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Monotone iterative method for differential systems with coupled integral boundary value problems
Boundary Value Problemsvolume 2013, Article number: 245 (2013)
By establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique, we investigate the differential systems with coupled integral boundary value problems. Sufficient conditions are established for the existence of an extremal system of solutions of the given problem.
We will devote the paper to considering the existence of a solution of coupled integral boundary value conditions for the second-order ordinary differential system (ODS for short)
where A and B are right continuous on , left continuous at and nondecreasing on , ; and denote the Riemann-Stieltjes integrals of u with respect to A and B, respectively.
The theory of differential system with coupled boundary value conditions is an important branch of nonlinear analysis. It is worth mentioning that a differential system with coupled boundary value conditions appears often in investigations connected with mathematical physics, mathematical biology, biochemical system and so on (see [1–3]). One of the basic problems considered in the theory of differential system with coupled boundary value conditions is to establish convenient conditions guaranteeing the existence of solutions of those equations. However, the theory of coupled boundary value problems for a differential system is still in the initial stages.
The monotone iterative technique combined with the method of upper and lower solutions is a powerful tool for proving the existence of solutions of differential equations/system (see [4–9] and the references therein), the advantage and importance of the technique needs no special emphasis [10–12]. The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences from the corresponding linear differential equations/system, and these sequences converge monotonically to the maximal and minimal solutions of the nonlinear differential equations/system. When the method is applied to a differential system with coupled boundary value conditions, it usually needs suitable differential inequalities as a comparison principle. The results in this paper are inspired by . Here, we establish differential inequalities as a comparison principle, i.e., Lemma 2.2. Then, we give a different proof for the existence and uniqueness of the solutions for linear coupled boundary value conditions for a differential system, i.e., Lemma 2.5. Finally, by use of the monotone iterative technique and the method of upper and lower solutions, we obtain the existence result of extremal solutions for (1.1).
2 Preliminary results
We make assumptions involving , and κ as follows.
(H1) , , .
Definition 2.1 is called a lower system of solutions of differential system (1.1) if
Analogously, is called an upper system of solutions of differential system (1.1) if
In what follows, we assume that
and define that sector
where the vectorial inequalities mean that the same inequalities hold between their corresponding components.
We present new comparison results and lemmas which are crucial for our discussion.
Suppose that . If there exist and satisfying
then , .
Lemma 2.2 Suppose that (H1) holds. Let satisfy
where . Then , , .
Proof Suppose the contrary. By Lemma 2.1, one easily sees that there are only three cases to consider:
Case 1. and . By Lemma 2.1, for all . Then , which contradicts .
Case 2. and . By Lemma 2.1, for all . So, we have , which contradicts .
Case 3. and . There are such that
It follows from (2.2) that
Hence, we have
The last two inequalities give
which implies that , a contradiction. Hence, , , . □
Lemma 2.3 Suppose that (H1) holds. Let satisfy
where . Then , .
The proof of Lemma 2.3 is easy, so we omit it.
Consider the differential system of BVPs
Lemma 2.4 Assume that (H1) holds. Then is a system of solutions of BVPs (2.3) if and only if is a system of solutions of the integral equation
Proof First, suppose that is a system of solutions of BVPs (2.3). It is easy to see that (2.3) is equivalent to the system of integral equations
Integrating (2.5) and (2.6) with respect to and respectively on gives
Substituting (2.7) into (2.5) and (2.6), we have
which is equivalent to system (2.4).
Conversely, assume that is a system of solutions of an integral equation. Direct differentiation on (2.4) implies
Making use of the fact for , we obtain
Simple computations yield
Hence, is a system of solutions of BVPs (2.3). □
Consider the linear differential system of BVPs
where and .
Lemma 2.5 Assume that (H1) holds. Then there exists a unique system of solutions to BVPs (2.8).
Proof It follows from Lemma 2.4 that (2.8) is equivalent to the operator equation
By using standard arguments, we can easily show that is linear completely continuous. By Lemma 2.3, the operator equation has only the zero solution. Then, by the Fredholm theorem, for given , the operator equation has only one solution in . Hence, (2.8) has exactly one system of solutions . □
3 Main results
In this section, on the basis of Lemma 2.2 and Lemma 2.5, using the monotone iterative technique, we shall show an existence theorem of a solution of (1.1).
We list the following assumptions for convenience.
(H2) is nondecreasing in y and there exists such that
where , .
(H3) is nondecreasing in x and there exists such that
where , .
Theorem 3.1 Assume that , are lower and upper systems of solutions of problem (1.1) such that (2.1) holds, and satisfies (H1)-(H3). Then there exist monotone iterative sequences , which converge uniformly on to the extremal solutions of problem (1.1) in the sector Ω.
Proof , consider (2.8) with
By Lemma 2.5, (2.8) has a unique system of solutions . Denote an operator by , and
Then the operator S has the following properties:
Let , and . By (H2) and (H3), we have that
which implies, by virtue of Lemma 2.2, that , , , i.e., . A similar argument shows that . (ii) S is nondecreasing. Let be such that . Suppose that . By (H2) and (H3), we have
which implies by virtue of Lemma 2.2 that , , , i.e., S is nondecreasing. This together with (i) implies that .
Now let , , . Following (i) and (ii), we have
Using the standard arguments, it is easy to show that and are uniformly bounded and equicontinuous in Ω. By (3.1) and the Arzela-Ascoli theorem, we have
uniformly on , and , satisfy (1.1). Moreover, . Thus, and are solutions of (1.1) in Ω.
Next, we prove that and are extremal solutions of (1.1) in Ω. In fact, we assume that is any solution of (1.1). That is,
By (H2) and (H3), and Lemma 2.2, it is easy by induction to show that
Now, letting in (3.2), we have . That is, and are extremal systems of solutions of (1.1) in Ω. □
Consider the following problems:
Take , , then
It shows that and are lower and upper systems of solutions of (4.1).
On the other hand, by (4.2), we have that
where , .
Thus, all the conditions of Theorem 3.1 are satisfied. Therefore, by Theorem 3.1, (4.1) has an extremal system of solutions , , which can be obtained by taking limits from some iterative sequences.
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The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province and Foundation of SDUST.
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.