The paper examines multiplicative ergodic theorems and the related multiplicative Poisson equation for an irreducible Markov chain on a countable state space. The partial products are considered for a realvalued function on the state space. If the function of interest satisfies a monotone condition, or is dominated by such a function, then (i) The mean normalized products converge geometrically...

Machine-learning solutions to classification, clustering and matching problems critically depend on the adopted metric, which in the past was selected heuristically. In the last decade, it has been demonstrated that an appropriate metric can be learnt from data, resulting in superior performance as compared with traditional metrics. This has recently stimulated a considerable interest in the to...

the purpose of this paper is to obtain necessary and suffcient conditionsfor existence approximate fixed point on geraghty-contraction. in this paper,denitions of approximate -pair fixed point for two maps tα , sα and theirdiameters are given in a metric space.

In this paper, we prove some common fixed point results for two self mappingsf and g on S-metric space such that f is a g.w.c.m with respect to g.

In this paper, we apply the idea of integral type contraction and prove some coupled fixed point theorems for such contractions in ordered $G$-metric space. Also, we support the main results by an illustrative example.

in this paper, a concept of generalized weakly contraction mappings in partially ordered fuzzy metric spaces is introduced and  coincidence point  theorems on partially ordered fuzzy metric spaces are proved. also, as the corollary of these theorems, some common fixed point theorems on partially ordered fuzzy metric spaces are presented.