Methods of Functional Analysis
and Topology

In this paper, we introduce new metric characteristics in the space of summable functions. Using these metric characteristics it is obtained Zigmund-type inequalities for the bisingular integral. It is constructed an invariant $T_p$ space for bisingular integral operator according to the inequality. Furthermore, the existence and uniqueness of the solution to the nonlinear bisingular integral equation within the invariant space $T_p$ are proven using the method of successive approximations.

In this paper, we introduce a new class of contractive mappings, called generalized $\psi$-Geraghty contractions, in the framework of b-complete metric spaces. We establish a unique fixed-point theorem that extends existing results in fixed-point theory. An illustrative example with a graphical representation demonstrates the validity of our findings. Furthermore, we apply the main result to an integral equation, highlighting its effectiveness in ensuring the existence and uniqueness of solutions. This work underscores the theoretical significance and practical applicability of generalized $\psi$-Geraghty contractions in mathematics, physics, and engineering.

In this paper, the inequality of Riemann-Stieltjes integral is defined for functions of Hilbert space. The concept of time scales is introduced to unify both discrete and continuous problems. Also, the definition and properties of Riemann-Stieltjes integral are used in the application of self-adjoint and unitary operators in Hilbert spaces. Thees results are obtained on time scales.

In this paper, we introduce the notion of $m$-quasi-$n$-power-totally-$(\alpha,\beta)$-normal operators on a Hilbert space $\mathscr{H}$ as : An operator $\mathcal{L}$ is called $m$-quasi-$n$-power-totally-$(\alpha,\beta)$-normal $(0\leq \alpha \leq 1 \leq \beta)$ if \begin{align*} \alpha^{2}\mathcal{L}^{m*}(\mathcal{L}-\lambda)^{*}(\mathcal{L}-\lambda )^{n}\mathcal{L}^{m}& \leq \mathcal{L}^{m*}(\mathcal{L}-\lambda)^{n}(\mathcal{L}-\lambda)^{*}\mathcal{L}^{m}\\ &\leq \beta^{2} \mathcal{L}^{m*}(\mathcal{L}-\lambda)^{*}(\mathcal{L}-\lambda )^{n}\mathcal{L}^{m} \end{align*} for natural numbers $m$ and $n$ and for all $\lambda \in \mathbb{C}$. This paper aims to study several properties of $m$-quasi-$n$-power-totally-$(\alpha,\beta)$-normal operators.