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Zhang, Xuemei; Ge, Weigao

Positive solutions for a class of boundary-value problems with integral boundary conditions. (English) Zbl 1189.34035

Comput. Math. Appl. 58, No. 2, 203-215 (2009).
Summary: This paper investigates the existence and multiplicity of positive solutions for a class of nonlinear boundary-value problems of fourth-order differential equations with integral boundary conditions. The arguments are based upon a specially constructed cone and the fixed-point theory in cone. The nonexistence of positive solutions is also studied.

Cited in 35 Documents

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

boundary-value problem; integral boundary conditions; positive solution; fixed-point theorem; existence
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\textit{X. Zhang} and \textit{W. Ge}, Comput. Math. Appl. 58, No. 2, 203--215 (2009; Zbl 1189.34035)
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References:

[1] Bitsadze, A. V., On the theory of nonlocal boundary value problems, Soviet Math. Dock., 30, 8-10 (1964) · Zbl 0586.30036
[2] Bitsadze, A. V.; Samarskii, A. A., Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 185, 739-740 (1969) · Zbl 0187.35501
[3] Yang, B., Positive solutions for a fourth order boundary value problem, E. J. Qual. Theory Diff. Equ., 3, 1-17 (2005)
[4] Il in, V.; Moiseev, E., Nonlocal boundary value problems of the second kind for a Sturm-Liouville operator, Differential Equations, 23, 979-987 (1987) · Zbl 0668.34024
[6] Gupta, C. P., Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015
[7] Gupta, C. P., Existence and uniqueness results for some fourth order fully quasilinear boundary value problem, Appl. Anal., 36, 169-175 (1990)
[8] Liu, B., Positive solutions of fourth-order two point boundary value problem, Appl. Math. Comput., 148, 407-420 (2004) · Zbl 1039.34018
[9] Graef, J. R.; Qian, C.; Yang, B., A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl., 287, 217-233 (2003) · Zbl 1054.34038
[10] Anderson, D. R.; Avery, R. I., A fourth-order four-point right focal boundary value problem, Rocky Mt. J. Math., 36, 2 (2006) · Zbl 1137.34008
[11] Ma, R., Positive solutions of fourth-order two point boundary value problems, Ann. Differential Equations, 15, 305-313 (1999) · Zbl 0964.34021
[12] Guo, Y.; Shan, W.; Ge, W., Positive solutions for a second order \(m\)-point boundary value problem, J. Comput. Appl. Math., 151, 415-424 (2003) · Zbl 1026.34016
[13] Guo, Y.; Tian, J., Positive solutions of \(m\)-point boundary value problems for higher order ordinary differential equations, Nonlinear Anal., 66, 1573-1586 (2007) · Zbl 1117.34025
[14] Ma, R.; Zhang, J.; Fu, S., The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 215, 415-422 (1997) · Zbl 0892.34009
[15] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019
[16] Bai, Z.; Wang, H., On the positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023
[17] Wang, H., On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl., 281, 287-306 (2003) · Zbl 1036.34032
[18] Jiang, D. Q.; Gao, W. J.; Wan, A., A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems, Appl. Math. Comput., 132, 411-421 (2002) · Zbl 1036.34020
[19] Feng, W.; Weeb, T. R.L., Solvability of a \(m\)-point boundary value problem with nonlinear growth, J. Math. Anal. Appl., 212, 467-480 (1997) · Zbl 0883.34020
[20] Wei, Z.; Pang, C., Positive solutions of some singular \(m\)-point boundary value problems at non-resonance, Appl. Math. Comput., 171, 433-449 (2005) · Zbl 1085.34017
[21] Cheung, W.; Ren, J., Positive solutions for \(m\)-point boundary-value problems, J. Math. Anal. Appl., 303, 565-575 (2005) · Zbl 1071.34020
[22] Feng, M.; Ji, D.; Ge, W., Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math., 222, 351-363 (2008) · Zbl 1158.34336
[23] Zhang, G.; Sun, J., Positive solutions of \(m\)-point boundary value problems, J. Math. Anal. Appl., 291, 406-418 (2004) · Zbl 1069.34037
[24] Zhang, Z.; Wang, J., The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, J. Comput. Appl. Math., 147, 41-52 (2002) · Zbl 1019.34021
[25] Feng, M.; Ge, W., Positive solutions for a class of \(m\)-point singular boundary value problems, Math. Comput. Modelling, 46, 375-383 (2007) · Zbl 1142.34012
[26] Aftabizadeh, A. R., Existence and uniqueness theorems for fourth-order boundary problems, J. Math. Anal. Appl., 116, 415-426 (1986) · Zbl 0634.34009
[27] Yang, Y., Fourth-order two-point boundary value problem, Proc. Amer. Math. Soc., 104, 175-180 (1988) · Zbl 0671.34016
[28] Del Pino, M. A.; Manasevich, R. F., Existence for fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc., 112, 81-86 (1991) · Zbl 0725.34020
[29] Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal., 62, 1251-1265 (2005) · Zbl 1089.34022
[30] Liu, Z.; Zhang, X., A class of two-point boundary value problems, J. Math. Anal. Appl., 254, 599-617 (2001) · Zbl 0983.34010
[31] Ma, H., Symmetric positive solutions for nonlocal boundary value problems of fourth order, Nonlinear Anal., 68, 645-651 (2008) · Zbl 1135.34310
[32] Gallardo, J. M., Second order differential operators with integral boundary conditions and generation of semigroups, Rocky Mt. J. Math., 30, 1265-1292 (2000) · Zbl 0984.34014
[33] Karakostas, G. L.; Tsamatos, P. Ch., Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations, 30, 1-17 (2002) · Zbl 0998.45004
[34] Lomtatidze, A.; Malaguti, L., On a nonlocal boundary-value problems for second order nonlinear singular differential equations, Georg. Math. J., 7, 133-154 (2000) · Zbl 0967.34011
[35] Corduneanu, C., Integral Equations and Applications (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0714.45002
[36] Agarwal, R. P.; O’Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordtreht · Zbl 1003.39017
[37] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0661.47045
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