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Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems

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Abstract

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.

1. Introduction

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 [1]. Many problems in the theory of elastic stability can also be modeled as Multipoint boundary value problem [2].

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 [7]. Since then the study of nonlinear three-point BVPs has attracted much attention of many researchers, see [811] and references therein for boundary value problems with ordinary differential equations and also [12] for boundary value problems on time scales. Recently, the study of singular BVPs has attracted the attention of many authors, see for example, [1318] and the recent monograph by Agarwal et al. [19].

The study of system of BVPs has also fascinated many authors. System of BVPs with continuous nonlinearity can be seen in [2022] and the case of singular nonlinearity can be seen in [8, 21, 2326]. Wei [25], developed the upper and lower solutions method for the existence of positive solutions of the following coupled system of BVPs:

(1.1)

where , and may be singular at , , and/or .

By using fixed point theorem in cone, Yuan et al. [26] studied the following coupled system of nonlinear singular boundary value problem:

(1.2)

are allowed to be superlinear and are singular at and/or . Similarly, results are studied in [8, 21, 23].

In this paper, we generalize the results studied in [25, 26] to the following more general singular system for three-point nonlocal BVPs:

(1.3)

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, 2426]. Throughout this paper, we assume that are continuous and may be singular at , , , and/or . We also assume that the following conditions hold:

(A1) and satisfy

(1.4)

(A2)There exist real constants such that , , and for all , ,

(1.5)

(A3)There exist real constants such that , , and for all , ,

(1.6)

for example, a function that satisfies the assumptions and is

(1.7)

where , , ; such that

(1.8)

The main result of this paper is as follows.

Theorem 1.1.

Assume that hold. Then the system (1.3) has at least one positive solution.

2. Preliminaries

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 [27]).

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.

Result.

Let such that . Let , and concave on . Then, for all .

Choose such that . For fixed and , the linear three-point BVP

(2.1)

has a unique solution

(2.2)

where is the Green's function and is given by

(2.3)

We note that as , where

(2.4)

is the Green's function corresponding the boundary value problem

(2.5)

whose integral representation is given by

(2.6)

Lemma 2.2 (see [9]).

Let . If and , then then unique solution of the problem (2.5) satisfies

(2.7)

where .

We need the following properties of the Green's function in the sequel.

Lemma 2.3 (see [11]).

The function can be written as

(2.8)

where

(2.9)

Following the idea in [10], we calculate upper bound for the Green's function in the following lemma.

Lemma 2.4.

The function satisfies

(2.10)

where

Proof.

For , we discuss various cases.

Case 1.

, ; using (2.3), we obtain

(2.11)

If , the maximum occurs at , hence

(2.12)

and if , the maximum occurs at , hence

(2.13)

Case 2.

, ; using (2.3), we have

(2.14)

Case 3.

, ; using (2.3), we have

(2.15)

Case 4.

, ; using (2.3), we have

(2.16)

For , the maximum occurs at , hence

(2.17)

For , the maximum occurs at , so

(2.18)

Now, we consider the nonlinear nonsingular system of BVPs

(2.19)

We write (2.19) as an equivalent system of integral equations

(2.20)

By a solution of the system (2.19), we mean a solution of the corresponding system of integral equations (2.20). Define a retraction by and an operator by

(2.21)

where operators are defined by

(2.22)

Clearly, if is a fixed point of , then is a solution of the system (2.19).

Lemma 2.5.

Assume that holds. Then is completely continuous.

Proof.

Clearly, for any , . We show that the operator is uniformly bounded. Let be fixed and consider

(2.23)

Choose a constant such that , , . Then, for every , using (2.22), Lemma 2.4, and , we have

(2.24)

which implies that

(2.25)

that is, is uniformly bounded. Similarly, using (2.22), Lemma 2.4, and , we can show that is also uniformly bounded. Thus, is uniformly bounded. Now we show that is equicontinuous. Define

(2.26)

Let such that . Since is uniformly continuous on , for any , there exist such that implies

(2.27)

For , using (2.22)–(2.27), we have

(2.28)

Hence,

(2.29)

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.

Now we show that is continuous. Let such that Then by using (2.22) and Lemma 2.4, we have

(2.30)

Consequently,

(2.31)

By Lebesgue dominated convergence theorem, it follows that

(2.32)

Similarly, by using (2.22) and Lemma 2.4, we have

(2.33)

From (2.32) and (2.33), it follows that

(2.34)

that is, is continuous. Hence, is completely continuous.

3. Main Results

Proof of Theorem 1.1.

Let . Choose a constant such that

(3.1)

Choose a constant such that , , , . For any , using (2.22), (3.1), , and , we have

(3.2)

Since,

(3.3)

it follows that

(3.4)

Similarly, using (2.22), (3.1), , and , we have

(3.5)

From (3.4), and (3.5), it follows that

(3.6)

Choose a real constant such that

(3.7)

Choose a constant such that , , , . For any , using (2.22), (3.7), , and , we have

(3.8)

We used the fact that

(3.9)

Thus,

(3.10)

Similarly, using (2.22), (3.7), and , we have,

(3.11)

From (3.10) and (3.11), it follows that

(3.12)

Hence by Lemma 2.1, has a fixed point , that is,

(3.13)

Moreover, by (3.4), (3.5), (3.10) and (3.11), we have

(3.14)

Since is a solution of the system (2.19), hence and are concave on . Moreover, and . For , using result (2.2) and (3.14), we have

(3.15)

which implies that is uniformly bounded on . Now we show that is equicontinuous on . Choose and and consider the integral equation

(3.16)

Using Lemma 2.3, we have

(3.17)

Differentiating with respect to t, we obtain

(3.18)

which implies that

(3.19)

In view of and (3.15), we have

(3.20)

which implies that

(3.21)

Similarly, consider the integral equation

(3.22)

using and (3.15), we can show that

(3.23)

In view of (3.21) and (3.23), is equicontinuous on . Hence by Arzelà-Ascoli theorem, the sequence has a subsequence converging uniformly on to . Let us consider the integral equation

(3.24)

Letting , we have

(3.25)

Differentiating twice with respect to t, we have

(3.26)

Letting , we have

(3.27)

Similarly, consider the integral equation

(3.28)

we can show that

(3.29)

Now, we show that also satisfies the boundary conditions. Since,

(3.30)

Similarly, we can show that

(3.31)

Equations (3.27)–(3.31) imply that is a solution of the system (1.3). Moreover, is positive. In fact, by (3.27) is concave and by Lemma 2.2

(3.32)

implies that for all . Similarly, for all . The proof of Theorem 1.1 is complete.

Example.

Let

(3.33)

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.

References

  1. 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.

  2. 2.

    Timoshenko T: Theory of Elastic Stability. McGraw-Hill, New York, NY, USA; 1971.

  3. 3.

    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.

  4. 4.

    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.

  5. 5.

    Bitsadze AV: On the theory of nonlocal boundary value problems. Soviet Mathematics—Doklady 1984, 30: 8–10.

  6. 6.

    Bitsadze AV: On a class of conditionally solvable nonlocal boundary value problems for harmonic functions. Soviet Mathematics—Doklady 1985, 31: 91–94.

  7. 7.

    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-H

  8. 8.

    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.2006.07.014

  9. 9.

    Ma R: Positive solutions of a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 1999, 1999(34):1–8.

  10. 10.

    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)00547-8

  11. 11.

    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.2007.11.037

  12. 12.

    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. 10.1080/10236190701840906

  13. 13.

    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.

  14. 14.

    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.2008.03.002

  15. 15.

    Chu J, Nieto JJ: Recent existence results for second order singular periodic differential equations. Boundary Value Problems. In press

  16. 16.

    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.2008.03.041

  17. 17.

    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/bdm110

  18. 18.

    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.2007.06.031

  19. 19.

    Agarwal RP, O'Regan D: Singular Differential and Integral Equations with Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+402.

  20. 20.

    Wang H: On the number of positive solutions of nonlinear systems. Journal of Mathematical Analysis and Applications 2003, 281(1):287–306.

  21. 21.

    Xie S, Zhu J: Positive solutions of boundary value problems for system of nonlinear fourth-order differential equations. Boundary Value Problems 2007, 2007:-12.

  22. 22.

    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.2005.07.014

  23. 23.

    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.2007.12.014

  24. 24.

    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.2004.08.003

  25. 25.

    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.2006.06.053

  26. 26.

    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.

  27. 27.

    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.

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Acknowledgment

Research of R. A. Khan is supported by HEC, Pakistan, Project 2- 3(50)/PDFP/HEC/2008/1.

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Correspondence to RahmatAli Khan.

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Keywords

  • Integral Equation
  • Couple System
  • Real Constant
  • Singular System
  • Order Ordinary Differential Equation