Some Generalizations of Fixed Point Theorems in Cone Metric Spaces

Abstract

We generalize, extend, and improve some recent fixed point results in cone metric spaces including the results of H.

Common fixed point of mappings satisfying implicit contractive conditions in TVS-valued ordered cone metric spaces

“…Our results generalize, extend and improve some recent fixed point results in K-metric spaces including the results of Abbas and Jungck [1], Olaleru [28], Huang and Zhang [14] and Rezapour and Hamlbarani [33]. It is worth mentioning that our results do not require the assumption that the cone is normal.…”

“…Fixed point theory in K-metric and K-normed spaces was developed by Perov et al [18,28,29], Mukhamadijev and Stetsenko [19], and Vandergraft [41]. For more details on this subject, we refer to Zabrejko [42].…”

Coupled fixed point results in cone metric spaces for "Equation missing" -compatible mappings

“…We establish b-coupled coincidence and b-common coupled fixed point theorems in such spaces. The presented theorems generalize and extend several well-known comparable results in the literature, in particular the recent results of Abbas et al [8] and the result of Olaleru [13]. Some examples and an application to non-linear integral equations are also considered.…”

“…Huang and Zhang [7] re-introduced such spaces under the name of cone metric spaces, and went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. Afterwards, many papers about fixed point theory in cone metric spaces were appeared (see, for example, [8][9][10][11][12][13][14][15]). …”

A hybrid class of expansive-contractive mappings in cone b-metric spaces

“…Expanding mappings have enjoyed a relatively lower popularity with the results of Wang et al [26], Daffer and Kaneko [7], Kumar and Garg [13] among others. However, with the generalization of metric spaces to b-metric spaces [3], cone metric spaces [10,15,16,20] and recently, cone b-metric spaces [11], fixed point theorems for expanding mappings have been proved in cone metric spaces (e.g. [1,23,25] The most recent and perhaps most general theorems on expanding mappings in cone metric spaces were initially proposed by Han and Xu [8] who considered a pair of surjective expansion mappings.…”