Nonlinear contractions on semimetric spaces

J. Jachymski, J. Matkowski and T. {S}wiatkowski

Abstract: Let $(X,d)$ be a Hausdorff semimetric ($d$ need not satisfy the triangle inequality) and $d$--Cauchy complete space. Let $f$ be a selfmap on $X$, for which $d(fx,fy)\leq\phi(d(x,y))$, $(x,y\in X)$, where $\phi$ is a non-- decreasing function from \R, the nonnegative reals, into \R\/ such that $\phi ^n(t)\ra 0$, for all $t\in\R$. We prove that $f$ has a unique fixed point if there exists an $r>0$, for which the diameters of all balls in $X$ with radius $r$ are equibounded. Such a class of semimetric spaces includes the Frechet spaces with a regular ecart, for which the Contraction Principle was established earlier by M. Cicchese \cite{Ci}, however, with some further restrictions on a space and a map involved. We also demonstrate that for maps $f$ satisfying the condition $d(fx,fy)\leq\phi(\max\{d(x,fx),d(y,fy)\})$, $(x,y\in X)$ (the Bianchini \cite{Bi} type condition), a fixed point theorem holds under substantially weaker assumptions on a distance function $d$.

Keywords: Fixed point, nonlinear contraction, semimetric,symmetric, space with a regular ecart, $E$-space, $d$-Cauchy completeness

Classification (MSC2000): 47H10, 54H25

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