Involutions with fixed points in 2-Banach spaces (Q687844)

Inspired by a condition of \textit{D. Delbosco} [Jñānābha 16, 1-11 (1986; Zbl 0632.54026)], the author extends a fixed point theorem of \textit{N. A. Assad} and the reviewer [Math. J. Toyama Univ. 14, 141-146 (1991; Zbl 0752.47020)] from a Banach space to a 2-Banach space \(X\) for involution maps \(T\) satisfying the following inequality: \[ d(Tx, Ty, a)\leq \varphi \bigl(d (x, y, a), d(x, Tx, a), d(y, Ty, a) \bigl) \] for all \(x\), \(y\), \(a\) in \(X\), where \(T: X\to X\), \(d: X\times X\to [0, +\infty)\) is a 2-norm and \(\varphi: [0, +\infty)^3\to [0, +\infty)\) is a continuous function such that \(\varphi (1,1,1)= k<2\) and for \(s\geq 0\), \(t\geq 0\) the inequality \(s\leq \varphi (t, 2t, s)\) implies \(s\leq kt\).