Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems
© N. A. Asif and R. A. Khan. 2009
Received: 27 February 2009
Accepted: 15 May 2009
Published: 22 June 2009
Existence of positive solutions for a coupled system of nonlinear three-point boundary value problems of the type , , , , , , , is established. The nonlinearities , are continuous and may be singular at , and/or , while the parameters , satisfy . An example is also included to show the applicability of our result.
Multipoint boundary value problems (BVPs) arise in different areas of applied mathematics and physics. For example, the vibration of a guy wire composed of parts with a uniform cross-section and different densities in different parts can be modeled as a Multipoint boundary value problem . Many problems in the theory of elastic stability can also be modeled as Multipoint boundary value problem .
The study of Multipoint boundary value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev, [3, 4], and extended to nonlocal linear elliptic boundary value problems by Bitsadze et al. [5, 6]. Existence theory for nonlinear three-point boundary value problems was initiated by Gupta . Since then the study of nonlinear three-point BVPs has attracted much attention of many researchers, see [8–11] and references therein for boundary value problems with ordinary differential equations and also  for boundary value problems on time scales. Recently, the study of singular BVPs has attracted the attention of many authors, see for example, [13–18] and the recent monograph by Agarwal et al. .
where , and may be singular at , , and/or .
where , , . We allow and to be singular at , , and also and/or . We study the sufficient conditions for existence of positive solution for the singular system (1.3) under weaker hypothesis on and as compared to the previously studied results. We do not require the system (1.3) to have lower and upper solutions. Moreover, the cone we consider is more general than the cones considered in [20, 21, 26].
By singularity, we mean the functions and are allowed to be unbounded at , , , and/or . To the best of our knowledge, existence of positive solutions for a system (1.3) with singularity with respect to dependent variable(s) has not been studied previously. Moreover, our conditions and results are different from those studied in [21, 24–26]. Throughout this paper, we assume that are continuous and may be singular at , , , and/or . We also assume that the following conditions hold:
The main result of this paper is as follows.
Assume that hold. Then the system (1.3) has at least one positive solution.
For each , we write . Let . Clearly, is a Banach space and is a cone. Similarly, for each , we write . Clearly, is a Banach space and is a cone in . For any real constant , define . By a positive solution of (1.3), we mean a vector such that satisfies (1.3) and , on . The proofs of our main result (Theorem 1.1) is based on the Guo's fixed-point theorem.
Lemma 2.1 (Guo's Fixed-Point Theorem ).
Let be a cone of a real Banach space , , be bounded open subsets of and . Suppose that is completely continuous such that one of the following condition hold:
(i) for and for ;
(ii) for and for .
Then, has a fixed point in .
The following result can be easily verified.
Let such that . Let , and concave on . Then, for all .
Choose such that . For fixed and , the linear three-point BVP
Lemma 2.2 (see ).
We need the following properties of the Green's function in the sequel.
Lemma 2.3 (see ).
Following the idea in , we calculate upper bound for the Green's function in the following lemma.
For , we discuss various cases.
For , the maximum occurs at , hence
For , the maximum occurs at , so
Now, we consider the nonlinear nonsingular system of BVPs
Clearly, if is a fixed point of , then is a solution of the system (2.19).
Assume that holds. Then is completely continuous.
which implies that is equicontinuous. Similarly, using (2.22)–(2.27), we can show that is also equicontinuous. Thus, is equicontinuous. By Arzelà-Ascoli theorem, is relatively compact. Hence, is a compact operator.
that is, is continuous. Hence, is completely continuous.
3. Main Results
Proof of Theorem 1.1.
implies that for all . Similarly, for all . The proof of Theorem 1.1 is complete.
where the real constants satisfy , with and the real constants satisfy , with . Clearly, and satisfy the assumptions . Hence, by Theorem 1.1, the system (1.3) has a positive solution.
Research of R. A. Khan is supported by HEC, Pakistan, Project 2- 3(50)/PDFP/HEC/2008/1.
- Moshinsky M: Sobre los problems de conditions a la frontiera en una dimension de caracteristicas discontinuas. Boletin Sociedad Matemática Mexicana 1950, 7: 1–25.MathSciNetGoogle Scholar
- Timoshenko T: Theory of Elastic Stability. McGraw-Hill, New York, NY, USA; 1971.Google Scholar
- Il'in VA, Moiseev EI: A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations. Differential Equations 1987, 23(7):803–810.MATHMathSciNetGoogle Scholar
- Il'in VA, Moiseev EI: A nonlocal boundary value problem of the second kind for the Sturm-Liouville operator. Differential Equations 1987, 23(8):979–987.MATHMathSciNetGoogle Scholar
- Bitsadze AV: On the theory of nonlocal boundary value problems. Soviet Mathematics—Doklady 1984, 30: 8–10.MATHGoogle Scholar
- Bitsadze AV: On a class of conditionally solvable nonlocal boundary value problems for harmonic functions. Soviet Mathematics—Doklady 1985, 31: 91–94.MATHGoogle Scholar
- Gupta CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. Journal of Mathematical Analysis and Applications 1992, 168(2):540–551. 10.1016/0022-247X(92)90179-HMATHMathSciNetView ArticleGoogle Scholar
- Liu B, Liu L, Wu Y: Positive solutions for singular systems of three-point boundary value problems. Computers & Mathematics with Applications 2007, 53(9):1429–1438. 10.1016/j.camwa.2006.07.014MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Positive solutions of a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 1999, 1999(34):1–8.Google Scholar
- Webb JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Analysis: Theory, Methods & Applications 2001, 47(7):4319–4332. 10.1016/S0362-546X(01)00547-8MATHMathSciNetView ArticleGoogle Scholar
- Zhao Z: Solutions and Green's functions for some linear second-order three-point boundary value problems. Computers & Mathematics with Applications 2008, 56(1):104–113. 10.1016/j.camwa.2007.11.037MATHMathSciNetView ArticleGoogle Scholar
- Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scales. Journal of Difference Equations and Applications 2008, 14(7):723–736. 10.1080/10236190701840906MATHMathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
- van den Berg M, Gilkey P, Seeley R: Heat content asymptotics with singular initial temperature distributions. Journal of Functional Analysis 2008, 254(12):3093–3122. 10.1016/j.jfa.2008.03.002MATHMathSciNetView ArticleGoogle Scholar
- Chu J, Nieto JJ: Recent existence results for second order singular periodic differential equations. Boundary Value Problems. In press
- Chu J, Franco D: Non-collision periodic solutions of second order singular dynamical systems. Journal of Mathematical Analysis and Applications 2008, 344(2):898–905. 10.1016/j.jmaa.2008.03.041MATHMathSciNetView ArticleGoogle Scholar
- Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bulletin of the London Mathematical Society 2008, 40(1):143–150. 10.1112/blms/bdm110MATHMathSciNetView ArticleGoogle Scholar
- Orpel A: On the existence of bounded positive solutions for a class of singular BVPs. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(4):1389–1395. 10.1016/j.na.2007.06.031MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, O'Regan D: Singular Differential and Integral Equations with Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+402.MATHView ArticleGoogle Scholar
- Wang H: On the number of positive solutions of nonlinear systems. Journal of Mathematical Analysis and Applications 2003, 281(1):287–306.MATHMathSciNetView ArticleGoogle Scholar
- Xie S, Zhu J: Positive solutions of boundary value problems for system of nonlinear fourth-order differential equations. Boundary Value Problems 2007, 2007:-12.Google Scholar
- Zhou Y, Xu Y: Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations. Journal of Mathematical Analysis and Applications 2006, 320(2):578–590. 10.1016/j.jmaa.2005.07.014MATHMathSciNetView ArticleGoogle Scholar
- Kang P, Wei Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):444–451. 10.1016/j.na.2007.12.014MATHMathSciNetView ArticleGoogle Scholar
- Lü H, Yu H, Liu Y: Positive solutions for singular boundary value problems of a coupled system of differential equations. Journal of Mathematical Analysis and Applications 2005, 302(1):14–29. 10.1016/j.jmaa.2004.08.003MATHMathSciNetView ArticleGoogle Scholar
- Wei Z: Positive solution of singular Dirichlet boundary value problems for second order differential equation system. Journal of Mathematical Analysis and Applications 2007, 328(2):1255–1267. 10.1016/j.jmaa.2006.06.053MATHMathSciNetView ArticleGoogle Scholar
- Yuan Y, Zhao C, Liu Y: Positive solutions for systems of nonlinear singular differential equations. Electronic Journal of Differential Equations 2008, 2008(74):1–14.MathSciNetGoogle Scholar
- Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar
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