Positive solutions of singular sublinear fourth-order boundary value problems. (English) Zbl 1085.34020

Ma, Ruyun; Tisdell, Christopher C.

Appl. Anal. 84, No. 12, 1199-1220 (2005).

Summary: We give some necessary and sufficient conditions for the existence of \(C^2[0, 1]\) and \(C^3[0, 1]\) positive solutions to the singular boundary value problem \[ y''''(t)= p(t)y^\lambda(t),\quad t\in (0,1),\qquad y(0)= y(1)= y'(0)= y'(1)= 0, \] where \(\lambda\in(0,1)\) is given; and \(p: (0, 1)\to[0, \infty)\) can be singular at both ends \(t= 0\) and \(t= 1\). We also give a sufficient condition for the existence of \(C^1[0,1]\) positive solutions to the above problem. The proofs are based upon the method of lower and upper solutions for singular fourth-order boundary value problems.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations

Keywords:

singular boundary value problem; existence; Schauder fixed-point theorem; Green’s function; lower and upper solutions

References:

[1]	DOI: 10.1016/0377-0427(84)90058-X · Zbl 0541.65055
[2]	DOI: 10.1016/S0362-546X(99)00308-9 · Zbl 0992.34011
[3]	DOI: 10.1016/S0022-247X(02)00071-9 · Zbl 1006.34023
[4]	DOI: 10.1090/S0002-9939-1991-1043407-9
[5]	DOI: 10.4064/ap77-2-3 · Zbl 0989.34014
[6]	Graef JR, Proceedings of Dynamic Systems and Applications 3 pp 217– (2000)
[7]	DOI: 10.1080/00036818808839715 · Zbl 0611.34015
[8]	DOI: 10.1016/S0898-1221(00)00158-9 · Zbl 0976.34019
[9]	DOI: 10.1017/S0308210500003140 · Zbl 1060.34014
[10]	DOI: 10.1016/S0362-546X(03)00127-5 · Zbl 1030.34025
[11]	DOI: 10.4064/ap81-1-7 · Zbl 1028.34025
[12]	DOI: 10.1080/00036819508840401 · Zbl 0841.34019
[13]	Ma R, Acta Mathematica Scientia. Series A 22 pp 244– (2002)
[14]	O’Regan D, Theory of singular boundary value problems (1994)
[15]	Rynne B, Topological Methods in Nonlinear Analysis 19 pp 303– (2002)
[16]	DOI: 10.1017/S0004972700020712 · Zbl 1032.34022
[17]	DOI: 10.1016/0362-546X(79)90057-9 · Zbl 0421.34021
[18]	Wei Z, Acta Mathematica Sinica 42 pp 715– (1999)
[19]	DOI: 10.1016/S0893-9659(04)90037-7 · Zbl 1072.34022
[20]	DOI: 10.1006/jmaa.1994.1243 · Zbl 0823.34030
[21]	DOI: 10.1137/S0036141093246087 · Zbl 0823.34031
[22]	DOI: 10.1016/S0377-0427(02)00390-4 · Zbl 1019.34021