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Fixed point theorems related to Ćirić’s contraction principle. (English) Zbl 0917.54047
Recently Kada, Suzuki and Takahashi introduced the notion of $$w$$-distance on a metric space as follows: let $$X$$ be a metric space with metric $$d$$. Then a function $$p:X\times X\to[0,\infty)$$ is called $$w$$-distance on $$X$$ if the following conditions are satisfied: (1) $$p(x,z)\leq p(x,y)+ p(y,z)$$ for any $$x,y,z\in X$$; (2) for any $$x\in X$$, $$p(x,.): X\to[0,\infty)$$ is lower semicontinuous; (3) for any $$\varepsilon>0$$, there exists $$\delta>0$$ such that $$p(z,x) \leq\delta$$ and $$p(z,y)\leq\delta$$ imply $$d(x,y) \leq\varepsilon$$. They also proved a fixed-point theorem for contractive mappings in such metric spaces. The main purpose of the present paper is to extend fixed point theorems due to Kannan, Ćirić, Kada, Suzuki, and Takahashi in metric spaces with $$w$$-distance.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
##### Keywords:
$$w$$-distance on a metric space; contractive mappings
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##### References:
 [1] Ćirić, L.J., A generalization of Banach’s contraction principle, Proc. amer. math. soc., 45, 267-273, (1974) · Zbl 0291.54056 [2] Kada, O.; Suzuki, T.; Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. japonica, 44, 381-391, (1996) · Zbl 0897.54029 [3] Kannan, R., Some results on fixed points, II, Amer. math. monthly, 76, 405-408, (1969) · Zbl 0179.28203
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