Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 489353, 12 pages

http://dx.doi.org/10.1155/2012/489353

## Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

^{1}Faculty of Science, Alexandria University, Alexandria, Egypt^{2}Faculty of Science, Garyounis University, Benghazi, Libya

Received 13 October 2011; Accepted 5 December 2011

Academic Editor: IstvΓ‘nΒ GyΓΆri

Copyright Β© 2012 A. M. A. El-Sayed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation , , with the nonlocal condition , , where , , and . As an application the integral and the nonlocal conditions , will be considered.

#### 1. Introduction

The nonlocal boundary value problems of ordinary differential equations arise in a variety of different areas of applied mathematics and physics.

The study of nonlocal boundary value problems was initiated by Ilβin and Moiseev [1, 2]. Since then, the non-local boundary value problems have been studied by several authors. The reader is referred to [3β22] and references therein.

In most of all these papers, the authors assume that the function is continuous. They all assume that These assumptions are restrictive, and there are many functions that do not satisfy these assumptions.

Here we assume that the function is measurable in for all and continuous in for almost all is and there exists an integrable function and a constant such that Our aim here is to study the existence of at least one monotonic positive solution for the nonlocal problem of the second-order functional differential equation with the nonlocal condition where .

As an application, the problem with the integral and nonlocal conditions is studied.

It must be noticed that the nonlocal conditions are special cases of our the nonlocal and integral conditions.

#### 2. Integral Equation Representation

Consider the functional differential equation (1.3) with the nonlocal condition (1.4) with the following assumptions.(i) is measurable in for all and continuous in for almost all and there exists an integrable function , and a constant such that (ii) is continuous.(iii).(iv) Now, we have the following Lemma.

Lemma 2.1. *The solution of the nonlocal problem (1.3)-(1.4) can be expressed by the integral equation
**
where .*

*Proof. *Integrating (1.3), we get
Integrating (2.4), we obtain
Let , in (2.5), we get
and we deduce that
Substitute from (2.7) into (2.5), we obtain
Let , in (2.4), we obtain
and we deduce that
Substitute from (2.10) into (2.8), we obtain
which proves that the solution of the nonlocal problem (1.3)-(1.4) can be expressed by the integral equation (2.3).

#### 3. Existence of Solution

We study here the existence of at least one monotonic nondecreasing solution for the integral equation (2.3).

Theorem 3.1. *Assume that (i)β(iv) are satisfied. Then the nonlocal problem (1.3)-(1.4) has at least one solution .*

*Proof. *Define the subset by . Clear the set which is nonempty, closed, and convex.

Let be an operator defined by

Let , then
then and is uniformly bounded in .

Also for such that , we have
Then
The above inequality shows that
Therefore is equicontinuous. By the ArzelΓ -Ascoli theorem, is relatively compact.

Since all conditions of the Schauder theorem hold, then has a fixed point in which proves the existence of at least one solution of the integral equation (2.3), where
To complete the proof, we prove that the integral equation (2.3) satisfies nonlocal problem (1.3)-(1.4). Differentiating (2.3), we get
Let in (2.3), we obtain
which proves
Also let in (3.7), we obtain
then
Let in (3.7), we obtain
Adding (3.12) and (3.13), we obtain
This implies that there exists at least one solution of the nonlocal problem (1.3) and (1.4). This completes the proof.

Corollary 3.2. *The solution of the problem (1.3)-(1.4) is monotonic nondecreasing.*

*Proof. *Let, we deduce from (2.3) that
which proves that the solution of the problem (1.3)-(1.4) is monotonic nondecreasing.

##### 3.1. Positive Solution

Let and , then the nonlocal problem condition (1.4) will be

Theorem 3.3. *Let the assumptions (i)β(iv) of Theorem 3.1 be satisfied. Then the solution of the nonlocal problem (1.3)β(3.16) is positive .*

*Proof. *Let and in the integral equation (2.3) and the nonlocal condition (1.4), then the solution of the nonlocal problem (1.3)β(3.16) will be given by the integral equation
where .

Let , then
Multiplying by , we obtain
and the solution of the nonlocal problem (1.3) and (3.16), given by the integral equation (3.17), is positive for . This complete the proof.

*Example 3.4. *Consider the nonlocal problem of the second-order functional differential equation (1.3) with two-point boundary condition
Applying our results here, we deduce that the two-point boundary value problem (1.3)β(3.20) has at least one monotonic nondecreasing solution represented by the integral equation
This the solution is positive with .

#### 4. Nonlocal Integral Condition

Let be the solution of the nonlocal problem (1.3) and (1.4).

Let and let , then From the continuity of the solution of the nonlocal problem (1.3) and (1.4), we obtain and the nonlocal condition (1.4) transformed to the integral condition and the solution of the integral equation (2.3) will be Now, we have the following theorem.

Theorem 4.1. *Let the assumptions (i)β(iv) of Theorem 3.1 be satisfied. Then the nonlocal problem
**
has at least one monotonic nondecreasing solution represented by (4.4).*

#### References

- V. A. Il'in and E. I. Moiseev, βA nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations,β
*Differentsial'nye Uravneniya*, vol. 23, no. 7, pp. 1198β1207, 1987. View at Google Scholar - V. A. Il'in and E. I. Moiseev, βA nonlocal boundary value problem of the second kind for the Sturm-Liouville operator,β
*Differentsial'nye Uravneniya*, vol. 23, no. 8, pp. 1422β1431, 1987. View at Google Scholar - Y. An, βExistence of solutions for a three-point boundary value problem at resonance,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 65, no. 8, pp. 1633β1643, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. F. Curtain and A. J. Pritchand,
*Functional Analysis in Modern Applied Mathematics*, Academic Press, 1977. - P. W. Eloe and Y. Gao, βThe method of quasilinearization and a three-point boundary value problem,β
*Journal of the Korean Mathematical Society*, vol. 39, no. 2, pp. 319β330, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. M. A. El-Sayed and Kh. W. Elkadeky, βCaratheodory theorem for a nonlocal problem of the differential equation ${x}^{\prime}=f(t,{x}^{\prime})$,β
*Alexandria Journal of Mathematics*, vol. 1, no. 2, pp. 8β14, 2010. View at Google Scholar - Y. Feng and S. Liu, βExistence, multiplicity and uniqueness results for a second order
*m*-point boundary value problem,β*Bulletin of the Korean Mathematical Society*, vol. 41, no. 3, pp. 483β492, 2004. View at Publisher Β· View at Google Scholar - K. Goebel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1990. - C. P. Gupta, βSolvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,β
*Journal of Mathematical Analysis and Applications*, vol. 168, no. 2, pp. 540β551, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Guo, Y. Ji, and J. Zhang, βThree positive solutions for a nonlinear nth-order m-point boundary value problem,β
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 68, no. 11, pp. 3485β3492, 2008. View at Publisher Β· View at Google Scholar - G. Infante and J. R. L. Webb, βPositive solutions of some nonlocal boundary value problems,β
*Abstract and Applied Analysis*, vol. 2003, no. 18, pp. 1047β1060, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. N. Kolmogorov and S. V. Fomin,
*Introductory Real Analysis*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1970. - F. Li, M. Jia, X. Liu, C. Li, and G. Li, βExistence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 8, pp. 2381β2388, 2008. View at Publisher Β· View at Google Scholar - R. Liang, J. Peng, and J. Shen, βPositive solutions to a generalized second order three-point boundary value problem,β
*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 931β940, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. Liu, βPositive solutions of a nonlinear three-point boundary value problem,β
*Computers & Mathematics with Applications. An International Journal*, vol. 44, no. 1-2, pp. 201β211, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. Liu, J. Qiu, and Y. Guo, βThree positive solutions for second-order
*m*-point boundary value problems,β*Applied Mathematics and Computation*, vol. 156, no. 3, pp. 733β742, 2004. View at Publisher Β· View at Google Scholar - R. Ma, βPositive solutions of a nonlinear three-point boundary-value problem,β
*Electronic Journal of Differential Equations*, vol. 34, pp. 1β8, 1999. View at Google Scholar Β· View at Zentralblatt MATH - R. Ma, βMultiplicity of positive solutions for second-order three-point boundary value problems,β
*Computers & Mathematics with Applications*, vol. 40, no. 2-3, pp. 193β204, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Ma, βPositive solutions for second-order three-point boundary value problems,β
*Applied Mathematics Letters*, vol. 14, no. 1, pp. 1β5, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Ma and N. Castaneda, βExistence of solutions of nonlinear
*m*-point boundary-value problems,β*Journal of Mathematical Analysis and Applications*, vol. 256, no. 2, pp. 556β567, 2001. View at Publisher Β· View at Google Scholar - S. K. Ntouyas, βNonlocal initial and boundary value problems: a survey,β in
*Handbook of Differential Equations: Ordinary Differential Equations. Vol. II*, A. Canada, P. Drabek, and A. Fonda, Eds., pp. 461β557, Elsevier, Amsterdam, The Netherlands, 2005. View at Google Scholar - Y. Sun and X. Zhang, βExistence of symmetric positive solutions for an m-point boundary value problem,β
*Boundary Value Problems*, vol. 2007, Article ID 79090, 14 pages, 2007. View at Publisher Β· View at Google Scholar