This article investigates two subclasses of p-valent analytic functions formulated within the framework of q-calculus: the q-spirallike class S^η ,q_p(ψ ) and its convex counterpart C^η ,q_p(ψ ) . Necessary and sufficient convolution (Hadamard product) characterisations are obtained for both subclasses. Building on these, we establish a collection of additional convolution-theoretic results, comprising coefficient-wise criteria for membership, stability of a distinguished subclass under Hadamard multiplication, monotone inclusion relations between parameter families, and pointwise estimates on Taylor coefficients. The q-Extension of the Dziok–Srivastava Operator ℌ^(q)_p; r,s(α _1) , constructed via the q-generalised hypergeometric function (in which r and s count the numerator and denominator parameters, respectively), plays a central role throughout. Every result presented here degenerates to its classical counterpart in the limiting case q→ 1^- .
Uniform exponential dichotomy is a fundamental notion in the qualitative theory of nonautonomous differential equations and evolution families. In this paper, we establish a new characterization of uniform exponential dichotomy for linear periodic nonautonomous systems in terms of bounded operator-valued oscillatory integral operators. More precisely, we show that the associated evolution family admits a uniform exponential dichotomy if and only if two families of forward and backward oscillatory integrals remain uniformly bounded with respect to the oscillation frequency. The analysis combines Floquet theory with a reduction to the monodromy operator and discrete spectral techniques, thereby linking continuous-time dynamics to the spectral properties of the corresponding discrete evolution. The proposed characterization provides an alternative to classical spectral and Floquet criteria and is well suited to operator-theoretic and perturbation methods. The theoretical results are complemented by illustrative examples and counterexamples demonstrating the sharpness of the assumptions. As applications, we derive robustness results under small periodic perturbations and establish bounded solvability criteria for periodically forced linear systems.
In this paper, we develop a unified framework for studying the spectral stability of operator pencils whose entries satisfy relative Fredholm conditions. In particular, we derive new inclusion relations and stability criteria for the M -essential spectra of the sum of two unbounded 2× 2 blocks operator matrix pencils.
In this work, we introduce a generalized class of fractional Dirac operators constructed through the Whittaker transform of the Mittag–Leffler function using the generalized Fox–Wright function. The framework extends the theory of Dirac operators to the fractional setting on Riemannian manifolds via Clifford algebras, spin representations, and the geometry of fiber bundles. Fundamental properties of these operators are established, showing that they combine conventional and fractional derivatives, thereby generalizing key results in Riemannian geometry. In particular, we extend the Schrödinger–Lichnerowicz formula to fractional order, revealing new scalar curvature terms governed by different fractional derivatives. This approach bridges fractional calculus and geometric analysis, offering potential applications in field theory and the study of manifolds with fractal dimensions. Furthermore, we analyze the fractional Dirac operator on Lorentzian cylinders under Atiyah–Patodi–Singer boundary conditions, deriving corresponding constraints on the fractional exponents. The formulation naturally leads to a fractional Schrödinger–Dirac equation encompassing higher-order derivative terms. Our findings open prospects for extending the fractional formalism to spectral geometry and noncommutative spaces, particularly through the development of a fractional spectral action principle, where the geometric action is constructed from the spectrum of a fractional Dirac operator, and the investigation of index theorems and algebraic geometric structures associated with fractional Dirac operators, which generalize the classical Dirac operator by incorporating fractional-order nonlocal effects.
This paper studies the left-definite theory of dynamic Sturm–Liouville problems in the regular case.
