# zbMATH — the first resource for mathematics

Successive iteration and positive solution for nonlinear second-order three-point boundary value problems. (English) Zbl 1096.34015
This paper is concerned with the existence of positive solutions to a boundary value problem of the type
$u''(t) + f(t, u(t), u'(t))=0, t \in [0,1]$ with $$u(0)=0,$$ $$u(1)= \alpha u(\eta)$$ and $$0 < \alpha <1/2,$$ $$\eta <1$$.
By transforming the differential problem into an equivalent integral problem $$u = T u,$$ where $$T$$ is a suitable integral operator, the author proves two theorems (for $$f(t,u)$$ and $$f(t,u,u')$$) providing sufficient conditions on the nonlinear function $$f$$ and the constants to ensure the existence of a concave solution. Further, a fixed-point iteration $$u_{n+1}= T u_n$$ with given $$u_0$$ is proposed to approximate the solution of the integral equation. Finally, two examples with some polynomial functions $$f$$ are presented to show the performance of the proposed iterative scheme.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations
Full Text:
##### References:
  Gupta, C.P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. math. anal. appl., 108, 540-551, (1992) · Zbl 0763.34009  Marano, S.A., A remark on a second order three-point boundary value problem, J. math. anal. appl., 183, 518-522, (1994) · Zbl 0801.34025  Gupta, C.P., A note on a second order three-point boundary value problem, J. math. anal. appl., 186, 277-281, (1994) · Zbl 0805.34017  Gupta, C.P., A sharper condition for the solvability of a three-point second-order boundary value problem, J. math. anal. appl., 205, 579-586, (1997) · Zbl 0874.34014  Ma, R., Positive solutions of a nonlinear three-point boundary value problem, Electronic J. diff. eqns., 34, 1-8, (1999)  Ma, R., Multiplicity of positive solutions for second-order three-point boundary value problems, Computers math. applic., 40, 2/3, 193-204, (2000) · Zbl 0958.34019  Calvert, B.; Gupta, C.P., Multiple solutions for s superlinear three-point boundary value problems, Nonlinear anal., 50, 115-128, (2002) · Zbl 1037.34010  Yao, Q., Existence and multiplicity of positive solutions for a class of second-order three-point nonlinear boundary value problems (in Chinese), Acta math. sinica, 45, 1057-1064, (2002) · Zbl 1113.34320  Yao, Q., Monotone iterative method for a class of nonlinear second-order three-point boundary value problems (in Chinese), Numerical math. J. Chinese univ., 25, 135-143, (2003) · Zbl 1046.65061  Jankowski, T., Existence of solutions of differential equations with nonlinear multipoint boundary conditions, Computers math. applic., 47, 1095-1103, (2004) · Zbl 1093.34006  Yao, Q., The positive solution of classical Gelfand model with coefficient that changes sign, Appl. math. and mech., 23, 1458-1463, (2002) · Zbl 1034.34026  Yao, Q., Iteration of positive solution for a second-order ordinary differential equation with change of sign, Annals of diff. eqns., 18, 410-416, (2002) · Zbl 1022.34019  Yao, Q., Monotone iterative technique and positive solutions of lidstone boundary value problems, Appl. math. and comput., 138, 1-9, (2003) · Zbl 1049.34028  Yao, Q., Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem, Computers math. applic., 47, 8/9, 1195-1200, (2004) · Zbl 1062.34024  Yao, Q., On the positive solutions of a nonlinear fourth-order boundary value problem with two parameters, Applicable analysis, 83, 97-107, (2004) · Zbl 1051.34018  Yao, Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. math. lett., 17, 237-243, (2004) · Zbl 1072.34022
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.