Let T be a self-mapping on a complete metric space (X, d). In this paper, we obtain new fixed point theorems assuming that T satisfies a contractive-type condition of the following form:$$\begin{aligned} \psi (d(Tx,Ty)) \le \varphi (d(x,y)) \end{aligned}$$or T satisfies a generalized contractive-type condition of the form$$\begin{aligned} \psi (d(Tx,Ty)) \le \varphi (m(x,y)), \end{aligned}$$where \({\psi ,\varphi :(0,\infty ) \rightarrow {\mathbb {R}}}\) and m(x, y) is defined by$$\begin{aligned} m(x,y) = \max \left\{ d(x,y), d(x,Tx), d(y,Ty), [d(x,Ty)+d(y,Tx)] / 2 \right\} . \end{aligned}$$In both cases, the results extend and unify many earlier results. Among the other results, we prove that recent fixed point theorems of Wardowski (2012) and Jleli and Samet (2014) are equivalent to a special case of the well-known fixed point theorem of Skof (1977).

中文翻译：

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度量空间中广义压缩映射的不动点定理

令T为完整度量空间（X，  d）上的自映射。在本文中，我们假设T满足以下形式的收缩型条件，得到新的不动点定理：$$ \ begin {aligned} \ psi（d（Tx，Ty））\ le \ varphi（d（x， y））\ end {aligned} $$或T满足$$ \ begin {aligned} \ psi（d（Tx，Ty））\ le \ varphi（m（x，y）形式的广义收缩型条件），\ end {aligned} $$，其中\（{\ psi，\ varphi：（0，\ infty）\ rightarrow {\ mathbb {R}}} \）和m（x，  y）定义为$$ \ begin {aligned} m（x，y）= \ max \ left \ {d（x，y），d（x，Tx），d（y，Ty），[d（x，Ty）+ d （y，Tx）] / 2 \ right \}。\ end {aligned} $$在两种情况下，结果都会扩展并统一许多以前的结果。在其他结果中，我们证明了Wardowski（2012）和Jleli and Samet（2014）的最新不动点定理等同于Skof（1977）的不动点定理的特例。