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Existence and uniqueness theorems for fourth-order boundary value problems. (English) Zbl 0634.34009

The differential equation (1) y (IV) =f(x,y,y '' ) is considered under the following types of boundary conditions:

(2) y(0)=y 0 , y(1)=y 1 , y '' (0)=y ¯ 0 , y '' (1)=y ¯ 1 ;

(3) y(0)=y 0 , y(1)=y 1 , y '' (0)=y ¯ 0 , y ''' (1)=y ¯ 1 ;

(4) y(0)=y(1)=y '' (0)=y '' (1)=0 or

(5) y(0)=y(1)=0; y ''' (0)-hy '' (0)=0, y ''' (1)+ky '' (1)=0, with h,k0, h+k>0.

Existence theorems for all of the boundary value problems (1-2)–(1-5) are obtained by application of Schauder’s fixed point theorem under continuity and boundedness hypotheses on f and its partial derivative f 3 . A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative f 2 . Further uniqueness results for the problem (1-2)–(1-5) are reduced to uniqueness questions for solutions of corresponding second order boundary value problems for the integrodifferential equation

u '' =f(x,y 0 +x(y 1 -y 0 )+ 0 1 G(x,t)u(t)dt,u)

obtained by setting y '' =u, with G(x,t) the Green’s function for the problem

u '' =0,u(0)=u(1)=0·

Boundary conditions of the form (2)–(5) have been less extensively studied than the familiar conjugate or focal-type problems. R. A. Usmani [Proc. Am. Math. Soc. 77, 329–335 (1979; Zbl 0424.34019)] has studied a problem of the form (1-2) in which the equation is linear and independent of y '' .

Reviewer: L.J.Grimm

MSC:
34B15Nonlinear boundary value problems for ODE