Let S be a self-mapping on a normed space ${\mathcal{X}}$ . In this paper, we introduce three new classes of mappings satisfying the following conditions: $$\begin{aligned} & \max_{ \substack{ 0\leq k \leq m \\ k \text{ even} } } \bigl\Vert S^{k}x-S^{k}y \bigr\Vert =\max_{ \substack{ 0\leq k\leq m \\ k \text{ odd} } } \bigl\Vert S^{k}x-S^{k}y \bigr\Vert , \\ & \max_{ \substack{ 0\leq k \leq m \\ k \text{ even} } } \bigl\Vert S^{k}x-S^{k}y \bigr\Vert \leq \max_{ \substack{ 0\leq k\leq m \\ k \text{ odd} } } \bigl\Vert S^{k}x-S^{k}y \bigr\Vert , \\ & \max_{ \substack{ 0\leq k \leq m \\ k \text{ even} } } \bigl\Vert S^{k}x-S^{k}y \bigr\Vert \geq \max_{ \substack{ 0\leq k\leq m \\ k \text{ odd} } } \bigl\Vert S^{k}x-S^{k}y \bigr\Vert , \end{aligned}$$ for all $x,y\in {\mathcal{X}}$ , where m is a positive integer. We prove some properties of these classes of mappings.

中文翻译：

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规范空间上的非线性（（m，\ infty）\）-等距和\（（m，\ infty）\）-扩张（压缩）映射

令S为规范空间$ {\ mathcal {X}} $上的自映射。在本文中，我们介绍了三种新的满足以下条件的映射类：$$ \ begin {aligned}和\ max_ {\ substack {0 \ leq k \ leq m \\ k \ text {偶数}}} \ bigl \ Vert S ^ {k} xS ^ {k} y \ bigr \ Vert = \ max_ {\ substack {0 \ leq k \ leq m \\ k \ text {奇数}}}} \ bigl \ Vert S ^ {k} xS ^ {k} y \ bigr \ Vert，\\和\ max_ {\ substack {0 \ leq k \ leq m \\ k \ text {even}}} \ bigl \ Vert S ^ {k} xS ^ {k} y \ bigr \ Vert \ leq \ max_ {\ substack {0 \ leq k \ leq m \\ k \ text {奇数}}}} \ bigl \ Vert S ^ {k} xS ^ {k} y \ bigr \ Vert， \\＆\ max_ {\ substack {0 \ leq k \ leq m \\ k \ text {偶数}}}} \ bigl \ Vert S ^ {k} xS ^ {k} y \ bigr \ Vert \ geq \ max_ { \ substack {0 \ leq k \ leq m \\ k \ text {奇数}}} \ bigl \ Vert S ^ {k} xS ^ {k} y \ bigr \ Vert，\ end {aligned} $$对于所有$ x，y \ in {\ mathcal {X}} $中，其中m是一个正整数。