We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation ${x}^{\mathrm{\prime \prime}}\left(t\right)=f(t,x(\varphi \left(t\right)\left)\right)$, $t\in \left(\mathrm{0,1}\right)$, with the nonlocal condition ${\sum}_{k=1}^{m}{a}_{k}x\left({\tau}_{k}\right)={x}_{0}$, ${x}^{\prime}\left(0\right)+{\sum}_{j=1}^{n}{b}_{j}{x}^{\prime}\left({\eta}_{j}\right)={x}_{1}$, where ${\tau}_{k}\in (a,d)\subset \left(\mathrm{0,1}\right)$, ${\eta}_{j}\in (c,e)\subset \left(\mathrm{0,1}\right)$, and ${x}_{0},{x}_{1}>0$. As an application the integral and the nonlocal conditions ${\int}_{a}^{d}x\left(t\right)dt={x}_{0}$, ${x}^{\prime}\left(0\right)+x\left(e\right)-x\left(c\right)={x}_{1}$ will be considered.

## References

*Differentsial'nye Uravneniya*, vol. 23, no. 7, pp. 1198–1207, 1987. MR903975 V. A. Il'in and E. I. Moiseev, “A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations,”

*Differentsial'nye Uravneniya*, vol. 23, no. 7, pp. 1198–1207, 1987. MR903975

*Differentsial'nye Uravneniya*, vol. 23, no. 8, pp. 1422–1431, 1987. MR909590 V. A. Il'in and E. I. Moiseev, “A nonlocal boundary value problem of the second kind for the Sturm-Liouville operator,”

*Differentsial'nye Uravneniya*, vol. 23, no. 8, pp. 1422–1431, 1987. MR909590

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 65, no. 8, pp. 1633–1643, 2006. 1104.34007 MR2248690 Y. An, “Existence of solutions for a three-point boundary value problem at resonance,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 65, no. 8, pp. 1633–1643, 2006. 1104.34007 MR2248690

*Journal of the Korean Mathematical Society*, vol. 39, no. 2, pp. 319–330, 2002. 1012.34014 MR1881995 10.4134/JKMS.2002.39.2.319 P. W. Eloe and Y. Gao, “The method of quasilinearization and a three-point boundary value problem,”

*Journal of the Korean Mathematical Society*, vol. 39, no. 2, pp. 319–330, 2002. 1012.34014 MR1881995 10.4134/JKMS.2002.39.2.319

*Alexandria Journal of Mathematics*, vol. 1, no. 2, pp. 8–14, 2010. A. M. A. El-Sayed and Kh. W. Elkadeky, “Caratheodory theorem for a nonlocal problem of the differential equation ${x}^{\prime }=f(t,{x}^{\prime })$,”

*Alexandria Journal of Mathematics*, vol. 1, no. 2, pp. 8–14, 2010.

*m*-point boundary value problem,”

*Bulletin of the Korean Mathematical Society*, vol. 41, no. 3, pp. 483–492, 2004. MR2081541 1065.34013 10.4134/BKMS.2004.41.3.483 Y. Feng and S. Liu, “Existence, multiplicity and uniqueness results for a second order

*m*-point boundary value problem,”

*Bulletin of the Korean Mathematical Society*, vol. 41, no. 3, pp. 483–492, 2004. MR2081541 1065.34013 10.4134/BKMS.2004.41.3.483

*Topics in Metric Fixed Point Theory*, vol. 28 of

*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1990. MR1074005 0708.47031 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. MR1074005 0708.47031

*Journal of Mathematical Analysis and Applications*, vol. 168, no. 2, pp. 540–551, 1992. 0763.34009 MR1176010 10.1016/0022-247X(92)90179-H C. P. Gupta, “Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,”

*Journal of Mathematical Analysis and Applications*, vol. 168, no. 2, pp. 540–551, 1992. 0763.34009 MR1176010 10.1016/0022-247X(92)90179-H

*Nonlinear Analysis: Theory, Methods and Applications*, vol. 68, no. 11, pp. 3485–3492, 2008. MR2401362 Y. Guo, Y. Ji, and J. Zhang, “Three positive solutions for a nonlinear nth-order m-point boundary value problem,”

*Nonlinear Analysis: Theory, Methods and Applications*, vol. 68, no. 11, pp. 3485–3492, 2008. MR2401362

*Abstract and Applied Analysis*, vol. 2003, no. 18, pp. 1047–1060, 2003. 1072.34014 MR2040990 10.1155/S1085337503301034 euclid.aaa/1070910529 G. Infante and J. R. L. Webb, “Positive solutions of some nonlocal boundary value problems,”

*Abstract and Applied Analysis*, vol. 2003, no. 18, pp. 1047–1060, 2003. 1072.34014 MR2040990 10.1155/S1085337503301034 euclid.aaa/1070910529

*Introductory Real Analysis*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1970. MR267052 0213.07305 A. N. Kolmogorov and S. V. Fomin,

*Introductory Real Analysis*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1970. MR267052 0213.07305

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 8, pp. 2381–2388, 2008. MR2398658 F. Li, M. Jia, X. Liu, C. Li, and G. Li, “Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 8, pp. 2381–2388, 2008. MR2398658

*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 931–940, 2008. 1140.34313 MR2388746 10.1016/j.amc.2007.07.025 R. Liang, J. Peng, and J. Shen, “Positive solutions to a generalized second order three-point boundary value problem,”

*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 931–940, 2008. 1140.34313 MR2388746 10.1016/j.amc.2007.07.025

*Computers & Mathematics with Applications. An International Journal*, vol. 44, no. 1-2, pp. 201–211, 2002. 1032.34020 MR1908281 B. Liu, “Positive solutions of a nonlinear three-point boundary value problem,”

*Computers & Mathematics with Applications. An International Journal*, vol. 44, no. 1-2, pp. 201–211, 2002. 1032.34020 MR1908281

*m*-point boundary value problems,”

*Applied Mathematics and Computation*, vol. 156, no. 3, pp. 733–742, 2004. MR2088135 1069.34014 10.1016/j.amc.2003.06.021 X. Liu, J. Qiu, and Y. Guo, “Three positive solutions for second-order

*m*-point boundary value problems,”

*Applied Mathematics and Computation*, vol. 156, no. 3, pp. 733–742, 2004. MR2088135 1069.34014 10.1016/j.amc.2003.06.021

*Electronic Journal of Differential Equations*, vol. 34, pp. 1–8, 1999. 0926.34009 MR1713593 R. Ma, “Positive solutions of a nonlinear three-point boundary-value problem,”

*Electronic Journal of Differential Equations*, vol. 34, pp. 1–8, 1999. 0926.34009 MR1713593

*Computers & Mathematics with Applications*, vol. 40, no. 2-3, pp. 193–204, 2000. 0958.34019 MR1763618 R. Ma, “Multiplicity of positive solutions for second-order three-point boundary value problems,”

*Computers & Mathematics with Applications*, vol. 40, no. 2-3, pp. 193–204, 2000. 0958.34019 MR1763618

*Applied Mathematics Letters*, vol. 14, no. 1, pp. 1–5, 2001. 0989.34009 MR1793693 10.1016/S0893-9659(00)00102-6 R. Ma, “Positive solutions for second-order three-point boundary value problems,”

*Applied Mathematics Letters*, vol. 14, no. 1, pp. 1–5, 2001. 0989.34009 MR1793693 10.1016/S0893-9659(00)00102-6

*m*-point boundary-value problems,”

*Journal of Mathematical Analysis and Applications*, vol. 256, no. 2, pp. 556–567, 2001. MR1821757 0988.34009 10.1006/jmaa.2000.7320 R. Ma and N. Castaneda, “Existence of solutions of nonlinear

*m*-point boundary-value problems,”

*Journal of Mathematical Analysis and Applications*, vol. 256, no. 2, pp. 556–567, 2001. MR1821757 0988.34009 10.1006/jmaa.2000.7320

*Handbook of Differential Equations: Ordinary Differential Equations. Vol. II*, A. Canada, P. Drabek, and A. Fonda, Eds., pp. 461–557, Elsevier, Amsterdam, The Netherlands, 2005. MR2182761 1098.34011 S. K. Ntouyas, “Nonlocal initial and boundary value problems: a survey,” in

*Handbook of Differential Equations: Ordinary Differential Equations. Vol. II*, A. Canada, P. Drabek, and A. Fonda, Eds., pp. 461–557, Elsevier, Amsterdam, The Netherlands, 2005. MR2182761 1098.34011

*Boundary Value Problems*, vol. 2007, Article ID 79090, 14 pages, 2007. MR2304524 1148.34020 10.1155/2007/79090 Y. Sun and X. Zhang, “Existence of symmetric positive solutions for an m-point boundary value problem,”

*Boundary Value Problems*, vol. 2007, Article ID 79090, 14 pages, 2007. MR2304524 1148.34020 10.1155/2007/79090