Methods of Functional Analysis
and Topology

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.

The primary objective of this paper is to present and investigate an inertial Krasnoselski-Mann (KM) type iterative method for approximating a common solution to a split monotone variational inclusion problem and a hierarchical fixed point problem for a finite family of $l$-strictly pseudocontractive non-self mappings. Additionally, we demonstrate that the iterative sequences provided by the proposed method converge weakly to a common solution to these problems. The methodology and conclusions described in this work extend and unify previously published findings in this domain. Finally, a numerical example is presented to demonstrate the suggested iterative method's convergence analysis of the sequences obtained. We also carried out a justification how the inertial term is useful.

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 establish conditions under which each positive Null almost L-weakly compact operator is Null almost M-weakly compact and conversely. Moreover, we provide the necessary and sufficient conditions under which any positive Null almost L-weakly compact operator $T: E\rightarrow F$ admits a Null almost M-weakly compact adjoint $T': F'\rightarrow E'$. Finally, we give some connections between the class of Null almost L-weakly compact (resp. Null almost M-weakly compact) operators and the class of L-weakly compact (resp. M-weakly compact).