For the $p$-localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that the homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$, with respect to any homotopy associative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$. We also prove that $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}= 1$ for all but a finite number of primes $p >3$. Then, for the loop space of the associated $\mathbb {S}^{2m-1}_{(p)}$-projective space $\mathbb {S}^{2m-1}_{(p)}P(n-1)$, with $m,n\ge 2$ and $m\mid p-1$, we derive that $\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$.

Let $M(A,n)$ be the Moore space of type $(A,n)$ for an Abelian group A and $n\ge 2$ . We show that the loop space $\Omega (M(A,n))$ is homotopy nilpotent if and only if A is a subgroup of the additive group $\mathbb {Q}$ of the field of rationals. Homotopy nilpotency of loop spaces $\Omega (M(A,1))$ is discussed as well.

A continuous selection and a coincidence theorem are proved in H-spaces which generalize the corresponding results of Ben-El-Mechaiekh-Deguire-Granas, Browder, Ko-Tan, Lassonde, Park, Simon and Takahashi to noncompact and/or nonconvex settings. By applying the two theorems, some intersection theorems concerning sets with H-convex sections are obtained which generalize the corresponding results of Fan, Lassonde and Shih-Tan to H-spaces. Some applications to minimax principle are given.

Matching theorems, fixed point theorems and minimax inequalities are obtained in H-spaces which generalize the corresponding results of Bae-Kim-Tan, Browder, Fan, Horvath, Kim, Ko-Tan, Shih-Tan, Takahashi, Tan and Tarafdar to non-compact and/or non-convex settings.