Solvability of some fourth (and higher) order singular boundary value problems. (English) Zbl 0795.34018

O’Regan, Donal

J. Math. Anal. Appl. 161, No. 1, 78-116 (1991).

This paper deals primarily with solutions of (1) \(y^{(4)}= f(t,y,y'')\), \(0<t<1\), satisfying boundary conditions of the type (2) \(y(0)= a\geq 0\), \(y''(0)= c\leq 0\), \(y'(1)=b\geq 0\), \(y'''(1)=0\), (3) \(y(0)= a\geq 0\), \(y'(0)= b\geq 0\), \(y''(1)=0\), \(y'''(1)=0\), (4) \(y(0)= a\geq 0\), \(y'(0)=0\), \(y'(1)=0\), \(y'''(1)=0\), or (5) \(y(0)=a\geq 0\), \(y''(0)=0\), \(y'(1)= b\geq 0\), \(y''(1)=0\), where \(f(t,y_ 1,y_ 2)\) may have singularities at \(t=0,1\) and at \(y_ 1=0\), \(y_ 2=0\).
The first results concern problem (1), (2), where \(f(t,y_ 1,y_ 2)\) has singularities at \(y_ 1=0\), not at \(y_ 2=0\). Growth assumptions are made on \(f\) such that one can conclude the existence of a priori bounds (independent of \(\lambda\)), on solutions of \(y^{(4)}= \lambda f(t,y,y'')\), \(0< t<1\), \(0\leq\lambda \leq 1\), satisfying \(y(0)= 1/n\), \(y''(0)= c\leq 0\), \(y'(1)= b\geq 0\), \(y'''(1)=0\). By applying the topological transversality theorem of Granas, one obtains a solution \(y\in C[0,1]\cap C^ 3(0,1) \cap C^ 4(0,1)\) of (1), (2). Later, boundary value problems are treated similarly for \(y^{(4)}= \psi(t) f(t,y,y'')\), where \(f(t,y_ 1,y_ 2)\) has singularities at \(y_ 1=0\), and \(\psi(t)\) is positive and improper integrable over (0,1).
Boundary value problems for equation (1) with any of the conditions (3), (4), or (5) are dealt with similarly, and then in the last section, results are given for boundary problems for \(y^{(n)}= f(t,y,y'')\), \(0<t<1\), where, as above, \(f(t,y_ 1,y_ 2)\) may have singularities at \(t=0,1\) and at \(y_ 1=0\), \(y_ 2=0\).

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

topological transversality theorem of Granas

References:

[1]	Aftabizadeh, A. R., Existence and uniqueness theorems for fourth-order boundary value problem, J. Math. Anal. Appl., 116, 415-426 (1986) · Zbl 0634.34009
[2]	Agarwal, R. P., Some new results on two point boundary value problems for higher order differential equations, Funkcial. Ekvac., 29, 197-212 (1986) · Zbl 0623.34019
[3]	Agarwal, R. P.; Krishnamoorthy, P., Boundary value problems for \(n\) th order differential equations, Bull. Inst. Math. Acad. Sinica, 7, 211-230 (1979) · Zbl 0413.34020
[4]	Bailey, P.; Shampine, L.; Waltman, P., Nonlinear Two Point Boundary Value Problems (1968), Academic Press: Academic Press New York · Zbl 0169.10502
[5]	Bobisud, L. E., Existence and Behaviour of positive solutions for a class of parabolic reaction-diffusion equations, Appl. Anal., 28, 135-149 (1988) · Zbl 0627.35051
[6]	Bobisud, L. E.; O’Regan, D., Existence of solutions to some singular initial value problems, J. Math. Anal. Appl., 133, 214-230 (1988) · Zbl 0646.34003
[7]	Bobisud, L. E.; O’Regan, D.; Royalty, W. D., Existence and nonexistence for a singular boundary value problem, Appl. Anal., 28, 245-256 (1988) · Zbl 0628.34025
[8]	Dugundji, J.; Granas, A., Fixed Point Theory, (Monographie Mathematyczne, Vol. I (1982), PNW: PNW Warsaw) · Zbl 1025.47002
[9]	Eloe, P. W.; Henderson, J., Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. Math. Anal., 15, 642-647 (1984) · Zbl 0547.34015
[10]	Granas, A., Sur la methode de continuité de Poincaré, C. R. Acad. Sci. Paris, 282, 983-985 (1976) · Zbl 0348.47039
[11]	Granas, A.; Guenther, R. B.; Lee, J. W., Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. Warsaw (1985) · Zbl 0476.34017
[12]	Granas, A.; Guenther, R. B.; Lee, J. W., Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math., 10, 35-58 (1980) · Zbl 0476.34017
[13]	Gupta, C. P., Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015
[14]	Jackson, L. K., Existence and uniqueness of solutions of boundary value problems for Lipschitz Equations, J. Differential Equations, 32, 76-90 (1979) · Zbl 0407.34018
[15]	Jackson, L. K., Boundary value problems for Lipschitz equations, (Ahmad, S.; Keener, M.; Lazer, A. C., Differential Equation (1980), Academic Press: Academic Press New York), 31-50
[17]	Luning, C. D.; Perry, W. L., Positive solutions of negative exponent generalized Emder-Fowler boundary value problems, SIAM J. Math. Anal., 12, 874-879 (1981) · Zbl 0478.34021
[18]	O’Regan, D., Topological Transversality: Applications to third order boundary value problems, SIAM J. Math. Anal., 18, 630-641 (1987) · Zbl 0628.34017
[19]	O’Regan, D., Fourth (and higher) order singular boundary value problems, Nonlinear Anal., 14, 1001-1038 (1990) · Zbl 0711.34026
[20]	Taliaferro, S. D., A nonlinear singular boundary value problem, Nonlinear Anal., 3, 897-904 (1979) · Zbl 0421.34021