# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Solvability of some fourth (and higher) order singular boundary value problems. (English) Zbl 0795.34018
This paper deals primarily with solutions of (1) $y\sp{(4)}= f(t,y,y'')$, $0<t<1$, 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 $f(t,y\sb 1,y\sb 2)$ may have singularities at $t=0,1$ and at $y\sb 1=0$, $y\sb 2=0$. The first results concern problem (1), (2), where $f(t,y\sb 1,y\sb 2)$ has singularities at $y\sb 1=0$, not at $y\sb 2=0$. Growth assumptions are made on $f$ such that one can conclude the existence of a priori bounds (independent of $\lambda$), on solutions of $y\sp{(4)}= \lambda f(t,y,y'')$, $0< t<1$, $0\leq\lambda \leq 1$, 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 $y\in C[0,1]\cap C\sp 3(0,1) \cap C\sp 4(0,1)$ of (1), (2). Later, boundary value problems are treated similarly for $y\sp{(4)}= \psi(t) f(t,y,y'')$, where $f(t,y\sb 1,y\sb 2)$ has singularities at $y\sb 1=0$, 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 $y\sp{(n)}= f(t,y,y'')$, $0<t<1$, where, as above, $f(t,y\sb 1,y\sb 2)$ may have singularities at $t=0,1$ and at $y\sb 1=0$, $y\sb 2=0$.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE
##### Keywords:
topological transversality theorem of Granas
Full Text:
##### 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 nth 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) · 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 (1982) · Zbl 0483.47038 [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 0615.34010 [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. Differential equation, 31-50 (1980) [16] J. W. Lee and D. O’Regan, Boundary value problems for nonlinear fourth order equations with applications to nonlinear beams, to appear. [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) [20] Taliaferro, S. D.: A nonlinear singular boundary value problem. Nonlinear anal. 3, 897-904 (1979) · Zbl 0421.34021