\((m,p)\)-hyperexpansive mappings on metric spaces (Q2831407)

Let \((X,d)\) be a metric space, and \(T:X\to X\) be a map.NEWLINENEWLINEThe following are the main results in this paper.NEWLINENEWLINETheorem 1. Let \(m\geq 3\) be such that \(T\) is \((m,p)\)-expansive and \((2,p)\)-expansive. Then, \(T\) is \((m-1,p)\)-expansive.NEWLINENEWLINETheorem 2. Suppose that there exist \(S:X\to X\) and \(m\geq 1\) such that (i) \(TS=I\) (= the identity) (ii) \(\Theta_m^{(p)}(d,S;x,y)\leq 0\), for all \(x,y\in\mathcal R(T^m)\).NEWLINENEWLINEThen, the following conclusions holdNEWLINENEWLINE(C1) \(T\) is \((m,p)\)-expansive if \(m\)=evenNEWLINENEWLINE(C2) \(T\) is \((m,p)\)-contractive, if \(m\)=odd.NEWLINENEWLINETheorem 3. Suppose that \(T\) is \((2,p)\)-expansive. Then, \(T^n\) is \((2,p)\)-expansive, for each \(n\geq 1\).NEWLINENEWLINEFinally, a partial extension of these facts to seminormed spaces is being performed.