Abstract
We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, that approximate (in some sense) eigenvalues of some linear operators, acting in some function spaces, must be eigenvalues while approximate eigenvectors are close to eigenvectors with the same eigenvalue. Our results also provide some natural generalizations and extensions of the classical Banach Contraction Principle.
Highlights
The notion of Ulam stability has motivated several generalizations of Banach Contraction Principle for various function spaces
We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space
To formulate it we need the following hypothesis for operators Λ : R+ E → R+ E (R+ stands for the set of nonnegative reals, E is a nonempty set and AB denotes the family of all functions mapping a nonempty set B into a nonempty set A): (C0) Ifn∈N is a sequence in R+ E with limn→∞ δn(t) = 0 for t ∈ E, nl→im∞(Λδn)(t) = 0, t ∈ E
Summary
The notion of Ulam stability (see [8,16,22] for details) has motivated several generalizations of Banach Contraction Principle for various function spaces (see, e.g., [3,4,6,7,9]). Let us recall for instance the main result in [9] (see [4, Theorem 2]) To formulate it we need the following hypothesis for operators Λ : R+ E → R+ E (R+ stands for the set of nonnegative reals, E is a nonempty set and AB denotes the family of all functions mapping a nonempty set B into a nonempty set A):. (C0) If (δn)n∈N is a sequence in R+ E with limn→∞ δn(t) = 0 for t ∈ E, nl→im∞(Λδn)(t) = 0, t ∈ E
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