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

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
Full Text:

### References:

  Aftabizadeh, A.R., Existence and uniqueness theorems for fourth-order boundary value problem, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009  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  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  Bailey, P.; Shampine, L.; Waltman, P., Nonlinear two point boundary value problems, (1968), Academic Press New York · Zbl 0169.10502  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  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  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  Dugundji, J.; Granas, A., Fixed point theory, () · Zbl 1025.47002  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  Granas, A., Sur la methode de continuité de Poincaré, C. R. acad. sci. Paris, 282, 983-985, (1976) · Zbl 0348.47039  Granas, A.; Guenther, R.B.; Lee, J.W., Nonlinear boundary value problems for ordinary differential equations, Dissertationes math. Warsaw, (1985) · Zbl 0476.34017  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  Gupta, C.P., Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015  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  Jackson, L.K., Boundary value problems for Lipschitz equations, (), 31-50  {\scJ. W. Lee and D. O’Regan}, Boundary value problems for nonlinear fourth order equations with applications to nonlinear beams, to appear.  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  O’Regan, D., Topological transversality: applications to third order boundary value problems, SIAM J. math. anal., 18, 630-641, (1987) · Zbl 0628.34017  O’Regan, D., Fourth (and higher) order singular boundary value problems, Nonlinear anal., 14, 1001-1038, (1990) · Zbl 0711.34026  Taliaferro, S.D., A nonlinear singular boundary value problem, Nonlinear anal., 3, 897-904, (1979) · Zbl 0421.34021
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