simulation function

In this manuscript, we consider the interpolative contractions mappings via simulation func-tions in the setting of complete metric space. We also express an illustrative example to show the validity of our presented results.

The purpose of this short note is to consider much shorter and nicer proofs about fixed point results on b-metric spaces via b-simulation function introduced very recently by Demma et al. [M. Demma, R. Saadati, P. Vetro, emph{Fixed point results on b-metric space via Picard sequences and b-simulation functions}, Iranian J. Math. Sci. Infor. 11 (1) (2016) 123--136].

In a recent paper, Khojasteh emph{et al.} [F. Khojasteh, S. Shukla, S. Radenovic, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189-–1194] presented a new class of simulation functions, say $mathcal{Z}$-contractions, with unifying power over known contractive conditions in the literature. Following this line of research, we extend and generalize their results on a $b$-metric context, by giving a new notion of $b$-simulation function. Then, we prove and discuss some fixed point results in relation with existing ones.