×
Zhang, Xuemei; Feng, Meiqiang

Green’s function and positive solutions for a second-order singular boundary value problem with integral boundary conditions and a delayed argument. (English) Zbl 1470.34171

Abstr. Appl. Anal. 2014, Article ID 393187, 9 p. (2014).
Summary: This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument \(- x^{\prime \prime} \left(t\right) - a \left(t\right) x' \left(t\right) + b \left(t\right) x \left(t\right) = \omega \left(t\right) f \left(t, x \left(\alpha \left(t\right)\right)\right), t \in \left(0, 1\right); x' \left(0\right) = 0, x \left(1\right) - \int_0^1 h \left(t\right) x \left(t\right) d t = 0\), where \(a \in \left(\left[0, 1\right], \left[0, + \infty\right)\right), b \in C \left(\left[0, 1\right], \left(0, + \infty\right)\right)\) and, \(\omega\) may be singular at \(t = 0\) or/and at \(t = 1\). Furthermore, several new and more general results are obtained for the existence of positive solutions for the above problem by using Krasnosel’skii’s fixed point theorem. We discuss our problems with a delayed argument, which may concern optimization issues of some technical problems. Moreover, the approach to express the integral equation of the above problem is different from earlier approaches. Our results cover a second-order boundary value problem without deviating arguments and are compared with some recent results.

Cited in 2 Documents

MSC:

34K10 Boundary value problems for functional-differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Cannon, J. R., The solution of the heat equation subject to the specification of energy, Quarterly of Applied Mathematics, 21, 155-160 (1963) · Zbl 0173.38404
[2] Ionkin, N. I., Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions, Differential Equations, 13, 294-304 (1977)
[3] Chegis, R. Y., Numerical solution of a heat conduction problem with an integral boundary condition, Lietuvos Matematikos Rinkinys, 24, 209-215 (1984) · Zbl 0578.65092
[4] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., On shifted Jacobi spectral method for high-order multi-point boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 17, 10, 3802-3810 (2012) · Zbl 1251.65112 · doi:10.1016/j.cnsns.2012.02.027
[5] Ma, R. Y., Existence of positive solutions for a nonlinear \(m\)-point boundary value problem, Acta Mathematica Sinica, 46, 4, 785-794 (2003) · Zbl 1056.34015
[6] Bai, Z.; Du, Z., Positive solutions for some second-order four-point boundary value problems, Journal of Mathematical Analysis and Applications, 330, 1, 34-50 (2007) · Zbl 1115.34016 · doi:10.1016/j.jmaa.2006.07.044
[7] Avery, R. I.; Chyan, C. J.; Henderson, J., Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Computers & Mathematics with Applications, 42, 3-5, 695-704 (2001) · Zbl 1006.34022 · doi:10.1016/S0898-1221(01)00188-2
[8] Zhao, Y.; Chen, H., Existence of multiple positive solutions for \(m\)-point boundary value problems in Banach spaces, Journal of Computational and Applied Mathematics, 215, 1, 79-90 (2008) · Zbl 1147.34019 · doi:10.1016/j.cam.2007.03.025
[9] Liu, Y.; O’Regan, D., Multiplicity results for a class of fourth order semipositone \(m\)-point boundary value problems, Applicable Analysis, 91, 5, 911-921 (2012) · Zbl 1255.34020 · doi:10.1080/00036811.2011.556623
[10] Du, Z.; Kong, L., Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems, Applied Mathematics Letters, 23, 9, 980-983 (2010) · Zbl 1200.34061 · doi:10.1016/j.aml.2010.04.021
[11] Gupta, C. P., A generalized multi-point boundary value problem for second order ordinary differential equations, Applied Mathematics and Computation, 89, 1-3, 133-146 (1998) · Zbl 0910.34032 · doi:10.1016/S0096-3003(97)81653-0
[12] Feng, M.; Ge, W., Positive solutions for a class of \(m\)-point singular boundary value problems, Mathematical and Computer Modelling, 46, 3-4, 375-383 (2007) · Zbl 1142.34012 · doi:10.1016/j.mcm.2006.11.009
[13] Ahmad, B.; Alsaedi, A.; Alghamdi, B. S., Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Analysis, Real World Applications, 9, 4, 1727-1740 (2008) · Zbl 1154.34311 · doi:10.1016/j.nonrwa.2007.05.005
[14] Infante, G.; Webb, J. R. L., Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations, Proceedings of the Edinburgh Mathematical Society, 49, 3, 637-656 (2006) · Zbl 1115.34026 · doi:10.1017/S0013091505000532
[15] Kong, L., Second order singular boundary value problems with integral boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 72, 5, 2628-2638 (2010) · Zbl 1188.34028 · doi:10.1016/j.na.2009.11.010
[16] Zhang, X.; Feng, M., Existence and dependence of positive solution on parameter for the one-dimensional singular \(p\)-Laplacian, Journal of Mathematical Analysis and Applications, 413, 566-582 (2014) · Zbl 1316.34029 · doi:10.1016/j.jmaa.2013.11.038
[17] Zhang, X.; Feng, M., Transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations, Journal of Computational and Applied Mathematics, 271, 117-129 (2014) · Zbl 1326.34045 · doi:10.1016/j.cam.2014.04.008
[18] Boucherif, A., Second-order boundary value problems with integral boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 70, 1, 364-371 (2009) · Zbl 1169.34310 · doi:10.1016/j.na.2007.12.007
[19] Feng, M.; Ji, D.; Ge, W., Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces, Journal of Computational and Applied Mathematics, 222, 2, 351-363 (2008) · Zbl 1158.34336 · doi:10.1016/j.cam.2007.11.003
[20] Yang, C.; Zhai, C.; Yan, J., Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument, Nonlinear Analysis: Theory, Methods & Applications, 65, 10, 2013-2023 (2006) · Zbl 1113.34048 · doi:10.1016/j.na.2005.11.003
[21] Jankowski, T., Solvability of three point boundary value problems for second order differential equations with deviating arguments, Journal of Mathematical Analysis and Applications, 312, 2, 620-636 (2005) · Zbl 1154.34367 · doi:10.1016/j.jmaa.2005.03.076
[22] Jankowski, T., Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives, Nonlinear Analysis, 87, 83-101 (2013) · Zbl 1286.34096 · doi:10.1016/j.na.2013.04.004
[23] Jankowski, T., Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions, Boundary Value Problems, 2013, article 8 (2013) · Zbl 1342.34036 · doi:10.1186/1687-2770-2013-8
[24] Jiang, D.; Wei, J., Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations, Nonlinear Analysis, 50, 7, Ser. A: Theory Methods, 885-898 (2002) · Zbl 1014.34049 · doi:10.1016/S0362-546X(01)00782-9
[25] Wang, G., Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments, Journal of Computational and Applied Mathematics, 236, 9, 2425-2430 (2012) · Zbl 1238.65077 · doi:10.1016/j.cam.2011.12.001
[26] Wang, G.; Zhang, L.; Song, G., Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 74, 3, 974-982 (2011) · Zbl 1223.34091 · doi:10.1016/j.na.2010.09.054
[27] Hu, C.; Liu, B.; Xie, S., Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments, Applied Mathematics and Computation, 222, 72-81 (2013) · Zbl 1333.34117 · doi:10.1016/j.amc.2013.07.048
[28] Bai, D.; Xu, Y., Positive solutions of second-order two-delay differential systems with twin-parameter, Nonlinear Analysis: Theory, Methods & Applications, 63, 4, 601-617 (2005) · Zbl 1091.34034 · doi:10.1016/j.na.2005.05.021
[29] Du, B.; Hu, X.; Ge, W., Positive solutions to a type of multi-point boundary value problem with delay and one-dimensional \(p\)-Laplacian, Applied Mathematics and Computation, 208, 2, 501-510 (2009) · Zbl 1170.34046 · doi:10.1016/j.amc.2008.12.020
[30] Jiang, D., Multiple positive solutions for boundary value problems of second-order delay differential equations, Applied Mathematics Letters, 15, 5, 575-583 (2002) · Zbl 1011.34055 · doi:10.1016/S0893-9659(02)80009-X
[31] Wang, W.; Shen, J., Positive solutions to a multi-point boundary value problem with delay, Applied Mathematics and Computation, 188, 1, 96-102 (2007) · Zbl 1125.34333 · doi:10.1016/j.amc.2006.09.093
[32] Wang, Y.; Zhao, W.; Ge, W., Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional \(p\)-Laplacian, Journal of Mathematical Analysis and Applications, 326, 1, 641-654 (2007) · Zbl 1119.34050 · doi:10.1016/j.jmaa.2006.03.028
[33] Jankowski, T., Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions, Nonlinear Analysis, 73, 5, 1289-1299 (2010) · Zbl 1200.34072 · doi:10.1016/j.na.2010.04.055
[34] Jankowski, T.; Jankowski, R., Multiple solutions of boundary-value problems for fourth-order differential equations with deviating arguments, Journal of Optimization Theory and Applications, 146, 1, 105-115 (2010) · Zbl 1208.34105 · doi:10.1007/s10957-010-9658-5
[35] Feng, M.; Zhang, X.; Yang, X., Positive solutions of nth-order nonlinear impulsive differential equation with nonlocal boundary conditions, Boundary Value Problems, 2011, article 19 (2011) · Zbl 1213.34046 · doi:10.1155/2011/456426
[36] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0661.47045
[37] Wang, H., Positive periodic solutions of functional differential equations, Journal of Differential Equations, 202, 2, 354-366 (2004) · Zbl 1064.34052 · doi:10.1016/j.jde.2004.02.018
[38] Sun, H.; Li, W., Existence theory for positive solutions to one-dimensional \(p\)-Laplacian boundary value problems on time scales, Journal of Differential Equations, 240, 2, 217-248 (2007) · Zbl 1139.34047 · doi:10.1016/j.jde.2007.06.004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.