Research Article | Open Access
Upper and Lower Solutions for -Point Impulsive BVP with One-Dimensional -Laplacian
Upper and lower solutions theories are established for a kind of -point impulsive boundary value problems with -Laplacian. By using such techniques and the Schauder fixed point theorem, the existence of solutions and positive solutions is obtained. Nagumo conditions play an important role in the nonlinear term involved with the first-order derivatives.
In this paper, we study the following -point impulsive boundary value problem with one-dimensional -Laplacian: where , . , (, where is a fixed positive integer) are fixed points with . . , . , , where and represent the right-hand limit and left-hand limit of at , respectively.
The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations in , see Lakshmikantham et al. , Baĭnov and Simeonov , Samoilenko and Perestyuk , and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see for instance [4–9] and the references therein).
At the same time, it is well known that the method of lower and upper solutions is a powerful tool for proving the existence results for a large class of boundary value problems. For a few of such works, we refer the readers to [10–14].
In , Cabada and Pouso considered by using the upper and lower method, the authors get the existence of solution to the above BVP.
In , Lü et al. studied by giving conditions on involving pairs of lower and upper solutions, they get the existence of at least three solutions to the above BVP.
In , Shen and Wang studied they prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.
Motivated by the works mentioned above, in this paper, we considered BVP (1), the main tool is upper and lower method, and the Schäuder fixed point theorem. We not only get the existence of solutions, but also the existence of positive solutions.
The main structure of this paper is as follows. In Section 2, we give the preliminary and present some lemmas in order to prove our main results. Section 4 presents the main theorems of this paper, and at the end of Section 4, we give an example to illustrate our main results.
Define is continuous at , , exists, left continuous}, and , is continuous at , and exists, , . Then, is a real Banach space with the norm , where .
Next, we define the Nagumo condition we are going to use. Note that the condition does not depend on the boundary data of the problem.
Definition 2. We say that satisfies a Nagumo condition relative to the pair and , with , in , if there exists a function such that where ; , and also that here, , .
From Definition 2, we can find a real number such that
In addition, we assume that the following three conditions hold:) with on and ;(); (), . is nondecreasing in for all .
Firstly, we define
One can find the next result, with its proof, in .
Lemma 3. For each , the next two properties hold:(i) exists for a.e. ,(ii)if , and in , then
We consider the following modified problem: with where is defined by
Thus is a continuous function on and satisfies for some constant . Moreover, we may choose so that .
Lemma 4. Suppose that , denote , then the following boundary value problem: has a unique solution as follows:
For , integrating (17) from to , we get
For , integrating (17) from to , we get
Combining (18), (19), and the boundary condition, for , we have
For , integrating (17) from to , and combining (20), we have
By induction, for , there holds Integrating (22) from to , we have For , integrating (22) from to , we have By the boundary condition , (23), and (24), for , we have Repeating in the above manner, for , we have By (26), we have By the boundary condition , we obtain Substituting (26) into (28), we have (16) which holds. The proof is complete.
Proof. Denote , we will only see that for every . An analogous reasoning shows that for all .
Otherwise, if , does not hold, then . Since , there are three cases.
Case 1. Denote , . There exists , such that . Then, . Let , . If for all , let , and if for all , let . Thus, As a consequence, for all and since is increasing and, in particular, one to one, we have Thus, , which is a contradiction.
Case 2. According to Case 1, if , then or , . Since , cannot be .
Next, we claim that for , if , then . Otherwise, . Suppose that and .
From the boundary condition and (29), we have Hence, Suppose that and is nonincreasing on some interval , where is sufficiently small such that on . For , Note that as . Hence, when is sufficiently small, , , which contradicts the assumption of monotonicity of . Thus, we obtain We use the preceding procedure and deduce by induction that which contradicts that . Essentially, by the same analysis, we can get that , cannot hold.
Case 3. . Easily, it holds that . While by the boundary condition and , we have which is also a contradiction.
Consequently, holds for all . The proof is complete.
Lemma 6. If is a solution of (11), then for all .
Proof. Let be a solution of (11). From Lemma 5, we have , and so By the mean value theorem, there exists with , , and as a result, Let . Suppose that there exists a point in the interval for which or . From the continuity of , we can choose , such that one of the following situations holds: (i), and for all ;(ii), and for all ;(iii), and for all ;(iv), and for all . Without loss of generality, suppose that for all . Then, and so As a result, Note also that for , so we have and thus , which leads to the following: a contradiction. The proof is complete.
Theorem 7 (Schäuder fixed point theorem). Let be a convex subset of a normed Linear space . Each continuous, compact map has a fixed point.
3. The Main Results
Theorem 8. Suppose that conditions ()–() hold. Then, BVP (1) has at least one solution such that
Proof. Solving (11) is equivalent to finding a which satisfies
where , , .
Now, define the following operator by It is obvious that is completely continuous.
By the Schäuder fixed point Theorem 7, we can easily obtain that has a fixed point , which is a solution of BVP (11). And by Lemmas 5 and 6, we know that , , then BVP (11) becomes BVP (1), therefore is a solution of BVP (1). The proof is complete.
Theorem 9. Suppose that conditions ()–() hold. Assume that there exist two lower solutions and and two upper solutions and for problem (1), satisfying the following:(i); (ii); (iii), which means that there exists such that ;(iv)if is a solution of (1) with , then on ;(v)if is a solution of (1) with , then on . If satisfies the Nagumo condition with respect to , , then problem (1) has at least three solutions , , and satisfying
Proof. We consider the following modified problem:
Now, define the following operator by
where , , and .
It is standard that is completely continuous.
Let , where , , , .
It is immediate from the argument above that . Thus, Let Since , , (i.e., choose such that , . It follows that , , and .
By assumptions (iv) and (v), there are no solutions in . Thus, We show that , then and there are solutions in , , as required.
We now show . The proof that is similar and hence omitted. We define , the extension to of the restriction of to as follows. Let where (in replace by ) and are previously defined. Thus, is a continuous function on and satisfies for some constants . Moreover, we may choose so that , .
Consider the following problem: Now, define the following operator: where , , and . Again, it is easy to check (from a previous argument and (v)) that is a solution of (57) if and (note that is compact).
Thus, . Moreover, it is easy to see that . By assumptions (iv) and (v), there are no solutions in . Thus, Thus there are three solutions, as required. The proof is complete.
A slight modification of the argument in Theorem 9 yields the next result.
Theorem 10. Suppose that conditions ()–() hold. Assume that there exist two lower solutions and and two upper solutions and for problem (1), satisfying(i); (ii); (iii); (iv)there exists such that all , the function , and are, respectively, lower and upper solution of (1);(v). If satisfies the Nagumo condition with respect to , , then problem (1) has at least three solutions , , and satisfying
Theorem 11. Let and satisfies (). Furthermore, the following conditions hold.(i)BVP (1) has a pair of positive upper and lower solutions , satisfying
(ii) satisfies a Nagumo condition relative to the pair and .
Then, BVP (1) has at least one positive solution such that
Proof. It can be proved easily, we omit it here.
Example 1. Consider the following BVP: where , . Let ; then, , which means that .
Clearly, . After substituting , into , , denote