Use fmincon.
Define the constraints
1<a<2; 2<b<3; 3<c<4; 4<d<5; 5<e<6
in lb and ub,
define the constraints
0<F(a,b,c,d,e)<=0.01, 0<G(a,b,c,d,e)<=0.01, 0.9<=H(a,b,c,d,e)<=0.95, I(a,b,c,d,e)=sqrt(1-H(a,b,c,d,e)^2)
in the function nonlcon for nonlinear constraints in the array c.
Then simply define the objective function as "0" since you don't want to minimize anything, but you only want to find a feasible point a,b,c,d,e for your problem.
