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Monotonic positive solutions of nonlocal boundary value problems for a second-order functional differential equation. (English) Zbl 1239.34077
Summary: We study the existence of at least one monotone positive solution for the nonlocal boundary value problem $$x''(t) = f(t, x(\phi(t))), ~t \in (0, 1)$$ with the nonlocal condition $$\sum^m_{k=1} a_kx(\tau_k) = x_0, ~x'(0) + \sum^n_{j=1} b_jx'(\eta_j) = x_1,$$ where $\tau_k \in (a, d) \subset (0, 1), \eta_j \in (c, e) \subset (0, 1)$, and $x_0, x_1 > 0$. As an application the integral and the nonlocal conditions $\int^d_a x(t)dt = x_0, ~x'(0) + x(e) - x(c) = x_1$ will be considered.

34K10Boundary value problems for functional-differential equations
34K12Growth, boundedness, comparison of solutions of functional-differential equations
Full Text: DOI
[1] 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. · Zbl 0636.34019
[2] 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. · Zbl 0668.34024
[3] 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. · Zbl 1104.34007 · doi:10.1016/j.na.2005.10.044
[4] R. F. Curtain and A. J. Pritchand, Functional Analysis in Modern Applied Mathematics, Academic Press, 1977.
[5] 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. · Zbl 1012.34014 · doi:10.4134/JKMS.2002.39.2.319
[6] 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.
[7] 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. · Zbl 1065.34013 · doi:10.4134/BKMS.2004.41.3.483
[8] 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. · Zbl 0708.47031
[9] 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. · Zbl 0763.34009 · doi:10.1016/0022-247X(92)90179-H
[10] 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. · Zbl 1156.34311 · doi:10.1016/j.na.2007.03.041
[11] 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. · Zbl 1072.34014 · doi:10.1155/S1085337503301034 · eudml:51675
[12] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1970. · Zbl 0213.07305
[13] 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. · Zbl 05253965 · doi:10.1016/j.na.2007.01.065
[14] 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. · Zbl 1140.34313 · doi:10.1016/j.amc.2007.07.025
[15] 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. · Zbl 1032.34020 · doi:10.1016/S0096-3003(02)00341-7
[16] 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. · Zbl 1069.34014 · doi:10.1016/j.amc.2003.06.021
[17] R. Ma, “Positive solutions of a nonlinear three-point boundary-value problem,” Electronic Journal of Differential Equations, vol. 34, pp. 1-8, 1999. · Zbl 0926.34009 · emis:journals/EJDE/Volumes/1999/34/abstr.html · eudml:120017
[18] 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. · Zbl 0958.34019 · doi:10.1016/S0898-1221(00)00153-X
[19] R. Ma, “Positive solutions for second-order three-point boundary value problems,” Applied Mathematics Letters, vol. 14, no. 1, pp. 1-5, 2001. · Zbl 0989.34009 · doi:10.1016/S0893-9659(00)00102-6
[20] 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. · Zbl 0988.34009 · doi:10.1006/jmaa.2000.7320
[21] 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. · Zbl 1098.34011
[22] 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. · Zbl 1148.34020 · doi:10.1155/2007/79090 · eudml:54646