Operators defined on \(n\)-modular spaces (Q843999)

Let \(X\) be a real vector space of dimension \(d\geq n\in\mathbb{N}\). An \(n\)-modular \(\rho(\cdot,\dots,\cdot)\) is defined on \(X^n\), generating an \(n\)-modular space \((X,\rho)\). Notions like \(n\)-modular convergence (boundedness, compactness) are introduced and investigated. Next, compact linear operators \(T: X\to Y\) between two \(n\)-modular spaces \((X,\rho)\) and \((Y,\sigma)\) are considered. The results are applied to prove a fixed point theorem for mappings \(T: X\to X\), where \((X,\rho)\) is an \(n\)-modular space satisfying suitable conditions.