In the last little more than a hundred years, many research mathematicians have tried to generalize
the mentioned Banach theorem. A large number of beautiful papers were written that
continue to motivate many mathematicians. One of the significant results that generalizes Banach's
famous theorem is also the result of the Polish mathematician D. Wardowski [11]. After
that result, more new ones were created, which now generalize Wardowski.
The aim of this paper is to significantly improved and supplement the recently
established results from the papers (R. Batra, S. Vashistha, Coincidence Point Theorem for a
New Type of Contraction on Metric Spaces, Int. Journal of Math. Analysis, Vol. 8, 2014,
no. 27, 1315 - 1320 and D. Wardowski, N. Van Dung, Fixed points of F−weak contractions
on complete metric spaces, Demonstratio Mathematica Vol. XLVII No 1. 2014, 146-155, about
F − g−contractions and F−weak contractions. In the entire paper, for the Wardowski function
F, we assume only its strict increasing on (0,+1) i.e., the property F1. In both papers, the
authors assume all three properties F1, F2 and F3 of the mappings F.