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Journal of Zhejiang University SCIENCE A 2004 Vol.5 No.3 P.353-357

http://doi.org/10.1631/jzus.2004.0353


Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative


Author(s):  YAO Qing-liu

Affiliation(s):  Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210003, China

Corresponding email(s):   yaoqingliu2002@hotmail.com

Key Words:  Nonlinear fourth-order equation, Two-point boundary value problem, Solution and positive solution, Existence, Fixed point theorem


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YAO Qing-liu. Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative[J]. Journal of Zhejiang University Science A, 2004, 5(3): 353-357.

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T1 - Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative
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Abstract: 
Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the “height” of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1] Aftabizadeh, A.R., 1986. Existence and uniqueness theorems for fourth-order boundary value problems.J. Math. Anal. Appl.,116:415-426.

[2] Del Peno, M.A., Manasevich, R.F., 1991. Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition.Proc. Amer. Math. Soc.,112:81-86.

[3] Ma, R., Zhang, J., Fu, S., 1997. The method of lower and upper solutions for fourth-order two-point boundary value problems.J. Math. Anal. Appl.,215:415-422.

[4] Usmani, R.A., 1979. A uniqueness theorem for a boundary value problem.Proc. Amer. Math. Soc.,77:327-335.

[5] Yang, Y., 1988. Fourth-order two-point boundary value problems.Proc. Amer. Math. Soc.,104:175-180.

[6] Yao, Q., Bai, Z., 1999. Existence of positive solutions of BVP foru(4)(t)-λh(t)f(u(t))=0.Chinese Annals of Math.,20A:575-578(in Chinese).

[7] Yao, Q., 2002. Existence and multiplicity of positive solutions for a class of second-order three-point nonlinear boundary value problems.Acta Math. Sinica, 46:1057-1064 (in Chinese).

[8] Yao, Q., 2003. The existence and multiplicity of positive solutions for a third-order three-point boundary value problem.Acta Math. Appl. Sinica, English Ser., 19:117-122.

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