### Fixed point theorems and tiling problems

#### Abstract

Let $(X, d)$ be a complete metric

space and let $f:X\to X$ satisfy $\inf\{\alpha(x, y)d(f^m(x),

f^m(y)): m\in\mathbb{J}\}\leq Kd(x, y)$ for all $x, y\in X$ and

some $K\in (0, 1)$ and $\alpha:X\times X\to [0, \infty)$, where

$\mathbb{J}$ is a set of positive integers. In this paper, we

prove fixed point theorems for this mapping $f$. We also discuss

the connection

with tiling problems and give a titling proof of a fixed point theorem.

space and let $f:X\to X$ satisfy $\inf\{\alpha(x, y)d(f^m(x),

f^m(y)): m\in\mathbb{J}\}\leq Kd(x, y)$ for all $x, y\in X$ and

some $K\in (0, 1)$ and $\alpha:X\times X\to [0, \infty)$, where

$\mathbb{J}$ is a set of positive integers. In this paper, we

prove fixed point theorems for this mapping $f$. We also discuss

the connection

with tiling problems and give a titling proof of a fixed point theorem.

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