A new approach to the study of fixed point theory for simulation functions in G-Metric spaces

Abstract

In this paper first of all, we introduce the mapping ζ : [0, ∞)×[0, ∞) → R, called the simulation function and the notion of Z-contraction with respect to ζ which generalize several known types of contractions. Secondly, we prove certain fixed point theorems using simulation functions in G-Metric spaces. An example is also given to support our results.

Solution of integral equations via new Z-contraction mapping in Gb-metric spaces

“…Finally, we apply our result to the existence of a solution of an Integral equation. The obtained results in this paper generalize, unify and improve the fixed point results of Liu et al, [17], Antonio-Francisco et al [23], Khojasteh et al [15], Kumar et al [16] and other results in this direction in the literature.…”

“…Motivated by the concept of b-metric and G-metric spaces [8,21], Aghajani et al in [3], introduced the notion of generalized b-metric space (G b − metric spaces), presented some properties of G b -metric spaces and prove some coupled coincidence fixed point theorems for (ψ, ϕ)-weakly con-tractive mappings in the frame work of partially ordered G b -metric spaces. Thereafter, several results and applications has been extended from metric spaces, b-metric spaces and G-metric spaces to G b -metric spaces, more so, a lot of results on the fixed point theory of various classes of mappings in the frame work of G b -metric spaces has been established by different researchers in this area (see [16] and the references therein). The notion of G b -metric spaces generalize, improves and unifies results in metric spaces, b-metric and G-metric.…”

“…Very recently, Kumar et al [16] introduced the concept of Z-contraction with respect to ζ in the frame work of G-metric spaces. They establish some fixed point results and gave an example to support their main result.…”

“…Motivated by the works of Liu et al [17], Kumar et al [16], Khojasteh et al [15], Antonio-Francisco [23] and the current research interest in this direction, we introduce the notions of b-C F simulation function, (α, β)admissible type B mapping and (α, β)-Z F -contraction mapping with re-spect to ζ in the frame work G b -metric spaces. Furthermore, we establish some fixed point results for (α, β)-Z F -contraction mapping in the frame work of complete G b -metric spaces and apply our results to establish the existence of solution of an integral equation.…”

New Fixed-Point Theorems on an S-metric Space via Simulation Functions

“…Motivated by the above results, in this paper, we prove some fixed-point theorems using the set of simulation functions on an S-metric space. To do this, we are inspired by the idea given in [5,12] using the simulation function approach on a metric and a G-metric space, respectively. Given the fact that not every S-metric space is generated by a metric space, since the classes of S-metric spaces and G-metric spaces are different, it is interesting to study new fixed-point results on S-metric spaces using the set Z.…”