This paper studies a generalized class of linear operators acting on spaces of analytic functions, defined by $P_n^{\psi ,\phi }\left(f\right)\left(z\right)={\displaystyle \sum _{j=0}^n}{\psi }_j\left(z\right)f^{\left(j\right)}\left(\phi \left(z\right)\right)$, where $\psi =\{{\psi }_0,{\psi }_1,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. This formulation encompasses several classical operators, including composition, weighted composition, differentiation–composition, and the Stević–Sharma operator. We focus on the action of $P_n^{\psi ,\phi }$ from $BMOA$ and analytic Besov spaces ${\mathcal{B}}_p$ into the Bloch space $\mathcal{B}$, and provide necessary and sufficient conditions for boundedness and compactness. These results unify and extend many previously known characterizations and demonstrate the flexibility of the $P_n^{\psi ,\phi }$ framework in the context of analytic operator theory.

We prove in this paper some existence and uniqueness results of weak solutions for some nonlinear degenerate parabolic problems arising from capillarity phenomena with the Dirichlet type boundary condition.

Let $H\left(\mathbb{D}\right)$ be the set of analytic functions on the open unit disk $\mathbb{D}$. For $n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$, define the iterated weighted-type Banach space ${\mathcal{V}}_n:=\left\{f\in H\left(\mathbb{D}\right):{sup}_{z\in \mathbb{D}}{(1-|z|}^2\left)\right|f^{\left(n\right)}\left(z\right)|<\infty \right\}.$ In this work, we study the boundedness and the essential norm of products of Volterra-type operators and composition operators on iterated weighted-type Banach spaces.

In this work, we characterize the boundedness of t-generalized composition operators from $F(p,q,s)$ spaces to iterated weighted-type Banach space. We also provide estimates of the norm and the essential norm of t-generalized composition operators from $F(p,q,s)$ spaces to iterated weighted-type Banach space. As corollaries, we obtain approximations of the essential norm of integral operators and generalized composition operators from $F(p,q,s)$ spaces to iterated weighted-type Banach space. Moreover, we conclude our work by discussing the t-generalized composition operators and the special cases of $F(p,q,s)$.

We investigate invariant finite-dimensional linear spaces for a class of nonlinear evolution equations with second-order time dependence of the type ${\partial }_{tt}u=-{\partial }_x\left(A(x,u)u\right)+{\partial }_{xx}\left(B(x,u)u\right)+f\left(u\right),$ where the coefficients may depend nonlinearly on the solution. Using direct algebraic verification and invariant subspace reductions, we derive several exact solutions and reduce the governing equations to finite-dimensional systems of ordinary differential equations. Various examples are presented to illustrate the applicability of the method to nonlinear transport, diffusion, and reaction models.