## Multiple solutions for fourth-order boundary value problem.(English)Zbl 1094.34012

A twopoint fourth-order nonlinear boundary value problem is studied of the form $x^{(4)}(t)=f(x(t),-x''(t)),\quad x(0)=x(1)=x''(0)= x''(1)=0.$
The function $$f$$ is continuous, satisfies $$f(0,0)=0$$, and such that $$f(u,v)\geq 0$$ for $$u>0$$, $$v>0$$, while $$f(u,v)=0$$ for $$u<0$$, $$v<0$$. Sufficient conditions are given for the existence of at least six nontrivial solutions: two positive, two negative, and two sign-changing. In the case of an odd function $$f(-u,-v)\equiv -f(u,v)$$, there exist at least eight nontrivial solutions. The proof employ the theory of fixed-point index in a cone and Leray-Schauder degree. Some corollaries are given, in particular, for the case if the right-hand side is independent of the second-order derivative.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H11 Degree theory for nonlinear operators
Full Text:

### References:

 [1] Cupta, C.P., Existence and uniqueness results for the bending of elastic beam equation at resonance, J. math. anal. appl., 135, 208-225, (1982) [2] Cupta, C.P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015 [3] Aftabizadeh, A.R., Existence and uniqueness theorems for fourth-order boundary value problems, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009 [4] Agarwal, R., On fourth-order boundary value problems arising in beam analysis, Differential integral equations, 2, 91-110, (1989) · Zbl 0715.34032 [5] 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 [6] Liu, B., Positive solutions of fourth-order two point boundary value problems, Appl. math. comput., 148, 407-420, (2004) · Zbl 1039.34018 [7] 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 [8] Bai, Z., The method of lower and upper solutions for a bending of an elastic beam equation, J. math. anal. appl., 248, 195-202, (2000) · Zbl 1016.34010 [9] Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016 [10] Yang, Y., Fourth-order two-point boundary valve problems, Proc. amer. math. soc., 104, 175-180, (1988) · Zbl 0671.34016 [11] De Coster, C.; Fabry, C.; Munyamarere, F., Nonresonance conditions for fourth order nonlinear boundary value problems, Internat. J. math. sci., 17, 725-740, (1994) · Zbl 0810.34017 [12] De Coster, C.; Sanchez, L., Upper and lower solutions, ambrosetti – prodi problem and positive solutions for fourth-order O.D.E., Riv. mat. pura appl., 14, 1129-1138, (1994) · Zbl 0979.34015 [13] 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 [14] Liu, Z., On Dancer’s conjecture and multiple solutions of elliptic differential equations, Northeast. math. J., 9, 388-394, (1993) · Zbl 0818.35031 [15] Xu, X., Multiple sign-changing solutions for some m-point boundary value problems, Electronic J. differential equations, 89, 1-14, (2004) [16] Wei, Z., A class of fourth-order singular boundary value problems, Appl. math. comput., 153, 865-884, (2004) · Zbl 1057.34006 [17] Wei, Z., Positive solutions of singular boundary value problems of fourth-order differential equations, Math. sin., 42, 715-722, (1999), (in Chinese) · Zbl 1022.34018 [18] Krasnoselskii, M.A.; Zabreiko, P.P., Geometrical method of nonlinear analysis, (1984), Springer-Verlag Berlin [19] Guo, D.; Lakskmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York
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.