We investigate the convergence of coupled Picard iterations in complete metric spaces using positively homogeneous Lyapunov functionals. We introduce a unified parameterization of admissible homogeneous invariants, namely continuous positively homogeneous functionals equivalent to the maximum norm, through continuous profile functions on the unit simplex. This framework includes many standard choices, such as the sum, maximum, weighted norms, and \(\ell_p\) norms. Building on this representation, we develop an abstract orbit-controlled convergence framework based on two complementary conditions: geometric decay of a homogeneous invariant and a projective ratio condition describing the asymptotic balance of the iteration. Under these assumptions, we prove convergence of the coupled Picard iteration to a coupled fixed point. For asymptotically linear systems, we identify a distinguished Lyapunov functional by incorporating the classical Perron–Frobenius optimization of induced absolute norms. The resulting Perron-weighted absolute \(\ell_1\) norm minimizes the induced contraction factor among absolute norms for positive linear operators, linking the homogeneous-functional framework with classical matrix analysis. A nonlinear example illustrates the applicability of the proposed approach.
Geometric optimization problems such as \(PA + k \cdot PB\) are commonly solved using reflection methods, projection techniques, or calculus-based optimization. In elementary treatments, these approaches often assume that signed algebraic expressions directly represent non-negative geometric distances across the entire real domain. When the variable extends beyond the traditional geometric boundary, this implicit assumption fails, creating what we describe as a representation legality gap. In this paper, we propose a parametric reconstruction framework that internalizes geometric admissibility constraints into the algebraic representation. Rather than partitioning the variable domain through piecewise case distinctions, we introduce an admissible auxiliary scaffold parameter \(L \ge x\) to decompose the objective function into a geometrically legal core path and an algebraically compensating residual. The auxiliary parameter \(L\) cancels identically in the reconstructed representation, ensuring global validity across \(\mathbb{R}\). We demonstrate this framework on a classical Hu Bugui-type optimization problem, establish the global minimum via dynamic projection inequalities, prove the exact invariance of the minimizer, and extend the result to an isomorphic parametric family.
We analyze both the semigroup and spectral properties of a semigroup of weighted composition operators on the weighted Dirichlet-type space of the unit disc. These composition semigroups are induced by the rotation class of the automorphisms of the upper half plane.
In this paper, we introduce and analyze a new two-step iterative scheme for approximating common fixed points of two enriched nearly asymptotically nonexpansive self-mappings defined on a closed convex subset of a uniformly convex Banach space. The proposed iteration combines averaged mappings with a carefully selected convex parameter to achieve an optimal contraction rate. We first establish that for enriched contractions, the new scheme converges with a rate factor equal to the contraction constant \(\kappa_t\), which is strictly faster than the factor \(k + \alpha_n(1-k)\) obtained for the existing Yao–-Chen iteration and for the classical Mann iteration. Under more general enriched nearly asymptotically nonexpansive conditions, we prove asymptotic regularity, provided the sequence parameters lie in a closed interval within \((0,1)\) and certain conditions hold on the nearly asymptotically nonexpansive sequences. Using this regularity, we establish the demiclosedness principle for \(I-\gamma\) at zero and, under Opial’s condition, Fréchet differentiability of the norm, or the Kadec–-Klee property of the dual space, we prove weak convergence of the iterates to a common fixed point. Furthermore, under condition \((A')\), we obtain strong convergence. A concrete example illustrates the applicability of the considered mapping class. The results extend and unify several recent findings in fixed point theory, particularly in the context of enriched and nearly asymptotically nonexpansive mappings.
In this work, we obtain new inequalities for the Laguerre-Bessel transform in the space \(L^{2}_{\alpha}(\mathbb{X}),\) by using generalized continuity modulus , where \(\mathbb{X}=[0,+\infty[\times[0,+\infty[\) and \(\alpha\geq0\).
The linear canonical Fourier transform, (LCFT), satisfies some uncertainty principales similar to classical Fourier transform. The aim of this paper is to prove a generalization of these principles for the LCFT.
