Open Journal of Mathematical Analysis (OMA)

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.

In this note, we prove sharp necessary conditions for the boundedness of the manifold-adapted Wolff potential between Zygmund spaces on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. The nonlinear homogeneity of the operator determines the natural form of the norm inequality, while the Bishop comparison theorem and localized ball tests determine the admissible Sobolev scaling. The boundedness assumption forces Euclidean lower volume growth, identifies the relation between the source and target integrability exponents, and gives the critical logarithmic constraint for the Zygmund indices. In the linear case, the conclusions reduce to the corresponding Riesz-potential scaling with the expected logarithmic refinement.

We construct a class of digit-defined Cantor sets \(C_{(d_k)}\subset[0,1]\) from a prescribed sequence \((d_k)_{k\ge1}\) by using the digit \(d_k\) to determine the removal ratio at level \(k\). When only finitely many digits are zero, the construction has a uniform separation property after a finite initial stage. This yields compact, perfect, totally disconnected, nowhere dense sets of Lebesgue measure zero. The natural probability measure \(\mu\) is obtained by assigning equal mass to the two children of each cylinder; equivalently, \(\mu\) is the pushforward of the fair Bernoulli measure on \(\{0,1\}^{\mathbb{N}}\) under the coding map. If
\[
\ell_k=\prod_{j=1}^{k}a_j,\qquad a_k=\frac12\Bigl(1-\frac{d_k}{10}\Bigr),
\]
then
\[
\dim_H C_{(d_k)}=\liminf_{k\to\infty}\frac{\log 2}{-\frac1k\sum\limits_{j=1}^{k}\log a_j}.
\]
The associated Cantor function \(F(x)=\mu([0,x])\) is Hölder continuous for every exponent below \(\dim_H C_{(d_k)}\), and no exponent above this dimension is possible. Cartesian products are also treated: under Ahlfors regularity of the factors, Hausdorff dimensions add and the product Cantor function has optimal Hölder threshold equal to the smallest one-dimensional dimension. A sufficient and verifiable condition for Ahlfors regularity is the bounded comparability \(2^{-k}\asymp \ell_k^{\alpha}\), where \(\alpha\) is the limiting similarity dimension. The results identify how the asymptotic distribution of decimal digits controls dimension, singular measure regularity, and product geometry.

We establish necessary and sufficient Boas-type criteria for generalized Lipschitz classes associated with the generalized Dunkl transform on the real line. The criteria are expressed through the growth of weighted spectral moments of \(\mathcal{F}_{D}(f)\) as the spectral radius tends to infinity. A symmetric generalized difference generated by the dual translation operators is used; its transform multiplier is \((1-j_{\alpha}(th))^{n}\), which gives the correct order \(2n\) at the origin and the required lower control away from the origin. Direct theorems are proved for the classes \(B_{\gamma}^{n}\) and \(b_{\gamma}^{n}\), while converse theorems are obtained under the standard one-sign condition on the transform. These results identify the precise connection between decay of the generalized Dunkl transform and \(Q\)-weighted smoothness measured by generalized translations.

We introduce the concept of a generalized sequential Yeh-Feynman integral for functionals defined on Yeh-Wiener space, formulated via stochastic process \(Z_h\) associated with a nonzero function \(h\). Existence theorems and evaluation formulas for generalized sequential Yeh-Feynman integral are established for functionals in the Banach algebra \(\hat{\mathcal S}(L_2(Q))\) and some related functionals. Furthermore, we show that the class of generalized sequential Yeh-Feynman integrable functionals is strictly larger than \(\hat{\mathcal S}(L_2(Q))\). Previous results on sequential Yeh-Feynman integral are recovered as corollaries of our results.

This study investigates a generalized fractional Hill differential equation subject to two-point separated homogeneous boundary conditions. First, we formulate the Green’s function and detail its fundamental characteristics. Subsequently, we derive a Lyapunov-type inequality for this specific boundary value problem. The article illustrates that these results encompass several established findings in existing literature as special cases. Finally, the work highlights potential avenues for future study through a series of open problems.

In this article, we present a new Hilbert-type integral inequality involving variable weight functions and an adjustable parameter. It can be described as a generalization of the Mingzhe integral inequality. Some other integral inequalities are also derived. These results provide a flexible framework for obtaining valuable bounds and facilitating further analytical applications.