Sahand Communications in Mathematical Analysis
Volume:21 Issue: 2, Spring 2024

Integral transformations are crucial for solving a variety of actual issues. The right choice of integral transforms aids in simplifying both integral and differential problems into a solution-friendly algebraic equation. In this paper, $\mathscr{M}$-transform is applied to establish the image formula for the multiplication of a family of polynomials and incomplete $I$-functions. Additionally, we discovered image formulations for a few significant and valuable cases of incomplete $I$-functions. Numerous previously unknown and novel conclusions can be reached by assigning specific values to the parameters involved in the primary conclusions drawn in this study.

In the present paper, we introduce  the generalized inverse operators, which have an exciting role in operator theory. We establish Douglas' factorization theorem type  for  the Hilbert pro-$C^{\ast}$-module.We introduce the notion of atomic system and $K$-frame in the Hilbert pro-$C^{\ast}$-module and study their relationship. We also demonstrate some properties of the $K$-frame by using Douglas' factorization theorem.Finally  we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{\ast}$-module with certain conditions is once again a $K$-frame.

This paper is devoted to study $\mathcal{I}$-convergent,$\mathcal{I-}$null, $\mathcal{I-}$bounded and bounded sequence spaces in gradual normed linear spaces, denoted by $c_{\| \cdot \|_G} ^\mathcal{I} ,c_{0 \| \cdot \|_G} ^\mathcal{I} ,\ell_{\infty \| \cdot \|_G} ^\mathcal{I}, \ell_{\infty \| \cdot\|_G}, m_{\| \cdot \|_G} ^\mathcal{I}$ and $m_{0 \| \cdot \|_G} ^\mathcal{I}$ respectively. We discussed some algebraic and topological properties of these classes. Also, we studied some inclusions involving these spaces.

New variants of the Hermite - Hadamard inequality within the framework of generalized fractional integrals for $(h,m,s)$-convex modified second type functions have been obtained in this article. To achieve these results, we used the Holder inequality and another form of it - power means. Some of the known results described in the literature can be considered as particular cases of the results obtained in our study.

The objective of this paper is to highlight the idea of $k$-weakly and $2k$-weakly soft compatible mappings and their utilization in proving the main results. For this aim, we establish some fixed point results for the Pre\v{s}i\'{c}'s type contractive mappings in the context of soft metric spaces, when the set of the parameter is finite. Also we give an example to show that the condition of finiteness on the set of parameter can't be omitted. Some examples are given to support main findings of this article. Finally, an application of a soft version of BCP in iterated soft function systems is established.

Using subordination, we introduce a new class of symmetric functions associated with a vertical strip domain. We have provided some interesting deviations or adaptation which are helpful in unification and extension of various studies of analytic functions. Inclusion relations, geometrical interpretation, coefficient estimates, inverse function coefficient estimates and solution to the Fekete-Szeg\H{o} problem of the defined class are our main results.  Applications of our main results are given as corollaries.

In this paper, we introduce and solve a system of bi-Drygas functional equations \begin{equation}\left\{\begin{aligned}        &f(x+y,z)+f(x-y,z)=2f(x,z)+f(y,z)+f(-y,-z)\nonumber\\        &f(x,y+z)+f(x, y-z)=2f(x,y)+f(x,z)+f(-x,-z)\nonumber\end{aligned}\right.\end{equation}for all $x,y,z\in X$. We will also investigate the Hyers-Ulam stability of the system of bi-Drygas functional equations.

In this paper, we derive Saigo fractional $q$-integrals of the general class of $q$-polynomials and demonstrate their application by investigating $q$-Konhouser biorthogonal polynomial,  $q$-Jacobi polynomials and basic analogue of the Kamp$\acute{e}$ de F$\acute{e}$riet function. We have also derived polynomials as a specific example of our significant findings.

We introduce a new subclass of convex functions as follows:\[  \mathcal{K}_{IC}:=\left\{f\in \mathcal{A}:{\rm  Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>\left|f'(z)-1\right|,\quad  |z|<1\right\},\]where $\mathcal{A}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. Some properties of this particular class, including subordination relation, integral representation, the radius of convexity, rotation theorem, sharp coefficients estimate and Fekete-Szeg\"{o} inequality associated with the $k$-th root transform, are investigated.

In this paper, we introduce a relatively new class, $\mathcal{\mathcal{B}}_{1}^\beta(\alpha)$, namely the class of Beta-Bazilevi\v{c} function is generated by the function Bazilevi\v{c} $\mathcal{B}_{1}(\alpha)$. We introduce the class in question by constructing the Alpha-Convex function, $\mathcal{M}(\alpha)$, introduced by Miller et al. \cite{15}. Using Lemmas of function with positive real part, we were given a sharp estimate of coefficient problems. The coefficient problems to be solved are the modulus of initial coefficients $f$, the modulus of inverse coefficients $f^{-1}$, the modulus of the Logarithmic coefficients $\log \frac{f(z)}{z}$, the Fekete-Szeg\"{o} problem and the second Hankel determinant problem.

In this paper, we introduce the concept of generalized difference lacunary weak convergence of sequences. Using the concept of difference operator, we have introduced some new classes of sequences. We investigated several of its algebraic and topological properties, such as solidness, symmetry and monotone. We gave appropriate examples and detailed discussions to validate our established  failure instances and definitions. Further, we have established some inclusion relations of the introduced sequence spaces with other sequence spaces, in particularly with the weak Ces`aro summable sequences.

We introduce the notions of $\mathcal{T}_{M}$-amenability and $\phi$-$\mathcal{T}_{M}$-amenability. Then, we characterize $\phi$-$\mathcal{T}_{M}$-amenability  in terms of $WAP$-diagonals and $\phi$-invariant means. Some concrete cases are also discussed.

In this paper, we obtain a $\varphi$-fixed point result concerning $w$-distance. There are three illustrative examples. In a separate section, we compare of the present result with that of the corresponding results prevalent in metric spaces and indicate certain new features obtained using $w$-distance. One such feature is that under certain circumstances, the fixed point can be a point of discontinuity, which is impossible in the metric case. We give an application to non-linear integral equations. The paper ends with a conclusion.

Our present work is the extension of the line of research in the context of $\phi$-metric spaces. We introduce the notion of fixed circle and obtain suitable conditions for the existence and uniqueness of fixed circles for self mappings. Additionally, we present some figures and examples in support of our  results.

In this research work, we investigate novel findings concerning the existence and uniqueness of intuitionistic fuzzy solutions for state-dependent delay intuitionistic fuzzy partial functional differential equations with local initial conditions in a new weighted intuitionistic fuzzy complete metric space under suitable assumptions. The main results of this paper are based on the Banach fixed point theorem. An illustrated example of our results is given with some numerical simulations for $\beta$-cuts of the intuitionistic fuzzy solutions.

In this paper, we study the concept of exponential convex functions with respect to $s$ and prove Hermite-Hadamard type inequalities for the newly introduced this class of functions. In addition, we get some refinements of    the Hermite-Hadamard (H-H) inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is exponential convex with respect to $s$. Our results coincide with the results obtained previously in special cases.

The objective of this paper is to examine integral inequalities related to multiplicatively differentiable functions. Initially, we establish a novel identity using the two-point Newton-Cotes formula for multiplicatively differentiable functions. Using this identity, we derive Companion of Ostrowski's inequalities for multiplicatively differentiable convex mappings. The work also provides the results' applications.

In this article, we introduce quasi-mixing operators and construct various examples. We prove that quasi-mixing operators exist on all finite-dimensional and infinite-dimensional  Banach spaces. We also prove that an invertible operator $T$ is quasi-mixing if and only if $T^{-1}$ is quasi-mixing. We state some sufficient conditions  under which an operator is quasi-mixing. Moreover, we prove that the direct sum of two operators is quasi-mixing if and only if any of them is quasi-mixing.

Let $(G,+)$ be an abelian group and $Y$  a linear space over the field $\Bbb{F}\in\{\Bbb{R}, \Bbb{C}\}$. In this paper, we investigate the conditional Cauchy functional equation \[f(x+y)\neq af(x)+bf(y)\quad\Rightarrow \quad f(x+y)=f(x)+f(y),\quad x,y\in G,\] for functions $f:G\to Y$, where  $a,b\in \Bbb{F}$ are fixed constants. The general solution and stability of this functional equation are described.

In the present paper, we study continuous frames in Hilbert $C^*$-modules and present some results of these frames. Next, we give the concept of dual continuous frames in Hilbert $C^*$-modules and investigate some properties of them. Also, by introducing the notion of the similarity of the continuous frames, characterizing it, and stating some of its properties, we refer to the investigation of the effect of similarity on the dual continuous frames in Hilbert $C^*$-modules.