Methods of Functional Analysis
and Topology

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.

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, 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.

This paper investigates the essential pseudospectra of closed linear operators in Banach spaces, focusing on perturbations induced by polynomially non-strictly singular operators, a class that extends the concept of polynomially strictly singular operators. New results are presented regarding the behavior of the essential pseudospectra under these perturbations. In particular, we explore the impact on the left (resp. right) Weyl and Fredholm essential pseudospectra. Additionally, we examine the essential pseudospectra of the sum of two bounded linear operators and apply the results to characterize the pseudo-Fredholm spectra of \( 2 \times 2 \) block operator matrices.