Sahand Communications in Mathematical Analysis
Volume:19 Issue: 2, Spring 2022

In this study, new Hermite-Hadamard type inequalities are generated for geometric-arithmetic functions with the help of an integral equation proved for differentiable functions. In proofs, some classical integral inequalities, such as H"{o}lder's inequality, basic definitions and known mathematical analysis procedures are used. The third part of the study includes various applications confirming the accuracy of the generated results. A brief conclusion of the study has been given in the last part of the paper.

In this paper, we give a definition of the $F$-Hardy-Rogers contraction of Nadler type by eliminating the conditions $(F3)$ and $(F4)$. And, we obtain some fixed point theorems for such mappings using Mann's iteration process in complete convex $b$-metric spaces. We also give an example in order to support the main results,  which generalize some results in [5,6].

Let $H(mathbb{D})$ be the space of all analytic functions on the open unit disc $mathbb{D}$ in the complex plane $mathbb{C}$. In this paper, we investigate the boundedness and compactness of the generalized integration operator$$I_{g,varphi}^{(n)}(f)(z)=int_0^z f^{(n)}(varphi(xi))g(xi) dxi,quad zinmathbb{D},$$ from Zygmund space into weighted Dirichlet type space, where $varphi$ is an analytic self-map of $mathbb{D}$, $ninmathbb{N}$ and $gin H(mathbb{D})$. Also we give an estimate for the essential norm of the above operator.

In this paper, we first show that the  induced topologies by Felbin and Bag-Samanta type fuzzy norms on a linear space $X$ are equivalent. So all results in Felbin-fuzzy normed linear spaces are valid in Bag-Samanta fuzzy normed linear spaces and vice versa. Using this, we will be able to define a fuzzy norm on $FB(X,Y)$, the space of all fuzzy bounded linear operators from $X$ into $Y$, where $X$ and $Y$ are fuzzy normed linear spaces.

In this paper, we establish some inequalities for generalized fractional integrals by utilizing the assumption that the second derivative of $phi (x)=varpi left( frac{kappa _{1}kappa _{2}}{mathcal{varkappa }}right)  $ is bounded. We also prove again a Hermite-Hadamard type inequality obtained in [34] under the condition $phi ^{prime }left( kappa_{1}+kappa _{2}-mathcal{varkappa }right) geq phi ^{prime }(mathcal{varkappa })$ instead of harmonically convexity of $varpi $. Moreover, some new inequalities for $k$-fractional integrals are given as special cases of main results.

In this corrigenda, we have pointed out that Example 2.7, Corollary 3.7 and Corollary 5.3 in the paper: $omega b-$Topological Vector Spaces, WSEAS Trans. Math. 19 (2020), $119-132$, by Latif are incorrect. We have also presented the corrected version of these results. Furthermore, we introduce and study some new classes of topological vector spaces.

In this paper, we proposed a new iterative process to approximate fixed point of generalized $alpha$-nonexpansivemappings and show that the coefficient used in the proposed iterative process play a fundamental role in the rate of convergence. We compare the speed of convergence of new iterative process  with other well-known iterative process by using numerical examples. Finally, by using new iterative process, we obtained some weak and strong convergence theorems for generalized $alpha$-nonexpansive mappings in a  Banach space.

In this study, we purpose to extend approximation properties of the $ (p,q)$-Bernstein-Chlodowsky operators from real function spaces to fuzzy function spaces. Firstly, we define fuzzy $ (p,q)$-Bernstein-Chlodowsky operators, and we give some auxiliary results. Later, we give a fuzzy Korovkin-type approximation theorem for these operators. Additionally, we investigate rate of convergence by using first order fuzzy modulus of continuity and Lipschitz-type fuzzy functions. Eventually, we give an estimate for fuzzy asymptotic expansions of the fuzzy $ (p,q)$-Bernstein-Chlodowsky operators.

In this paper, we establish further improvements  of the Young inequality and its reverse. Then, we assert operator versions corresponding them. Moreover, an application including positive linear mappings is given. For example, if $A,Bin {mathbb B}({mathscr H})$ are two invertible positive operators such that $0begin{align*}& Phi ^{2} bigg(A nabla _{nu} B+ rMm left( A^{-1}+A^{-1} sharp_{mu} B^{-1} -2 left(A^{-1} sharp_{frac{mu}{2}} B^{-1} right)right)\& qquad +left(frac{nu}{mu} right) Mm bigg(A^{-1}nabla_{mu} B^{-1} -A^{-1} sharp_{mu} B^{-1}bigg)bigg) \& quad leq left( frac{K(h)}{ Kleft( sqrt{{h^{'}}^{mu}},2 right)^{r^{'}}} right) ^{2} Phi^{2} (A sharp_{nu} B),end{align*}where $r=min{nu,1-nu}$, $K(h)=frac{(1+h)^{2}}{4h}$,  $h=frac{M}{m}$, $h^{'}=frac{M^{'}}{m^{'}}$ and $r^{'}=min{2r,1-2r}$. The results of this paper generalize the results of recent years.

Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space $K$ of systems of scalars are considered in this work. The concepts of  uncountable $K$-Bessel, $K$-Hilbert systems, $K$-frames and  $K^{*} $-Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable $K$-Besselianness, $K$-Hilbertianness for systems, $K$-frames and unconditional $K^{*} $-Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of $K$-Besselian systems which are not minimal are given. It is proved that every $K$-Hilbertian systems is minimal. The case where $K$ is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.