By Zhang B.

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**Extra info for 2 T -periodic solution for m order neutral type differential equations with time delays**

**Sample text**

11) The following theorem holds. 3. 11) is quasiconcave and it is concave if and only if α ≤ 1. Proof. The function g(x) = min{ xaii } is concave as the minimum of a ﬁnite i number of concave functions, so that f is an increasing transformation of g and thus it is quasiconcave. 1. , k, Consider the function z(x) = i=1 are positive concave functions on a convex set S ⊆ n . Since log z(x) = k αi log fi (x) is a concave function (as a positive linear combination of i=1 concave functions), the function z(x) = elog z(x) is quasiconcave.

More exactly, we have the following theorem. 6. Let f be a diﬀerentiable function on an open convex set S ⊆ n. (i) If f is pseudoconvex on S, then f is quasiconvex on S; (ii) If ∇f (x) = 0, ∀x ∈ S, then f is pseudoconvex on S if and only if it is quasiconvex on S. Proof. (i) Assume that f is not quasiconvex. Then, there exist x1 , x2 ∈ S with f (x1 ) ≥ f (x2 ) such that ∇f (x1 )T (x2 − x1 ) > 0. Consider the restriction ϕ(t) = f (x1 + t(x2 − x1 )), t ∈ [0, 1]. Since ϕ (0) = ∇f (x1 )T (x2 − x1 ) > 0, ϕ attains its maximum value at an interior point t0 ∈ (0, 1), so that ϕ(t0 ) = f (x0 ) > f (x1 ) = ϕ(0) ≥ f (x2 ) = ϕ(1) and ϕ (t0 ) = ∇f (x0 )T (x2 − x1 ) = 0, where x0 = x1 + t0 (x2 − x1 ).

1 may be speciﬁed in the case where ϕ is a lower semicontinuous function. 2. Let ϕ be a lower semicontinuous function deﬁned on the interval [a, b] ⊆ . Then, ϕ is quasiconvex if and only if there exists t0 ∈ [a, b] such that ϕ is non-increasing in [a, t0 ] and non-decreasing in [t0 , b], where one of the two subintervals may be reduced to a point. Proof. Assume that ϕ is quasiconvex. The lower semicontinuity of ϕ on [a, b] implies the existence of its minimum value m. Set A = {t ∈ [a, b] : ϕ(t) = m}; A is a closed interval since ϕ is quasiconvex and lower semicontinuous.