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## On generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality

### Fixed Point Theory and Applications (2013-09-23) 2013: 1-15 , September 23, 2013

Generalized asymptotically quasi-*ϕ*-nonexpansive mappings and a Ky Fan inequality are investigated. A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-*ϕ*-nonexpansive mappings and a Ky Fan inequality is established in a Banach space.

*MSC:*47H05, 47J25, 90C33.

## Hybrid methods for common solutions in Hilbert spaces with applications

### Journal of Inequalities and Applications (2014-05-13) 2014: 1-16 , May 13, 2014

In this paper, hybrid methods are investigated for treating common solutions of nonlinear problems. A strong convergence theorem is established in the framework of real Hilbert spaces.

## Approximations of ternary Jordan homomorphisms and derivations in multi-C ∗ ternary algebras

### Acta Mathematica Hungarica (2012-01-01) 134: 99-114 , January 01, 2012

Using fixed point methods, we prove the generalized Hyers–Ulam stability of homomorphisms in multi-*C*^{∗} ternary algebras and of derivations on multi-*C*^{∗} ternary algebras for the additive functional equation
$$\sum_{i=1}^{m}f \bigg(mx_i+\sum_{j=1,\ j\ne i}^{m}x_j\bigg)+ f\bigg(\sum_{i=1}^{m}x_i\bigg)= 2f\bigg(\sum_{i=1}^{m}mx_i\bigg) \quad (m\in {\mathbb{N}},\ m\geqq2).$$

## A fixed point theorem and the Hyers-Ulam stability in Riesz spaces

### Advances in Difference Equations (2013-05-14) 2013: 1-12 , May 14, 2013

We prove a fixed point theorem and show its applications in investigations of the Hyers-Ulam type stability of some functional equations (in single and many variables) in Riesz spaces.

*MSC:*39B82, 47H10.

## An altering distance function in fuzzy metric fixed point theorems

### Fixed Point Theory and Applications (2015-05-14) 2015: 1-19 , May 14, 2015

The aim of this paper is to improve conditions proposed in recently published fixed point results for complete and compact fuzzy metric spaces (Ćirić in Chaos Solitons Fractals 42:146-154, 2009; Shen *et al.* in Appl. Math. Lett. 25:138-141, 2012). For this purpose, the altering distance functions are used. Moreover, in some of the results presented the class of *t*-norms is extended by using the theory of countable extensions of *t*-norms. The mentioned generalizations are illustrated by examples.

## Orthogonal Stability of an Additive-Quadratic Functional Equation

### Fixed Point Theory and Applications (2011-10-25) 2011: 1-11 , October 25, 2011

Using the fixed point method and using the direct method, we prove the Hyers-Ulam stability of an orthogonally additive-quadratic functional equation in orthogonality spaces.

*(2010) Mathematics Subject Classification:* Primary 39B55; 47H10; 39B52; 46H25.

## Fuzzy quasi-triangular spaces, fuzzy sets of Pompeiu-Hausdorff type, and another extensions of Banach and Nadler theorems

### Fixed Point Theory and Applications (2016-03-12) 2016: 1-49 , March 12, 2016

Let
$\mathcal{A}$
be an index set, and
$C=\{C_{\alpha}\}_{\alpha\in \mathcal{A}}\in{}[1;\infty)^{\mathcal{A}}$
. Fuzzy quasi-triangular space is defined to be
$(X,\mathcal{M}_{C;\mathcal{A}},\ast)$
, where *X* is a nonempty set, a fuzzy family
$\mathcal{M}_{C;\mathcal{A}}=\{M_{\alpha }:X\times X\times(0;\infty)\rightarrow(0;1],\alpha\in\mathcal{A}\}$
satisfies
$\forall_{\alpha\in\mathcal{A}}\forall_{x,y,z\in X}\forall _{t,s\in(0;\infty)}\{M_{\alpha}(x,y,t)\ast M_{\alpha}(y,z,s)\leq M_{\alpha}(x,z,C_{\alpha}(t+s))\}$
, and ∗ is the continuous *t*-norm
$\ast:[0;1]\times{}[0;1]\rightarrow{}[0;1]$
. In
$(X,\mathcal{M}_{C;\mathcal{A}},\ast)$
, left (right)
$\mathcal{G}$
-families and
$\mathcal{W}$
-families
$\mathcal{K}_{C;\mathcal{A}}$
generated by
$\mathcal{M}_{C;\mathcal{A}}$
(
$\mathcal{K}_{C;\mathcal{A}}$
generalize
$\mathcal {M}_{C;\mathcal{A}}$
) are defined and described. Using families
$\mathcal {K}_{C;\mathcal{A}}$
, three kinds of left (right) fuzzy sets of Pompeiu-Hausdorff type on
$2^{X}\times2^{X}\times(0;\infty)$
are introduced. Using these fuzzy sets, three kinds of left (right) set-valued fuzzy contractions
$T:X\rightarrow2^{X}$
are constructed, and for such fuzzy contractions, conditions guaranteeing the existence of periodic points and left (right)
$\mathcal {M}_{C;\mathcal{A}}$
-convergence to these periodic points of dynamic processes
$(w^{m}:m\in\{0\}\cup\mathbb{N})$
,
$w^{m}\in T(w^{m-1})$
for
$m\in \mathbb{N}$
, starting at
$w^{0}\in X$
, are established. Moreover, in
$(X,\mathcal {M}_{C;\mathcal{A}},\ast)$
, using left (right)
$\mathcal{G}$
-families and
$\mathcal{W}$
-families
$\mathcal{K}_{C;\mathcal{A}}$
generated by
$\mathcal{M}_{C;\mathcal{A}}$
, two kinds of left (right) single-valued fuzzy contractions
$T:X\rightarrow X$
are constructed, and for such fuzzy contractions, the convergence, existence, approximation, uniqueness, periodic point, and fixed point result is also obtained. Examples are provided.

## Permutation Diagrams, Fixed Points and Kazhdan-Lusztig R-Polynomials

### Annals of Combinatorics (2006-12-01) 10: 369-387 , December 01, 2006

### Abstract.

In this paper, we give an algorithm for computing the Kazhdan-Lusztig *R*-polynomials in the symmetric group. The algorithm is described in terms of permutation diagrams. In particular we focus on how the computation of the polynomial is affected by certain fixed points. As a consequence of our methods, we obtain explicit formulas for the *R*-polynomials associated with some general classes of intervals, generalizing results of Brenti and Pagliacci.

## Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions

### Fixed Point Theory and Applications (2011-07-29) 2011: 1-16 , July 29, 2011

In this article, we introduce a new mapping generated by an infinite family of *κ*_{i}*-* strict pseudo-contractions and a sequence of positive real numbers. By using this mapping, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Strong convergence theorem of the purposed iteration is established in the framework of Hilbert spaces.

## Strong convergence theorems for modifying Halpern-Mann iterations for a quasi-ϕ-asymptotically nonexpansive multi-valued mapping in Banach spaces

### Fixed Point Theory and Applications (2013-05-17) 2013: 1-12 , May 17, 2013

The purpose of this paper is to introduce modifying Halpern-Mann’s iterations sequence for a quasi-*ϕ*-asymptotically nonexpansive multi-valued mapping. Under suitable limit conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results of Chang (Appl. Math. Comput. 218:6489-6497, 2012).

*MSC:*47J05, 47H09, 49J25.