Existence of periodic solutions for a class of first order delay differential equations dealing with vectors

By the weak linking theorem and the local linking theorem, we study the existence of periodic solutions for the following delay non-autonomous systems equation(1)u′(t)+A(t)u(t)+μf(t,u(t−r))=0u′(t)+A(t)u(t)+μf(t,u(t−r))=0 where r>0r>0, 0≠μ∈R0≠μ∈R, f∈C(R×Rn,Rn)f∈C(R×Rn,Rn) satisfies f(t+r,x)=f(t,x)f(t+r,x)=f(t,x) for all x∈Rnx∈Rn, and A(t)=(aij(t))n×nA(t)=(aij(t))n×n is an anti-symmetric matrix-valued function with aij(t)∈C(R,R)aij(t)∈C(R,R) which is rr-periodic in tt for all i,j=1,2,…,ni,j=1,2,…,n. The variational principle is given, some existence theorems of 4r4r-periodic solutions of (1) are obtained.