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Triple positive solutions of a class of fourth-order two-point boundary value problems. (English) Zbl 1193.34048

Summary: By using the Leggett-Williams fixed point theorem, we establish an existence criterion for triple positive solutions of the nonlinear fourth-order two-point boundary value problem

$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=g\left(t\right)f\left(t,u\left(t\right),{u}^{\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right)\hfill \\ u\left(0\right)={u}^{\text{'}}\left(1\right)={u}^{\text{'}\text{'}}\left(0\right)={u}^{\text{'}\text{'}\text{'}}\left(1\right)=0·\hfill \end{array}\right\$

An example is also included to demonstrate the result we obtained.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
##### Keywords:
positive solutions; Green function; fixed point; cone
##### References:
 [1] Bai, Ch.: Triple positive solutions of three-point boundary value problems of fourth-order differential equations, Comput. math. Appl. 56, 1364-1371 (2008) · Zbl 1155.34311 · doi:10.1016/j.camwa.2008.02.033 [2] Bai, Z.; Wang, Y.; Ge, W.: Triple positive solutions for a class of two-point boundary value problems, Electron. J. Differential equations 06, 1-8 (2004) · Zbl 1055.34046 · doi:emis:journals/EJDE/Volumes/2004/06/abstr.html [3] Guo, Y.; Liu, X.; Qiu, G.: Three positive solutions for higher order m-point boundary value problems, J. math. Anal. appl. 289, 545-553 (2004) · Zbl 1046.34028 · doi:10.1016/j.jmaa.2003.08.038 [4] Liu, B.: Positive solutions of fourth-order two-point boundary value problems, Appl. math. Comput. 148, 407-420 (2004) · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3 [5] Leggett, R. I.; Williams, L. R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. Math. J. 28, 673-688 (1979) · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046 [6] Liu, X.; Qiu, G.; Guo, Y.: Three positive solutions for second-order m-point boundary value problems, Appl. math. Comput. 156, 733-742 (2004) · Zbl 1069.34014 · doi:10.1016/j.amc.2003.06.021 [7] Sun, J.; Li, W.; Zhao, Y.: Three positive solutions of a nonlinear three-point boundary value problem, J. math. Anal. appl. 288, 708-716 (2003) · Zbl 1045.34006 · doi:10.1016/j.jmaa.2003.09.019 [8] Avery, R. I.: A generalization of the Leggett–Williams fixed point theorem, Math. sci. Res. hot-line 2, 9-14 (1998) · Zbl 0965.47038 [9] Avery, R. I.; Peterson, A. C.: Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. math. Appl. 42, 313-322 (2001) · Zbl 1005.47051 · doi:10.1016/S0898-1221(01)00156-0