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