In this work, we investigate the Ulam–Hyers stability and generalized Ulam–Hyers stability for a class of hybrid differential equations involving both proportional fractional derivatives and classical derivatives, coupled with the μ-Laplacian operator and subject to fractional integral boundary conditions. Building upon recent existence results for this class of equations, we establish sufficient conditions under which solutions exhibit Ulam–Hyers stability. The analysis relies on nonlinear analysis techniques and properties of the μ-Laplacian operator. Two illustrative examples are provided: one for classical Ulam–Hyers stability and another for uniform stability under a variable perturbation described by a power-type control function.
Fixed point theory plays a central role in functional analysis and provides a unifying framework for the analysis of iterative methods arising in differential equations, dynamical systems, and inverse problems. Motivated by recent developments in inertial and viscosity-based schemes, this paper introduces a novel inertial viscosity Z-algorithm for approximating a common fixed point of an infinite collection of nonexpansive maps in a real Hilbert space. The algorithm effectively combines the inertial term with a viscosity approximation, thereby guaranteeing strong convergence via a contraction map. Under mild and verifiable control conditions on the algorithmic parameters, strong convergence of the generated sequence is rigorously established. The framework is then extended to solve monotone inclusion problems and convex minimization problems of the form arg minx{h(x) + g(x)}, leading to a specialized forward–backward algorithm for image deblurring. To highlight the practical relevance of the theoretical findings, the proposed algorithm is applied to image deblurring problems formulated via variational and fixed point models. Numerical examples demonstrate that incorporating both inertial and viscosity effects leads to accelerated convergence and improved reconstruction quality when compared with existing methods. These results confirm that the proposed scheme provides an effective bridge between abstract fixed point theory and concrete applications in image restoration.
We compare Cuntz–Krieger and Brauer relations inside the same path algebra when every configuration vertex has valency two. Choosing one arrow from each successor pair produces a directed graph, and the Cuntz–Krieger quotient is its Leavitt path algebra. The Brauer quotient is the usual Brauer configuration algebra; it belongs to the Brauer graph class when all polygons are 2-gons. We also determine the two mixed quotients. One can be described for relaxed data and disappears under the standard Brauer axioms, while the other is always zero. In the converse direction, any finite directed graph can be recovered from a standard Brauer configuration and a cut, with isolated vertices accounting for scalar summands. The choice of cut affects more than growth: for diamond-tail and complete binary-tree configurations it changes the Grothendieck group. In the balanced cases we obtain closed spectral formulas and a determinant recurrence for the orders of the finite groups that occur.
On spaces of homogeneous type, we obtain the pointwise sparse domination principle for a class of multilinear operators. As a result, we can conclude that if T is an m-linear ω-Calderón-Zygmund operator with ω ∈ Dini, then it admits the pointwise sparse domination principle.
We study unit-speed general helices on a class of generalized cylinders in Euclidean 3-space whose base curve is planar and whose ruling direction is orthogonal to the base plane. Instead of relying only on the classical curvature–torsion characterization of general helices, we formulate the helix condition in terms of the surface parametrization and the Darboux frame. This leads to a first-order differential system which gives a local characterization of such helices and provides a constructive method for producing explicit examples. We also obtain a Darboux-data recognition criterion for deciding whether a given unit-speed curve on the cylinder is a general helix. A reduced equation is derived, and its first integral is shown to have a direct geometric interpretation in terms of the component of the helix axis along the ruling direction. As consequences, we obtain restrictions on the first-integral constant and identify the geodesic, asymptotic, and line-of-curvature cases. Several examples, including constant- and nonconstant-curvature base curves, illustrate the theory.
