We obtain an extended Reich fixed point theorem for the setting of generalized cone rectangular metric spaces without assuming the normality of the underlying cone. Our work is a generalization of the main result in cite{AAB} and cite{JS}.
Given an ideal $mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $mathcal{I}$-statistically convergent to $ell$ provided that $$ textstyle left{n in mathbf{N}: frac{1}{n}|{k le n: x_k otin U}| ge varepsilonright} in mathca
l{I} $$ for all neighborhoods $U$ of $ell$ and all $varepsilon>0$. First, we show that $mathcal{I}$-statistical convergence coincides with $mathcal{J}$-convergence, for some unique ideal $mathcal{J}=mathcal{J}(mathcal{I})$. In addition, $mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $mathcal{I}$ is the summable ideal ${Asubseteq mathbf{N}: sum_{a in A}1/a<infty}$ or the density zero ideal ${Asubseteq mathbf{N}: lim_{nto infty} frac{1}{n}|Acap [1,n]|=0}$ then $mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $mathcal{I}$ is maximal.
The aim of this paper is to provide characterizations of a Meir-Keeler type mapping and a fixed point theorem for the mapping in a metric space endowed with a transitive relation.
We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point xinH, the hyperplane through Tx whose normal is x-Tx alwa
ys cuts the space into two half-spaces one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these we conduct convergence analysis of successive iteration algorithms.
We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space $X$ into $X$, and the second with an application
of the continuation method to the case where they satisfy the Leray-Schauder boundary condition in Banach spaces.
We extend to binary relational systems the notion of compact and normal structure, introduced by J.P.Penot for metric spaces, and we prove that for the involutive and reflexive ones, every commuting family of relational homomorphisms has a common fix
ed point. The proof is based upon the clever argument that J.B.Baillon discovered in order to show that a similar conclusion holds for bounded hyperconvex metric spaces and then refined by the first author to metric spaces with a compact and normal structure. Since the non-expansive mappings are relational homomorphisms, our result includes those of T.C.Lim, J.B.Baillon and the first author. We show that it extends the Tarskis fixed point theorem to graphs which are retracts of reflexive oriented zigzags of bounded length. Doing so, we illustrate the fact that the consideration of binary relational systems or of generalized metric spaces are equivalent.