Applications of Banach Limit in Ulam Stability

We show how to get new results on Ulam stability of some functional equations using the Banach limit. We do this with the examples of the linear functional equation in single variable and the Cauchy equation.

General methods of convergence and summability

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ N ; and that if X is not complete, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\mathbb{N} $ N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ c ( X ) is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ c ( X ) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$ HB ( lim ) : = { T ∈ B ( ℓ ∞ ( X ) , X ) : T | c ( X ) = lim and ∥ T ∥ = 1 } and prove that $\mathcal{HB}(\lim )$ HB ( lim ) is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$ B L X 0 if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $ L X 0 : = { T ∈ B ( ℓ ∞ ( X ) , X ) : c 0 ( X ) ⊆ ker ( T ) } . Finally, we study the multipliers associated with series for the above methods of convergence.

Banach Limit in the Stability Problem of a Linear Functional Equation

We present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.

Characterizations of fixed points of nonexpansive mappings

Using the notion of Banach limits, we discuss the characterization of fixed points of nonexpansive mappings in Banach spaces. Indeed, we prove that the two sets of fixed points of a nonexpansive mapping and some mapping generated by a Banach limit coincide. In our discussion, we may not assume the strict convexity of the Banach space.

Asymptotic Behavior of Generalized Nonexpansive Sequences and Mean Points

Let 𝐸 be a real Banach space with norm ‖ · ‖ and let {𝑥𝑛}𝑛≥0 be a generalized nonexpansive sequence in 𝐸 (i.e., ‖ 𝑥𝑖+1 – 𝑥𝑗+1 ‖2 ≤ ‖𝑥𝑖 – 𝑥𝑗‖2 + (ε(𝑖+1, 𝑗+1) – ε(𝑖, 𝑗))2 for all 𝑖, 𝑗 ≥ 0, where the series of nonnegative terms is convergent). Let . We deal with the mean point of concerning a Banach limit μ. If 𝐸 is reflexive and 𝑑 = 𝑑(0, 𝐾), then we show that and there exists a point 𝑧0 with ‖ 𝑧0 ‖= 𝑑 such that . In the sequel, this result is applied to obtain the weak and strong convergence of .

Almost none of the sequences of0's and1's are almost convergent

We establish that, in the sense of the Law of Large Numbers, almost none of the sequences of0’s and1’s are assigned the same value by every Banach limit.