We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second-order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.

## References

F. Q. Zhang, “Boundary value problems for second order differential equations with piecewise constant arguments,”

*Annals of Differential Equations*, vol. 9, no. 3, pp. 369–374, 1993. 0782.34075 MR1244632 F. Q. Zhang, “Boundary value problems for second order differential equations with piecewise constant arguments,”*Annals of Differential Equations*, vol. 9, no. 3, pp. 369–374, 1993. 0782.34075 MR1244632 M. U. Akhmet, “On the reduction principle for differential equations with piecewise constant argument of generalized type,”

*Journal of Mathematical Analysis and Applications*, vol. 336, no. 1, pp. 646–663, 2007. 1134.34048 MR2348532 10.1016/j.jmaa.2007.03.010 M. U. Akhmet, “On the reduction principle for differential equations with piecewise constant argument of generalized type,”*Journal of Mathematical Analysis and Applications*, vol. 336, no. 1, pp. 646–663, 2007. 1134.34048 MR2348532 10.1016/j.jmaa.2007.03.010 M. U. Akhmet and C. Büyükadal\i, “Differential equations with state-dependent piecewise constant argument,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 11, pp. 4200–4210, 2010. 1195.34103 MR2606777 M. U. Akhmet and C. Büyükadal\i, “Differential equations with state-dependent piecewise constant argument,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 11, pp. 4200–4210, 2010. 1195.34103 MR2606777 M. U. Akhmet, C. Büyükadal\i, and T. Ergenç, “Periodic solutions of the hybrid system with small parameter,”

*Nonlinear Analysis: Hybrid Systems*, vol. 2, no. 2, pp. 532–543, 2008. 1170.34049 MR2400967 10.1016/j.nahs.2006.07.005 M. U. Akhmet, C. Büyükadal\i, and T. Ergenç, “Periodic solutions of the hybrid system with small parameter,”*Nonlinear Analysis: Hybrid Systems*, vol. 2, no. 2, pp. 532–543, 2008. 1170.34049 MR2400967 10.1016/j.nahs.2006.07.005 K.-S. Chiu and M. Pinto, “Periodic solutions of differential equations with a general piecewise constant argument and applications,”

*Electronic Journal of Qualitative Theory of Differential Equations*, no. 46, pp. 1–19, 2010. 1211.34082 MR2678388 K.-S. Chiu and M. Pinto, “Periodic solutions of differential equations with a general piecewise constant argument and applications,”*Electronic Journal of Qualitative Theory of Differential Equations*, no. 46, pp. 1–19, 2010. 1211.34082 MR2678388 M. Z. Liu, J. F. Gao, and Z. W. Yang, “Preservation of oscillations of the Runge-Kutta method for equation ${x}^{\prime }(t)+ax(t)+{a}_{1}x([t-1])=0$,”

*Computers & Mathematics with Applications. An International Journal*, vol. 58, no. 6, pp. 1113–1125, 2009. 1189.65143 MR2554343 M. Z. Liu, J. F. Gao, and Z. W. Yang, “Preservation of oscillations of the Runge-Kutta method for equation ${x}^{\prime }(t)+ax(t)+{a}_{1}x([t-1])=0$,”*Computers & Mathematics with Applications. An International Journal*, vol. 58, no. 6, pp. 1113–1125, 2009. 1189.65143 MR2554343 J. J. Nieto and R. Rodríguez-López, “Green's function for second-order periodic boundary value problems with piecewise constant arguments,”

*Journal of Mathematical Analysis and Applications*, vol. 304, no. 1, pp. 33–57, 2005. 1078.34046 MR2124647 10.1016/j.jmaa.2004.09.023 J. J. Nieto and R. Rodríguez-López, “Green's function for second-order periodic boundary value problems with piecewise constant arguments,”*Journal of Mathematical Analysis and Applications*, vol. 304, no. 1, pp. 33–57, 2005. 1078.34046 MR2124647 10.1016/j.jmaa.2004.09.023 M. Pinto, “Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments,”

*Mathematical and Computer Modelling*, vol. 49, no. 9-10, pp. 1750–1758, 2009. 1171.34321 MR2532088 10.1016/j.mcm.2008.10.001 M. Pinto, “Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments,”*Mathematical and Computer Modelling*, vol. 49, no. 9-10, pp. 1750–1758, 2009. 1171.34321 MR2532088 10.1016/j.mcm.2008.10.001 G. Seifert, “Second order scalar functional differential equations with piecewise constant arguments,”

*Journal of Difference Equations and Applications*, vol. 8, no. 5, pp. 427–445, 2002. 1010.34069 MR1897067 10.1080/10236190290017469 G. Seifert, “Second order scalar functional differential equations with piecewise constant arguments,”*Journal of Difference Equations and Applications*, vol. 8, no. 5, pp. 427–445, 2002. 1010.34069 MR1897067 10.1080/10236190290017469 G. Seifert, “Second-order neutral delay-differential equations with piecewise constant time dependence,”

*Journal of Mathematical Analysis and Applications*, vol. 281, no. 1, pp. 1–9, 2003. 1035.34093 MR1980070 10.1016/S0022-247X(02)00303-7 G. Seifert, “Second-order neutral delay-differential equations with piecewise constant time dependence,”*Journal of Mathematical Analysis and Applications*, vol. 281, no. 1, pp. 1–9, 2003. 1035.34093 MR1980070 10.1016/S0022-247X(02)00303-7 P. Yang, Y. Liu, and W. Ge, “Green's function for second order differential equations with piecewise constant arguments,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 64, no. 8, pp. 1812–1830, 2006. 1111.34048 MR2197362 P. Yang, Y. Liu, and W. Ge, “Green's function for second order differential equations with piecewise constant arguments,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 64, no. 8, pp. 1812–1830, 2006. 1111.34048 MR2197362 Z. W. Yang, M. Z. Liu, and J. J. Nieto, “Runge-Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments,”

*Journal of Computational and Applied Mathematics*, vol. 233, no. 4, pp. 990–1004, 2009. 1180.65094 MR2557290 10.1016/j.cam.2009.08.105 Z. W. Yang, M. Z. Liu, and J. J. Nieto, “Runge-Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments,”*Journal of Computational and Applied Mathematics*, vol. 233, no. 4, pp. 990–1004, 2009. 1180.65094 MR2557290 10.1016/j.cam.2009.08.105 R. Yuan, “Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 41, no. 7-8, pp. 871–890, 2000. 1024.34068 MR1764025 R. Yuan, “Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 41, no. 7-8, pp. 871–890, 2000. 1024.34068 MR1764025 R. Yuan, “On the second-order differential equation with piecewise constant argument and almost periodic coefficients,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 52, no. 5, pp. 1411–1440, 2003. 1029.34058 MR1942569 R. Yuan, “On the second-order differential equation with piecewise constant argument and almost periodic coefficients,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 52, no. 5, pp. 1411–1440, 2003. 1029.34058 MR1942569 A. Cabada, J. B. Ferreiro, and J. J. Nieto, “Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments,”

*Journal of Mathematical Analysis and Applications*, vol. 291, no. 2, pp. 690–697, 2004. 1057.34089 MR2039079 10.1016/j.jmaa.2003.11.022 A. Cabada, J. B. Ferreiro, and J. J. Nieto, “Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments,”*Journal of Mathematical Analysis and Applications*, vol. 291, no. 2, pp. 690–697, 2004. 1057.34089 MR2039079 10.1016/j.jmaa.2003.11.022 J. J. Nieto, “A comparison result for a linear differential equation with piecewise constant delay,”

*Glasnik Matematički. Serija III*, vol. 39(59), no. 1, pp. 73–76, 2004. 1050.34097 MR2055387 10.3336/gm.39.1.07 J. J. Nieto, “A comparison result for a linear differential equation with piecewise constant delay,”*Glasnik Matematički. Serija III*, vol. 39(59), no. 1, pp. 73–76, 2004. 1050.34097 MR2055387 10.3336/gm.39.1.07 Y. Liu, P. Yang, and W. Ge, “Periodic solutions of higher-order delay differential equations,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 63, no. 1, pp. 136–152, 2005. 1088.34061 MR2167321 Y. Liu, P. Yang, and W. Ge, “Periodic solutions of higher-order delay differential equations,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 63, no. 1, pp. 136–152, 2005. 1088.34061 MR2167321 L. Pan, “Periodic solutions for higher order differential equations with deviating argument,”

*Journal of Mathematical Analysis and Applications*, vol. 343, no. 2, pp. 904–918, 2008. 1160.34065 MR2417111 10.1016/j.jmaa.2008.01.096 L. Pan, “Periodic solutions for higher order differential equations with deviating argument,”*Journal of Mathematical Analysis and Applications*, vol. 343, no. 2, pp. 904–918, 2008. 1160.34065 MR2417111 10.1016/j.jmaa.2008.01.096 R. Rodríguez-López, “Nonlocal boundary value problems for second-order functional differential equations,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 18, pp. 7226–7239, 2011. MR2833706 R. Rodríguez-López, “Nonlocal boundary value problems for second-order functional differential equations,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 18, pp. 7226–7239, 2011. MR2833706