This paper introduces and studies a new class of operators on Hilbert spaces, called polynomially quasi- $m$-complex Helton operators, which extend classical Helton-type operator classes through polynomial perturbations and higher-order structural relations. Motivated by recent developments in operator theory, an operator $\mathcal{A}\in \mathcal{L}(\mathcal{K})$ is said to belong to this class if there exists a nonconstant polynomial $p$ such that $p(\mathcal{A}{)}^{\ast }\mathrm{\Theta }(\mathcal{B},\mathcal{A},C{)}^m(I)p(\mathcal{A})=0,$ where $\mathrm{\Theta }(\mathcal{B},\mathcal{A},C)$ represents the corresponding $m$-complex Helton-type operator expression. We establish fundamental properties of this class and investigate its structural behavior, including stability under direct sums, invariance under closed invariant subspaces, and stability under powers of operators. In particular, we show that, under suitable conditions, the order of the class may increase from $m$ to higher levels of the hierarchy. Several equivalent characterizations are also obtained in terms of kernels of operator polynomials, providing deeper insight into the underlying algebraic structure.

In this paper, we consider a nonlinear $n$-term fractional quadratic integral equation. Our investigation is located in the space $C(J,\mathbb{R}).$ We prove the existence and uniqueness of the solution for that problem by applying some fixed point theorems. Next, we establish the continuous dependence of the unique solution for that problem on some functions. Finally, we present some particular cases for $n$-term fractional quadratic integral equation and an example to illustrate our results.