There have been 9 years, we are familiar with control function F in the definition of F-contraction. In the literature, we read only four examples of this function, it shows that this function is NARROW by nature. On the other hand, fixed point theorists have published a huge number of papers on that topic. However, thanks to Proinov (2020), who has ignored F2, F3 and proved fixed point theorem using only F1. In this way, made F versatile.
Dear M. Nazam. I partially agree with you, however, in my opinion Proinov's main fixed point result cannot be deduced by considering condition F1 only. In his main theorem (see Theorem 3.6 Proinov) Proinov used condition (iii) to prove that T is asymptotically regular whereas Wardowski used condition F2 to prove that T is asymptotically regular (equivalent to equation 6" (see Theorem 2.1 Wardowski)). Please note that by assuming, \phi = \psi - \tau , the condition (iii) of Proinov's main result cannot be removed. In my opinion Proinov's deduction to Wardowski's result cannot guarantee, mapping T to be "asymptotically regular and Cauchy" without considering condition (iii).
Moreover, interestingly, only first example is sufficient. There are numerous important applications in which we may use F= lnt to weaken conditions of differential equation.
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