Fixed point theorems for generalized contractive mappings in metric spaces

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: ψ(d(Tx,Ty))≤φ(d(x,y))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \psi (d(Tx,Ty)) \le \varphi (d(x,y)) \end{aligned}$$\end{document}or T satisfies a generalized contractive-type condition of the form ψ(d(Tx,Ty))≤φ(m(x,y)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \psi (d(Tx,Ty)) \le \varphi (m(x,y)), \end{aligned}$$\end{document}where ψ,φ:(0,∞)→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\psi ,\varphi :(0,\infty ) \rightarrow {\mathbb {R}}}$$\end{document} and m(x, y) is defined by m(x,y)=maxd(x,y),d(x,Tx),d(y,Ty),[d(x,Ty)+d(y,Tx)]/2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m(x,y) = \max \left\{ d(x,y), d(x,Tx), d(y,Ty), [d(x,Ty)+d(y,Tx)] / 2 \right\} . \end{aligned}$$\end{document}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).