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We determine nontrivial intervals $${I \subset(0,+\infty)}$$ , numbers $${\alpha\in\mathbb R}$$ and continuous bijections $${f \colon I \to I}$$ such that f( x) f( x) = x for every $${x\in I}$$ . [ABSTRACT FROM AUTHOR]

We study the functional equation f( x) f( x) = x imposing no continuity assumptions on its bijective solutions defined on an interval. All the continuous bijections satisfying the equation were determined by the second author in (Aequat. Math. (in print), ) when solving the problem (Problem posed during the forty-ninth International Symposium on Functional Equations ) posed by Brillouët-Belluot. [ABSTRACT FROM AUTHOR]

We solve the problem posed by Nicole Brillouët-Belluot (During the Forty-ninth International Symposium on Functional Equations, ) of determining all continuous bijections f : I → I satisfying where I is an arbitrary subinterval of the real line. [ABSTRACT FROM AUTHOR]

Let $${I\subset\mathbb{R}}$$ be a nonempty open interval and let $${L:I^2\to I}$$ be a fixed strict mean. A function $${M:I^2\to I}$$ is said to be an L-conjugate mean on I if there exist $${p,q\in{]}0,1]}$$ and a strictly monotone and continuous function φ such thatfor all $${x,y\in I}$$ . Here L( x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means. [ABSTRACT FROM AUTHOR]

For every $${\alpha\in\mathbb R}$$ we determine all increasing bijections f : (0, + ∞)→ (0, + ∞) such that $${f(1)\neq1}$$ and f( x) f( x) = x for every $${x\in (0,+\infty)}$$ . [ABSTRACT FROM AUTHOR]