Multivalued almost contractions in metric space endowed with a graph

Author: Ishak Altun / Ozlem Acar

Abstract: The main goal of this paper is to introduce a multivalued almost contraction
on a metric space with a graph. In terms of this new contraction, we
establish some fixed point results on graph.
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On a functional equation arising in mathematical biology and theory of learning

Author: Vasile Berinde / Abdul Rahim Khan

Abstract: V. IstrÄƒÈ›escu [IstrÄƒÈ›escu, V. I., On a functional equation, J. Math. Anal. Appl., 56 (1976), No. 1, 133--136] used the Banach contraction mapping principle to establish an existence and approximation result for the solution of the functional equation
$$
\varphi(x)=x \varphi((1-\alpha)x+\alpha)+(1-x)\varphi((1-\beta)x), \,x\in [0,1],\,(0<\alpha\leq \beta <1),
$$
which is important for some mathematical models arising in biology and theory of learning.
This equation has been studied by Lyubich and Shapiro [A. P. Lyubich, Yu. I. and Shapiro, A. P., On a functional equation (Russian), Teor. Funkts., Funkts. Anal. Prilozh. 17 (1973), 81--84] and subsequently, by Dmitriev and Shapiro [Dmitriev, A. A. and Shapiro, A. P., On a certain functional equation of the theory of learning (Russian), Usp. Mat. Nauk 37 (1982), No. 4 (226), 155--156].
The main aim of this note is to solve this functional equation with more general arguments for $\varphi$ on the right hand side, by using appropriate fixed point tools.
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Convergence properties of Ibragimov-Gadjiev-Durrmeyer operators

Author: Emre Deniz / Ali Aral

Abstract: The purpose of the present paper is to study the local and global direct
approximation properties of the Durrmeyer type generalization of Ibragimov
Gadjiev operators defined in [Aral, A. and Acar, T., On Approximation Properties of Generalized Durrmeyer Operators, (submitted)]. The results obtained in
this study consist of Korovkin type theorem which enables us to approximate
a function uniformly by new Durrmeyer operators, and estimate for
approximation error of the operators in terms of weighted modulus of
continuity. These results are obtained for the functions which belong to
weighted space with polynomial weighted norm by new operators which act on
functions defined on the non compact interval $[0.\infty )$. We finally
present a direct approximation result.
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A survey on the stability of mean value points and functional equations involving some special functions

Author: Sorinel Dumitrescu / Mihai Moneaand Cristinel Mortici

Abstract: The aim of this work is to put together some of the
recent and classical results in the theory of stability. In the first part,
we recall the results regarding the intermediary points arising from various
Mean Value Theorems, then we study the stability of some functional
equations involving the gamma and beta functions.
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Abstract: A pair $(x,y)$ of elements $x$ and $y$ of a semigroup $S$ is said
to be a commuting regular pair, if there exists an element $z\in
S$ such that $xy=(yx)z(yx)$. In a finite semigroup $S$, the
probability that the pair $(x,y)$ of elements of $S$ is commuting
regular will be denoted by $dcr(S)$ and will be called the
Commuting Regularity degree of $S$. Obviously if $S$ is a group,
then $dcr(S)=1$. However for a semigroup $S$, getting an upper
bound for $dcr(S)$ will be of interest to study and to identify
the different types of non-commutative semigroups. In this paper,
we calculate this probability for certain classes of finite
semigroups. In this study we managed to present an interesting
class of semigroups where the probability is $\frac{1}{2}$. This
helps us to estimate a condition on non-commutative semigroups
such that the commuting regularity of $(x,y)$ yields the commuting
regularity of $(y,x)$.
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Abstract: Let $S$ be a set of size $n\geq 8$ and $V$ be the set of all subsets of $S$
of size $3$. Three types of intersection graphs $G_{i}(n),i=0,1,2$, can be
defined with the vertex set $V$ whose Wiener indices will be calculated in
this paper.
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Abstract: We introduce and explore weak separation axioms namely
$\gamma$-semi-$T_{i}$( for $i = 0, 1, 2$) spaces. We also define
and discuss $\gamma$-semi-$D_{i}$( for $i = 0, 1, 2$) spaces and
develop the relations between these spaces. Moreover, we initiate
the concept of $\gamma$-S-continuous function and discuss the
behavior of $\gamma$-semi-$D_{1}$ space under
$\gamma$-S-continuous function.
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Author: Adrian D. Indrea / Anamaria Indrea / Petru I. Braica

Abstract: The aim of this paper is to introduce a class of operators of Schurer-Stancu-type with the property that the test functions $e_0$ and $e_1$ are reproduced. Also, in our approach, a theorem of error approximation and a Voronovskaja-type theorem for this operators are obtained. Finally, we study the convergence of the iterates for our new class of operators.
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Coefficient inequality for transforms of starlike and convex functions

Author: D. Vamshee Krishna / B. Venkateswarlu / T. RamReddy

Abstract: The objective of this paper is to obtain an upper bound for the second Hankel functional associated with the $k^{th}$ root transform $\left[ f(z ^k ) \right] ^{\frac{1}{k}}$ of normalized analytic function $f(z)$ belonging to starlike and convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.
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Common fixed point and best approximation results for subcompatible mappings in hyperbolic ordered metric spaces

Author: Savita Rathee / Reetu

Abstract: In the present paper we establish a common fixed point theorem and apply it to find new best approximation results for ordered subcompatible mappings in the hyperbolic ordered metric space. Our results unify, generalize and complement various known results.
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Abstract: In this paper, the notions of locally $\mu$-regular closed
sets, $\widehat{\mu}$-$t$-sets, $\widehat{\mu}$-$B$-sets have been
introduced. Using these concepts, the decomposition of some weak
forms of continuity have been studied.
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Author: M. R. Ebrahimi Vishki / K. Mirzavaziri / M. Mirzavaziri

Abstract: In this paper we extend the notion of the binomial coefficient $n\choose k$ into a new notion $[G]\choose k$, where $[G]$ is an unlabelled graph with $n$ vertices and $0\leqslant k\leqslant n$. We call $[G]\choose k$ as the graph binomial coefficient and a version of the graph binomial expansion is also studied.
As an application of this notion, we enumerate the number of ways to color vertices of a path and beads of a necklace.
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A Banach algebra which is generated by idempotents

Author: A. Zivari-Kazempour

Abstract: In this paper we show that the Banach algebra $C_0(X)$, where $ X$ is a locally compact Hausdorff space, is generated by idempotents if and only if $X$ is totally disconnected.
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