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Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems. (English) Zbl 0713.34025

The paper deals with the problem of existence and uniqueness of solutions for the boundary value problems

(1)d 4 u dx 4 +f(x,u(x),u ' (x),u '' (x),u ''' (x))=e(x),0<x<1,
(2)u(0)=u(1)=u '' (0)=u '' (1)=0or(3)u(0)=u ' (1)=u '' (0)=u ''' (1)=0

where f: [0,1]×R 4 R is a function satisfying Caratheodory’s conditions and e: [0,1]R is in L 1 (0,1). There are 4 theorems concerning the existence of the solutions and 2 theorems concerning the uniqueness of the solutions. We give the following theorem as an example:

Theorem 5. Suppose that there exist functions a(x), b(x), c(x) and d(x) in L [0,1] such that (f(x,u 1 ,v 1 ,w 1 ,y 1 )-f(x,u 2 ,v 2 ,w 2 ,y 2 ).

(w 1 -w 2 )a(x)(w 1 -w 2 ) 2 +b(x)|u 1 -u 2 ||w 1 -w 2 |+c(x)|v 1 -v 2 w 1 -w 2 |+d(x)|y 1 -y 2 ||w 1 -w 2 |

for all (u i ,v i ,w i ,y i )R 4 , i=1,2 and a.e. x[0,1]. Suppose further that there exist an L 2 -Caratheodory function β:[0,1]×R 3 R and γ(x)L 1 [0,1] such that |f(x,u,v,w,y)|β(x,u,v,w)|y|+γ(x) for all (u,v,w)R 3 , yR and a.e. x[0,1]. Then for e(x)L 1 [0,1]:

1. (1),(2) has exactly one solution if πa +b +πc +π 2 d <π 3 ;

2. (1), (3) has exactly one solution if 4π 2 a +16b +8πc +2π 3 d <π 4 ·

We note that the proof of the existence of the solutions was made by applying the version of Leray-Schauder continuation theorem given by Mawhin. The paper is a continuation and complementation of an earlier paper of C. P. Gupta [Appl. Anal. 26, No.4, 289-304 (1988; Zbl 0611.34015)].

Reviewer: M.Švec

MSC:
34B15Nonlinear boundary value problems for ODE
34G20Nonlinear ODE in abstract spaces