Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential EquationsView this Special Issue
Computation of Positive Solutions for Nonlinear Impulsive Integral Boundary Value Problems with -Laplacian on Infinite Intervals
This paper deals with the existence and iteration of positive solutions for nonlinear second-order impulsive integral boundary value problems with -Laplacian on infinite intervals. Our approach is based on the monotone iterative technique.
The theory of impulsive differential equations has been emerging as an important area of investigation in recent years. It has been extensively applied to biology, biologic medicine, optimum control in economics, chemical technology, population dynamics, and so on. It is much richer because all the structure of its emergence has deep physical background and realistic mathematical model and coincides with many phenomena in nature. For an introduction of the basic theory of impulsive differential equations in , the reader is referred to see Lakshmikantham et al. [1, 2], Samoĭlenko and Perestyuk , and the references therein.
Boundary value problems on infinite intervals arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium; see [4–7], for example. In a recent paper , by means of a fixed-point theorem due to Avery and Peterson, Li and Nieto obtained some new results on the existence of multiple positive solutions for the following multipoint boundary value problem with a finite number of impulsive times on an infinite interval: where , , , , , and .
Boundary value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. Moreover, boundary value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems. They include two-point, three-point, and multipoint boundary value problems as special cases; see [9–14]. For boundary value problems with other integral boundary conditions and comments on their importance, we refer the reader to the papers [11–20] and the references therein.
There are relatively few papers available for integral boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times up to now. In , Zhang et al. investigated the existence of minimal nonnegative solution for the following second-order impulsive differential equation where , , , , , , and with . denotes the jump of at , that is, where and represent the right-hand limit and left-hand limit of at , respectively. has a similar meaning to .
In the past few years, the existence and the multiplicity of bounded or unbounded positive solutions to nonlinear differential equations on infinite intervals have been studied by several different techniques; we refer the reader to [5–8, 21–29] and the references therein. However, most of these papers only considered the existence of positive solutions under various boundary value conditions. Seeing such a fact, a natural question which arises is “how can we find the solutions when they are known to exist?” More recently, Ma et al.  and Sun et al. [31, 32] established iterative schemes for approximating the solutions for some boundary value problems defined on finite intervals by virtue of the iterative technique.
However, to the author's knowledge, the corresponding theory for impulsive integral boundary value problems with -Laplacian operator and infinite impulsive times on infinite intervals has not been considered till now. Motivated by previous papers, the purpose of this paper is to obtain the existence of positive solutions and establish a corresponding iterative scheme for the following impulsive integral boundary value problem of second-order differential equation with -Laplacian on an infinite interval where , , , , , , , and with , , and .
It is clear that Throughout this paper, we adopt the following assumptions.() on any subinterval of , and when are bounded, is bounded on . () is a nonnegative measurable function defined in and does not identically vanish on any subinterval of , and (), and there exist such that with .
If , , , then BVP (4) reduces to the following two-point boundary value problem: which has been studied in .
Compared with [8, 21], the main features of the present paper are as follows. Firstly, second-order differential operator is replaced by a more general -Laplacian operator. Secondly, in this paper, in boundary value conditions may not be zero which will bring about computational difficulties. Thirdly, by applying monotone iterative techniques, we construct successive iterative schemes starting off with simple known functions. It is worth pointing out that the first terms of our iterative schemes are simple functions. Therefore, the iterative schemes are significant and feasible.
The rest of this paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is presented to illustrate the main results.
2. Preliminaries and Several Lemmas
Definition 1. Let be a real Banach space. A nonempty closed set is said to be a cone provided that (1) for all and all ,(2) implies that .
Definition 2. A map is said to be concave on , if for all and .
Let is a map from into such that is continuous at , left continous at and exists for , exists and is continuous at , left continous at and exists for Obviously, . It is clear that is a Banach space with the norm and is also a Banach space with the norm where . Let , . Define a cone by
Remark 3. If satisfies (4), then , and which implies that is nonincreasing on ; that is, is also nonincreasing on . Thus, is concave on . Moreover, if , then , , and so is monotone increasing on .
Lemma 4. Let conditions hold. Then, with is a solution of BVP (4) if and only if is a fixed point of the following operator equation:
Proof. Suppose that with is a solution of BVP (4). For , integrating (4) from to , we have
which implies that
If , integrating (17) from 0 to , we get that
Integrating (17) from to , we obtain
Adding (18) and (19) together, we have
Repeating previous process, we get that
It follows that
Substituting (22) into (21), we get that
For , there exists such that . Set , and we have by and that By (6), (24), we have It follows from (24) and (25) that the right term in (23) is well defined. Thus, we have proved that is a fixed point of the operator defined by (14).
Conversely, suppose that is a fixed point of the operator equation (14). Evidently, Direct differentiation of (14) implies that, for , which means that . It is easy to verify that , . The proof of Lemma 4 is complete.
To obtain the complete continuity of , the following lemma is still needed.
Lemma 5 (see [33, 34]). Let be a bounded subset of . Then, is relatively compact in if and are both equicontinuous on any finite subinterval for any , and for any , there exists such that uniformly with respect to as , where , , .
This lemma is a simple improvement of the Corduneanu theorem in [33, 34].
Lemma 6. Let hold. Then is completely continuous.
Proof. For any , by (14), we have
It follows from (14), (29), and that , , , that is, . Now, we prove that is continuous and compact respectively. Let , as , then there exists such that . Let . By and , we have
It follows from (30) and dominated convergence theorem that
which implies that
By (30)–(32), and dominated convergence theorem, we get that
It follows from (33) that is continuous.
Let be any bounded subset. Then, there exists such that for any . Obviously, From (34), , and , we know that is bounded.
For any , with , by the absolute continuity of the integral, we have Thus, we have proved that is equicontinuous on any .
Next, we prove that for any , there exits sufficiently large such that For any , we have It follows from (37) that On the other hand, we arrive at Thus, (36) can be easily derived from (38) and (39). So, by Lemma 5, we know that is relatively compact. Thus, we have proved that is completely continuous.
3. Main Results
For notational convenience, we denote that
Theorem 7. Assume that hold, and there exists
such that for any ,
, for any .
Then, the boundary value problem (4) admits positive, nondecreasing on and concave solutions and such that , and , where and , , where .
Proof. We only prove the case that . Another case can be proved in a similar way. By Lemma 6, we know that is completely continuous. From the definition of , , and , we can easily get that for any with , . Denote that
In what follows, we first prove that . If , then . By (6), (40), (42), (44), , , and , we get that
Thus, we get that . Hence, we have proved that .
Let , , then . Let , , then by Lemma 6, we have that and . Denote that Since , we have that It follows from the complete continuity of that is a sequentially compact set. We assert that has a convergent subsequence , and there exists such that .
By (51), , we get that So, by (53) we have By induction, we get that Hence, we claim that as . Applying the continuity of and , we get that .
Let , , then . Let . By Lemma 6, we have that and . Denote Since , we have that . It follows from the complete continuity of that is a sequentially compact set. And, we assert that has a convergent subsequence and there exists such that .
Since , we have By , we have By induction, we get that Hence, we claim that as . Applying the continuity of and , we get that .
Since , then the zero function is not the solution of BVP (4). Thus, is a positive solution of BVP (4). By Lemma 4 we know that and are positive, nondecreasing on and concave solutions of the BVP (4).
We can easily get that Theorem 7 holds for in a similar manner.
Remark 8. The iterative schemes in Theorem 7 are , , and , , . They start off with a known simple linear function and the zero function respectively. This is convenient in application.
Theorem 9. Assume that hold, and there exist such that for any , , .