Methods of Functional Analysis
and Topology

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.

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).

Consider the nonlinear parabolic system $$\left\{\begin{array}{lll} \frac{\partial b_i(x,u_i)}{\partial t}-\mathop{\rm div}\Big(\mathcal{A}(x,t,u_i,\nabla u_i)+\Phi_i(x,t,u_i)\Big)+f_i(x,u_1,u_2)= 0 &\mbox{ in } Q_T \\ u_i=0 &\mbox{ on } \Gamma \\ b_i(x,u_i)(t=0)=b_i(x,u_{i,0})&\mbox{ in } \Omega,\end{array}\right.$$ where $ i=1,2$. In this paper we deal with the renormalized solution for the above system in Orlicz-Sobolev spaces where $f_i$ is a Carath\'{e}odory function satisfying some growth assumptions. The main term which contains the space derivatives and a non-coercive lower order term are considered in divergence form satisfying only the original Orlicz growths.