinary differential equations and difference equations have been studied in [7–9] and the references therein. Recently, there is much attention paid to question of positive solutions of boundary value problems for second-order dynamic equations on time scales, see [10–13]. In particular, we would like to mention some results of Agarwal and O'Regan , Chyan and Henderson , Song and Weng , Sun and Li , and Liu , which motivate us to consider the -Laplacian boundary value problem on time scales.
The aim of this paper is to establish some simple criterions for the existence of positive solutions of the -Laplacian BVP (1.1)-(1.2). This paper is organized as follows. In Section 2 we first present the solution and some properties of the solution of the linear -Laplacian BVP corresponding to (1.1)-(1.2). Consequently we define the Banach space, cone and the integral operator to prove the existence of the solution of (1.1)-(1.2). In Section 3, we state the fixed point theorems in order to prove the main results and we get the existence of at least one and two positive solutions for nonlinear -Laplacian BVP (1.1)-(1.2). Finally, using the monotone method, we prove the existence of solutions for -Laplacian BVP in Section 4.
2. Preliminaries and Lemmas
is a positive continuous function, therefore, has a minimum on , hence one supposes that there exists such that for .
It is easily seen that is continuous on .
Then, from condition (H2), we have that the function is strictly monoton nondecreasing on and , the function is strictly monoton nonincreasing on and , which implies
- (i). It follows from the concavity of that each point on the chard between and is below the graph of , thus(2.7)then(2.8)
this means for
- (ii). If , similarly, we have(2.9)If , similarly, we have(2.10)
this means for .
- (iii). Similarly we have(2.11)then(2.12)this means for From the above, we know(2.13)
which is a contradiction.
Throughout this paper, we assume that .
we get , for and , for , thus the operator is monotone increasing on and monotone decreasing on and also is the maximum point of the operator . So the operator is concave on and . Therefore, .
Suppose that the conditions (H1)–(H3) hold. is completely continuous.
Then, is bounded.
By the Arzela-Ascoli theorem, we can easily see that is completely continuous operator.
In order to follow the main results of this paper easily, now we state the fixed point theorems which we applied to prove Theorems 3.1–3.4.
Theorem 2.6 (see  (Krasnoselskii fixed point theorem)).
hold. Then has a fixed point in .
Theorem 2.7 (see  (Schauder fixed point theorem)).
Let be a Banach space, and let be a completely continuous operator. Assume is a bounded, closed, and convex set. If , then has a fixed point in .
Theorem 2.8 (see  (Avery-Henderson fixed point theorem)).
for all . Suppose that there exist positive numbers such that for all and
and for all ,
3. Main Results
In this section, we will prove the existence of at least one and two positive solution of -Laplacian BVP (1.1)-(1.2). In the following theorems we will make use of Krasnoselskii, Schauder, and Avery-Henderson fixed point theorems, respectively.
Assume that (H1)–(H3) are satisfied. In addition, suppose that satisfies
where and . Then the -Laplacian BVP (1.1)-(1.2) has a positive solution such that .
We define two open subsets and of such that and .
- (i)If , thus for , by and Lemma 2.1, we have(3.3)
- (ii)If , thus for , by and Lemma 2.1, we have(3.4)
- (iii)If , thus for , by and Lemma 2.1, we have(3.5)
Therefore, we have ,
Then, has a fixed point . Obviously, is a positive solution of the -Laplacian BVP (1.1)-(1.2) and .
Existence of at least one positive solution is also proved using Schauder fixed point theorem (Theorem 2.7). Then we have the following result.
then the -Laplacian BVP (1.1)-(1.2) has at least one positive solution.
which implies . The compactness of the operator follows from the Arzela-Ascoli theorem. Hence has a fixed point in .
If is continuous and bounded on , then the -Laplacian BVP (1.1)-(1.2) has a positive solution.
The function is positive and continuous on . Therefore, has a minimum on . Hence we suppose there exists such that
We observe here that, for every , and from Lemma 2.2, . Also, for ,
Define the cone as in (2.5). From Lemmas 2.2 and 2.3 and the conditions (H1) and (H2), we can obtain . Also from Lemma 2.5, we see that is completely continuous.
We now show that the conditions of Theorem 2.8 are satisfied.
- (a)If , we have(3.14)
- (b)If , we have(3.15)
- (c)If , we have(3.16)
Thus we have and condition (i) of Theorem 2.8 holds. Next we will show condition (ii) of Theorem 2.8 is satisfied. If , then .
we have , for .
Then (ii) yields for .
So condition (ii) of Theorem 2.8 holds.
- (a)If , we have(3.19)
- (b)If , we have(3.20)
- (c)If , we have(3.21)
Thus we have .
4. Monotone Method
In this section, we will prove the existence of solution of -Laplacian BVP (1.1)-(1.2) by using upper and lower solution method. We define the set
We will prove when the lower and the upper solutions are given in the well order, that is, , the -Laplacian BVP (1.1)-(1.2) admits a solution lying between both functions.
Assume that (H1)–(H3) are satisfied and and are, respectively, lower and upper solutions for the -Laplacian BVP (1.1)-(1.2) such that on . Then the -Laplacian BVP (1.1)-(1.2) has a solution on .
which is a contradiction and thus cannot be an element of .
which is a contradiction.
which is a contradiction. Thus we have on .
Similarly, we can get on . Thus is a solution of -Laplacian BVP (1.1)-(1.2) which lies between and .
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