Fixed Points of Geraghty‐Type Mappings in Various Generalized Metric Spaces

Abstract

Fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in the frame of partial metric spaces, ordered partial metric spaces, and metric-type spaces. Examples are given showing that these results are proper extensions of the existing ones.

Ordered b-metric spaces and Geraghty type contractive mappings

“…The following example shows that, in general, a b-metric does not necessarily need to be a metric, see also (Aghajani, et al, 2014), (Abbas, et al, 2016(Abbas, et al, , pp.1413(Abbas, et al, -1429, (Ansari, et al, 2016), (Ding, et al, 2016, pp.151-164), (Djukić, et al, 2011), (Huang, et al, 2015a, pp.808-815), (Huaping, et al, 2015), (Huang, et al, 2015b, pp.800-807), (Hussain, et al, 2012), (Hussain, et al, 2013), (Jleli, et al, 2012, pp.175-192), (Jovanović, et al, 2010), (Kadelburg, et al, 2015, pp.57-67), (Khamsi, Hussain, 2010, pp.3123-3129), , (Roshan, et al, 2015), (Roshan, et al, 2014, pp.229-245), (Zabihi, Razani, 2014 (Aghajani, et al, 2014, pp.941-960), (Abbas, et al, 2016(Abbas, et al, , pp.1413(Abbas, et al, -1429, (Ansari, et al, 2016), (Djukić, et al, 2011), (Huang, et al, 2015a, pp.808-815), (Huaping, et al, 2015), (Huang, et al, 2015b, pp.800-807), (Hussain, et al, 2012), (Hussain, et al, 2013), (Jleli, et al, 2012, pp.175-192), (Jovanović, et al, 2010), (Kadelburg, et al, 2015, pp.57-67), (Khamsi, Hussain, 2010, pp.3123-3129), , (Roshan, et al, 2015), (Roshan, et al, 2014, pp.229-245), (Zabihi, Razani, 2014) and the references mentioned therein. Also, for the concepts such as partial order, comparable, well ordered, nondecreasing, increasing, dominated, dominating and other, w...…”

On some new fixed point results for rational Geraghty contractive mappings in ordered b-metric spaces

“…Since then, several papers dealt with fixed point theory for single-valued and multi-valued operators in b-metric spaces (see, e.g., [7,9,12,13,17,19,21]). Definition 1.1.…”

“…For some useful examples of b-metric spaces, the reader may refer to [3,7,9,12,14,17,19,21]. Also, the notions such as b-convergent sequence, b-Cauchy sequence and b-complete space are defined by an obvious way.…”

“…It is clear that if f is a map which has a fixed point x, then x is also a fixed point of f n for every n ∈ N, but the converse does not hold. If F (f ) = F (f n ) for every n ∈ N, then f is said to have property P. For more details, we refer the reader to [2], [7] and [11] and their references therein.…”

Coupled fixed point of generalized contractive mappings on partially ordered G-metric spaces

“…This principle has been generalized in different directions in different spaces by mathematicians over the years (see [2][3][4][5][6][7][8][9][10] and references mentioned therein). On the other hand, fixed point theory has received much attention in metric spaces endowed with a partial ordering.…”