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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 $$\cases u^{(4)}(t)=g(t)f(t,u(t),u'(t)),\quad t\in(0,1)\\ u(0)=u'(1)=u''(0)=u'''(1)=0.\endcases$$ An example is also included to demonstrate the result we obtained.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[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 · emis:journals/EJDE/Volumes/2004/06/abstr.html
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[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
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[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