- Open Access
Monotone iterative method for differential systems with coupled integral boundary value problems
© Cui and Zou; licensee Springer. 2013
- Received: 12 July 2013
- Accepted: 18 October 2013
- Published: 19 November 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.
- monotone iterative technique
- upper and lower solutions
- coupled integral boundary conditions
- Fredholm theorem
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).
We make assumptions involving , and κ as follows.
(H1) , , .
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.
then , .
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 .
which implies that , a contradiction. Hence, , , . □
where . Then , .
The proof of Lemma 2.3 is easy, so we omit it.
which is equivalent to system (2.4).
Hence, is a system of solutions of BVPs (2.3). □
where and .
Lemma 2.5 Assume that (H1) holds. Then there exists a unique system of solutions to BVPs (2.8).
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 . □
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.
where , .
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.
which implies by virtue of Lemma 2.2 that , , , i.e., S is nondecreasing. This together with (i) implies that .
uniformly on , and , satisfy (1.1). Moreover, . Thus, and are solutions of (1.1) in Ω.
Now, letting in (3.2), we have . That is, and are extremal systems of solutions of (1.1) in Ω. □
It shows that and are lower and upper systems of solutions of (4.1).
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.
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.
- Aronson DG: A comparison method for stability analysis of nonlinear parabolic problems. SIAM Rev. 1978, 20: 245-264. 10.1137/1020038MATHMathSciNetView ArticleGoogle Scholar
- Leung A: A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: equilibrium and stability. Indiana Univ. Math. J. 1982, 31: 223-241. 10.1512/iumj.1982.31.31020MATHMathSciNetView ArticleGoogle Scholar
- Zettl A Math. Surveys Monogr. 121. In Sturm-Liouville Theory. Am. Math. Soc., Providence; 2005.Google Scholar
- Cabada A: Extremal solutions and Green’s functions of higher order periodic boundary value problems in time scales. J. Math. Anal. Appl. 2004, 290: 35-54. 10.1016/j.jmaa.2003.08.018MATHMathSciNetView ArticleGoogle Scholar
- Lakshmikanthan V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Anal. TMA 2008, 69: 2677-2682. 10.1016/j.na.2007.08.042View ArticleMathSciNetMATHGoogle Scholar
- Nieto JJ, Rodriguez-Lopez R: Monotone method for first-order functional differential equations. Comput. Math. Appl. 2006, 52: 471-484. 10.1016/j.camwa.2006.01.012MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodriguez-Lopez R: Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl. 2008, 55: 2715-2731. 10.1016/j.camwa.2007.10.019MATHMathSciNetView ArticleGoogle Scholar
- Wei Z, Li Q, Che J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 2010, 367: 260-272. 10.1016/j.jmaa.2010.01.023MATHMathSciNetView ArticleGoogle Scholar
- Zhang S: Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal. 2009, 71: 2087-2093. 10.1016/j.na.2009.01.043MATHMathSciNetView ArticleGoogle Scholar
- Coster CD, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solution. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
- Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston; 1985.MATHGoogle Scholar
- Zanolin F: Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations. Springer, New York; 1996.View ArticleMATHGoogle Scholar
- Asif NA, Khan RA: Positive solutions to singular system with four-point coupled boundary conditions. J. Math. Anal. Appl. 2012, 386: 848-861. 10.1016/j.jmaa.2011.08.039MATHMathSciNetView ArticleGoogle Scholar
- Protter MH, Wiinberger HF: Maximum Principles in Differential Equations. Prentice Hall International, Englewood Cliffs; 1967.Google Scholar
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