Abstract
In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202,
2018
), (Fisher in Math. Mag. 48(4):223–225,
1975
), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13,
2019
). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225,
1975
) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202,
2018
) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13,
2019
). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.