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.
Highlights
Introduction and preliminariesThe fixed point theory is an essential tool in the field of nonlinear analysis, and in almost all branches of mathematics
We provide the same family of k-dimensional extensions for Fisher type contractions and develop similar infinite dimensional extension in this case
Proof We can prove the existence of a solution of infinite dimensional Fredholm integral equation if we show that the operator T defined by (37) has a fixed point
Summary
We discuss some of the preliminaries which will be needed later for proving our main theorems. Definition 2.7 (Picard operator) Let (X, d) be a metric space. Inspired by the Picard operator definition, we extend this notion for the k-dimensional and infinite dimensional cases as follows. The k-Picard sequence with respect to the operator T based on the base point set {x1, x2, . Example 2.6 If we fix k = 1, the base point set is singleton and the infinite 1- Picard sequence with respect to T based on the base point {x0} is basically the sequence defined by xn := T((xn–1)∞ n=1) ∀n ≥ 1 for some x0 ∈ X. Let us give our first main fixed point result, which is a generalization of the Banach contraction principle with respect to the infinite-dimensional notion introduced in our paper.
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