Sahand Communications in Mathematical Analysis
Volume:18 Issue: 1, Winter 2021

In this paper, we establish a sequential characterisation of Lebesgue fuzzy metric and explore the relationship between Lebesgue, weak $G$-complete and compact fuzzy metric spaces. We also discuss the Lebesgue property of several well-known fuzzy metric spaces.

In this manuscript, we study the relation between K-fusion frame and its local components which leads to the definition of a $C$-controlled $K$-fusion frames, also we extend a theory based on K-fusion frames on Hilbert spaces, which prepares exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we define the analysis, synthesis and frame operator for $C$-controlled $K$-fusion frames, which even yield a reconstruction formula. Also, we define dual of $C$-controlled $K$-fusion frames and study some basic properties and perturbation of them.

In this paper,  we have improved some of the results in [C. Choi and   B. Lee, Stability of Mixed Additive-Quadratic and Additive--Drygas Functional Equations. Results Math.  75  no. 1 (2020), Paper No. 38]. Indeed, we investigate the Hyers-Ulam stability problem of the following   functional equationsbegin{align*}  2varphi(x + y) + varphi(x - y) &= 3varphi(x)+ 3varphi(y)  \  2psi(x + y) + psi(x - y) &= 3psi(x) + 2psi(y) + psi(-y).end{align*}We also consider the Pexider type functional equation [2psi(x + y) + psi(x - y) = f(x) + g(y),] and the additive functional equation[2psi(x + y) + psi(x - y) = 3psi(x) + psi(y).]

The duals of Gabor frames have an essential role in reconstruction of signals. In this paper we find a necessary and sufficient  condition for two Gabor systems $left(chi_{left[c_1,d_1right)},a,bright)$ and $left(chi_{left[c_2,d_2right)},a,bright)$ to form dual frames for $L_2left(mathbb{R}right)$, where $a$ and $b$ are positive numbers and $c_1,c_2,d_1$ and $d_2$ are real numbers such that $c_1<d_1$ and $c_2<d_2$.

Let $K$ be a bounded operator. $K$-frames are ordinary frames for the range $K$. These frames are a generalization of ordinary frames and are certainly different from these frames. This research introduces a new concept of bases for the range $K$. Here we define the  $K$-orthonormal basis and the $K$-Riesz basis, and then we describe their properties. As might be expected, the $K$-bases differ from the ordinary ones mentioned in this article.

In this paper, we establish some Trapezoid and Midpoint type inequalities for generalized fractional integrals by utilizing the functions whose second derivatives are bounded . We also give some new inequalities for $k$-Riemann-Liouville fractional integrals as special cases of our main results. We also obtain some Hermite-Hadamard type inequalities by using the condition $f^{prime }(a+b-x)geq f^{prime }(x)$ for all $xin left[ a,frac{a+b}{2}right] $ instead of convexity.

In this work, we discuss bi-Hamiltonian structures on a family of integrable systems on 4-dimensional real Lie groups. By constructing the corresponding control matrix  for this family of bi-Hamiltonian structures, we obtain an explicit process for finding  the variables of separation and the separated relations in detail.

In this paper, we generalize the concepts of weakly Kannan, weakly Chatterjea and weakly Zamfirescu for fuzzy metric spaces. Also, we investigate Banach's fixed point theorem for the mentioned classes of functions in these spaces. Moreover, we show that the class of weakly Kannan and weakly Chatterjea maps are subclasses of the class of weakly Zamfirescu maps.

The aim of this paper is to prove some coupled fixed point  theorems of  a self mapping satisfying a certain rational type contraction along with  strict mixed monotone property in an ordered metric space. Further, a result  is presented for the uniqueness of a coupled fixed point under an order relation in a space. These results generalize and extend known existing results in the literature.