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Upper and lower solution method for fourth-order four-point boundary value problems. (English) Zbl 1102.65084
This paper deals with the upper and lower solution method for system
\[ u^{(4)}(t) =f(t,u(t),u''(t)),\quad 0<t<1,\qquad u(0)=u(1)=0, \] \[ au''(x_1)-bu'''(x_1)=0,\qquad cu''(x_2) +du'''(x_2)=0, \] and establish some new existence resuls. Also a monotone iterative technique is presented.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[1] Aftabizadeh, A.R., Existence and uniqueness theorems for fourth-order boundary value problems, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009
[2] Agarwal, R.P., Focal boundary value problems for differential and difference equations, (1998), Kluwer Academic Dordrecht · Zbl 0914.34001
[3] Del Pino, M.A.; Manasevich, R.F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. amer. math. soc., 112, 81-86, (1991) · Zbl 0725.34020
[4] Graef, J.R.; Yang, B., On a nonlinear boundary value problem for fourth order equations, Appl. anal., 72, 439-448, (1999) · Zbl 1031.34017
[5] Hartman, P., Ordinary differential equations, (1982), Birkhäuser Boston · Zbl 0125.32102
[6] Leela, S., Monotone method for second order periodic boundary value problems, Nonlinear anal., 7, 349-355, (1983) · Zbl 0524.34023
[7] Ma, T.F., Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. numer. math., 47, 189-196, (2003) · Zbl 1068.74038
[8] Ma, R.Y.; Zhang, J.H.; Fu, S.M., 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
[9] Rachunkova, I., Upper and lower solutions and topological degree, J. math. anal. appl., 234, 311-327, (1999) · Zbl 1086.34017
[10] Shanthi, V.; Ramanujam, N., A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations, Appl. math. comput., 129, 269-294, (2002) · Zbl 1025.65044
[11] Wong, P.J.Y., Triple positive solutions of conjugate boundary value problems, Comput. math. appl., 36, 19-35, (1998) · Zbl 0936.34018
[12] Yang, Y., Fourth-order two-point boundary value problem, Proc. amer. math. soc., 104, 175-180, (1988) · Zbl 0671.34016
[13] Zhang, Z.X.; 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, (2003)
[14] Zill, D.G.; Cullen, M.R., Differential equations with boundary-value problems, (2001), Brooks Cole
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