x(4)=f(t,x,x′,x″,x) ,
boundary value condition
were considered and results were obtained, here the boundary value condition is one of the followings:
x(0)=A,x(1)=B,x″(0)=,x″(1)=,
x(0)=A,x(1)=B,x″(0)=,x(1)=,
x(0)=A,x(1)=B,x(0)=,x″(1)=,
x(0)=A,x′(1)=B,x″(0)=,x″(1)=,
x(0)=A,x′(1)=B,x″(0)=,x(1)=,
x(0)=A,x′(1)=B,x(0)=,x″(1)=,
x′(0)=A,x(1)=B,x″(0)=,x″(1)=,
x′(0)=A,x(1)=B,x″(0)=,x(1)=,
x′(0)=A,x(1)=B,x(0)=,x″(1)=.
These results were given to assume that the function f(t,x,y,p,r) satisfies the following condition. There are pairs (four or eight) of suitable constants such that f(t,x,y,p,r) does not change sign on sets of the form ［0,1］×Dx×Dy×Dp×I, where Dx, Dy, Dp are closed bounded intervals, I is a closed set in R and bounded by some pairs of constants, mentioned above.