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Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function $f\in C_{[-1,1]}^k$, with $f^{\left(k\right)}\left(x\right)\ge 0$ for x ∈ [0,1] and $f^{\left(k\right)}\left(x\right)\le 0$ for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where $e_n^{\left(k\right)}{\left(f\right)}_p$ is the best approximation of degree n to f in $L^p$ by polynomials which are comonotone with f, that is, polynomials P so that $P^{\left(k\right)}\left(x\right)f^{\left(k\right)}\left(x\right)\ge 0$ for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution...

This paper is concerned with periodic solutions of first-order nonlinear functional differential equations with deviating arguments. Some new sufficient conditions for the existence of periodic solutions are obtained. The paper extends and improves some well-known results.