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.
References
Almuthaybiri, S. S. and C. C. Tisdell. ``Global existence theory for fractional differential equations: New advances via continuation methods for contractive maps''. Analysis, 39(4):117–128, 2019. doi:10.1515/anly-2019-0027
Almuthaybiri, S. S. and C. C. Tisdell. ``Sharper existence and uniqueness results for solutions to third-order boundary value problems, mathematical modelling and analysis''. Math. Model. Anal. 25(3):409–420, 2020. doi:10.3846/mma.2020.11043
Banach, S. ``Sur les operations dans les ensembles abstraits et leur application aux equations integrales''. Fund. Math., 3:133–181 1922. doi:10.4064/fm-3-1-133-181
Brouwer, L. E. J. ``Ueber Abbildungen von Mannigfaltigkeiten''. Math. Ann. 71:598, 1912. doi:10.1007/BF01456812
Bryant, V. W. ``A remark on a fixed point theorem for iterated mappings'' Am. Math. Month. 75: 399–400, 1968. doi:10.2307/2313440
Caccioppoli, R. ``Un teorema generale sullesistenza de elemente uniti in una transformazione funzionale''. Rend. Acad. Naz. Linzei. 11:31–49, 1930.
Goebel, K., and W. A. Kirk. Topics in metric fixed point theory. Cambridge University Press, 1990, doi:10.1017/CBO9780511526152
Leray, J., and J. Schauder. ``Topologie et equations fonctionnelles''. Ann. Sci. Ecole Norm. Sup. 51:45–78, 1934. doi:10.24033/asens.836
O'Regan, D. and R. Precup. Theorems of Leray–Schauder type and applications, Series in Mathematical Analysis and Applications, Vol. 3. CRC Press, London, 2002. doi:10.1201/9781420022209
Rus, I. A. ``On a fixed point theorem of Maia''. Studia Univ. Babes-Bolyai Math. 22:40–42, 1977.
Schaefer, H. H. ``Ueber die Methode der a priori-Schranken''. Math. Ann. 129:415–416, 1955. doi:10.1007/bf01362380
Tisdell, C. C. ``When do fractional differential equations have solutions that are bounded by the Mittag-Leffler function?'' Fract. Calc. Appl. Anal. 18(3):642–650, 2015. doi:10.1515/fca-2015-0039
Tisdell, C. C. ``A note on improved contraction methods for discrete boundary value problems.'' J. Diff. Eq. Appl. 18(10):1773–1777, 2012. doi:10.1080/10236198.2012.681781
Tisdell, C. C. ``On the application of sequential and fixed-point methods to fractional differential equations of arbitrary order.'' J. Int. Eq. Appl. 24(2):283–319, 2012. doi:10.1216/JIE-2012-24-2-283
Ehrnstrom, M., Tisdell, C. C. and E. Wahlen. ``Asymptotic integration of second-order nonlinear difference equations.'' Glasg. Math. J. 53(2):223–243, 2011. doi:10.1017/S0017089510000650
Erbe, L., A. Peterson and C. C. Tisdell. ``Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales.'' Nonlin. Anal. 69(7):2303–2317, 2008. doi:10.1016/j.na.2007.08.010
Tisdell, C. C. and A. Zaidi. ``Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling.'' Nonlin. Anal. 68(11):3504–3524, 2008. doi:10.1016/j.na.2007.03.043
Tisdell, C. C. ``Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions.'' Int.. J. Math. Ed. Sci. Tech. 48(7):1087–1095, 2017. doi:10.1080/0020739X.2017.1298856
Tisdell, C. C. ``On Picard's iteration method to solve differential equations and a pedagogical space for otherness.'' Int. J. Math. Ed. Sci. Tech. 50(5):788–799, 2019. doi:10.1080/0020739X.2018.1507051
Zeidler, E. Nonlinear functional analysis and its applications. Springer-Verlag, New York, 1986. doi:10.1007/978-1-4612-4838-5
Some Results on Fractional m-Point Boundary Value Problems