Let K be a subset of a real normed linear space E and let T be a self-mapping on K . T is said to be nonexpansive provided ‖Tx−Ty‖ ‖x− y‖ for all x, y ∈ K . Fixed-point iteration process for nonexpansive mappings in Banach spaces including Mann and Ishikawa iteration processes has been studied extensively by many authors to solve the nonlinear operator equations in Hilbert spaces and Banach spa...

In this lecture, we discuss splitting algorithms for convex minimization problems with objective given by the sum of two nonsmooth functions. We start with the fixed point property of such problems and derive a general scheme of splitting algorithm based on fixed point iteration. This covers Douglas-Rachford splitting and Peaceman-Rachford splitting algorithms. We also discuss the convergence r...

In this paper, we prove the stability in random normed spaces via fixed point method for the functional equation f(x+ 2y)− 2f(x+ y) + 2f(x− y)− f(x− 2y) = 0.

We investigate the following generalized Cauchy functional equation f(αx+ βy) = αf(x) + βf(y) where α, β ∈ R \ {0}, and use a fixed point method to prove its generalized Hyers–Ulam–Rassias stability in Banach modules over a C∗-algebra.

we show that the variational inequality $vi(c,a)$ has aunique solution for a relaxed $(gamma , r)$-cocoercive,$mu$-lipschitzian mapping $a: cto h$ with $r>gamma mu^2$, where$c$ is a nonempty closed convex subset of a hilbert space $h$. fromthis result, it can be derived that, for example, the recentalgorithms given in the references of this paper, despite theirbecoming more complicated, are not...

a power digraph, denoted by $g(n,k)$, is a directed graph with $z_{n}={0,1,..., n-1}$ as the set of vertices and $l={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. in this paper, the structure of $g(2q+1,k)$, where $q$ is a sophie germain prime is investigated. the primality tests for the integers of the form $n=2q+1$ are established in terms of th...

In this paper, we establish a fixed point theorem for multi-valued operators in a complete b−metric space using the concept of Berinde and Berinde [9] on multi-valued weak contractions for the Picard iteration in a metric space. Our main result generalizes, extends and improves some of the recent results of Berinde and Berinde [9] as well as those of Daffer and Kaneko [17] and also unifies seve...

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not always possible to form the iterated homotopy fixed point spectrum (ZhH)hK/H , where Z is a continuous G-spectrum. However, we show that, if G = Gn, the extended Morava stabilizer group, and Z = L̂(En ∧ X), where L̂ is Bousfield localization with respect to Morava K-theory, En is the Lubin-Tate spectrum, a...

Let X,Y be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping f : X → Y satisfies f(x+ iy) + f(x− iy) = 2f(x)− 2f(y) (1) for all x, y ∈ X, then the mapping f : X → Y satisfies f(x+ y) + f(x− y) = 2f(x) + 2f(y) for all x, y ∈ X. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation (1) in complex Banach spaces. In this paper, we wi...