The purpose of this work is to advance the current state of mathematical knowledge regarding fixed point theorems of functions. Such ideas have historically enjoyed many applications, for example, to the qualitative and quantitative understanding of differential, difference and integral equations. Herein, we extend an established result due to Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42] that involves two metrics to ensure wider classes of functions admit a unique fixed point. In contrast to the literature, a key strategy herein involves placing assumptions on the iterations of the function under consideration, rather than on the function itself. In taking this approach we form new advances in fixed point theory under two metrics and establish interesting connections between previously distinct theorems, including those of Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42], Caccioppoli [Rend. Acad. Naz. Linzei. 11, 1930, 31–49] and Bryant [Am. Math. Month. 75, 1968, 399–400]. Our results make progress towards a fuller theory of fixed points of functions under two metrics. Our work lays the foundations for others to potentially explore applications of our new results to form existence and uniqueness of solutions to boundary value problems, integral equations and initial value problems.
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