In this paper, we present some properties of KKM maps and then use them to obtain a variety of collectively coincidence results for multivalued maps.

Let B be a Banach algebra, and let Ccb(B) denote the set of all closed convex bounded subsets of B. Assume that ⪰ is a partial order defined on Ccb(B), and define D := {A ∈ Ccb(B) : A ≻ 0}, where 0 denotes the zero element of Ccb(B). Furthermore, suppose that for every A,B ∈ D, the set A ⊗ B also belongs to D, where A ⊗ B means the closure of the product set AB. In this paper, general solutions F : D → D of the multiplicative set-valued functional equation F(X ⊗ Y ) = F(X) ⊗ F(Y ) for all X, Y ∈ D are determined. These solutions are closely involved with some set-valued mappings. Moreover, its stability is also proved on Banach algebras. The results not only generalize classical findings in functional equations but also open avenues for further exploration in nonlinear analysis and set-valued operator theory.

The classical Markov inequality asserts that the $n$-th Chebyshev polynomial $T_n(x)=\cos n\arccos x$, $x\in [-1,1]$, has the largest $C[-1,1]$-norm of its derivatives within the set of algebraic polynomials of degree at most $n$ whose absolute value in $[-1,1]$ does not exceed one. In 1941 R.J. Duffin and A.C. Schaeffer found a remarkable refinement of Markov inequality, showing that this extremal property of $T_n$ persists in the wider class of polynomials whose modulus is bounded by one at the extreme points of $T_n$ in $[-1,1]$. Their result gives rise to the definition of DS-type inequalities, which are comparison-type theorems of the following nature: inequalities between the absolute values of two polynomials of degree not exceeding $n$ on a given set of $n+1$ points in $[-1,1]$ induce inequalities between the $C[-1,1]$-norms of their derivatives. Here we apply the approach from a 1992 paper of A. Shadrin to prove some DS-type inequalities where Jacobi polynomials are extremal. In particular, we obtain an extension of the result of Duffin and Schaeffer.

We apply suitable maximum principles to obtain nonexistence and rigidity results for complete mean curvature flow solitons in certain warped product spaces. We also provide applications to self-shrinkers in Euclidean space, as well as to mean curvature flow solitons in real projective, pseudo-hyperbolic, Schwarzschild, and Reissner-Nordstr\"{o}m spaces. Furthermore, we establish new Moser-Bernstein type results for entire graphs constructed over the fiber of the ambient space that are mean curvature flow solitons.

One of the pillars of mathematical analysis is the Hardy-Hilbert integral inequality. In this article, we advance the theory by introducing several new modifications of this inequality. They have the property of incorporating an adjustable parameter and different power functions, allowing for greater flexibility and broader applicability. Notably, one modification has a logarithmic structure, offering a distinctive extension to the classical framework. For the main results, the optimality of the corresponding constant factors is shown. Additional integral inequalities of various forms and scopes are also established. Thus, this work contributes to the ongoing development of Hardy-Hilbert-type inequalities by presenting new generalizations and providing rigorous mathematical justification for each result.

In this article, we construct two Halpern-type relaxed algorithms with alternated and multi-step inertial extrapolation steps for split feasibility problems in infinite-dimensional Hilbert spaces. The first is the most general inertial method that employs three inertial steps in a single algorithm, one of which is an alternated inertial step, while the others are multi-step inertial steps, representing the recent improvements over the classical inertial step. Besides the inertial steps, the second algorithm uses a three-term conjugate gradient-like direction, which accelerates the sequence of iterates toward a solution of the problem. In proving the convergence of the second algorithm, we dispense with some of the restrictive assumptions in some conjugate gradient-like methods. Both algorithms employ a self-adaptive and monotonic step-length criterion that does not require knowledge of the norm of the underlying operator or the use of any line search procedure. Moreover, we formulate and prove some strong convergence theorems for each of the algorithms based on the convergence theorem of an alternated inertial Halpern-type relaxed algorithm with perturbations in real Hilbert spaces. Further, we analyse their applications to classification problems for some real-world datasets based on the extreme learning machine (ELM) with the $\ell_{1}$-regularization approach (that is, the Lasso model) and the $\ell_{1}-\ell_{2}$ hybrid regularization approach. Furthermore, we investigate their performance in solving a constrained minimization problem in infinite-dimensional Hilbert spaces. Finally, the numerical results of all experiments show that our proposed methods are robust, computationally efficient and achieve better generalization performance and stability than some existing algorithms in the literature.

We study the injectivity of normed ideals of weighted holomorphic mappings. To be more precise, the concept of injective hull of normed weighted holomorphic ideals is introduced and characterized in terms of a domination property. The injective hulls of those ideals -- generated by the procedures of composition and dual -- are described and these descriptions are applied to some examples of such ideals. A characterization of the closed injective hull of an operator ideal in terms of an Ehrling-type inequality -- due to Jarchow and Pelczy\'nski-- is established for weighted holomorphic mappings.

This paper extends the mapping properties of the general Hardy-type operators to local Morrey spaces built on ball quasi-Banach function spaces. As applications of the main result, we establish the two weight norm inequalities of the Hardy operators to the local Morrey spaces, the mapping properties of the Riemann-Liouville integrals on local Morrey spaces built on rearrangement-invariant quasi-Banach function spaces, the Hardy inequalities on the local Morrey spaces with variable exponents.

The main aim of this article is to propose a multidimensional quadratic-phase Fourier transform (MQFT) that generalises the well-known and recently introduced quadratic-phase Fourier transform (as well as, of course, the Fourier transform itself) to higher dimensions. In addition to the definition itself, some crucial properties of this new integral transform will be deduced. These include a Riemann-Lebesgue lemma for the MQFT, a Plancherel lemma for the MQFT and a Hausdorff-Young inequality for the MQFT. A second central objective consists of obtaining different uncertainty principles for this MQFT. To this end, using techniques that include obtaining various auxiliary inequalities, the study culminates in the deduction of $L^p$-type Heisenberg-Pauli-Weyl uncertainty principles and $L^p$-type Donoho-Stark uncertainty principles for the MQFT.

In the present paper, is proposed a method to approximate the Hilbert transform of a given function $f$ on $(0,\infty)$ employing truncated de la Vallée discrete polynomials recently studied in [25]. The method generalizes and improves in some sense a method based on truncated Lagrange interpolating polynomials introduced in [24], since is faster convergent and simpler to apply. Moreover, the additional parameter defining de la Vallée polynomials helps to attain better pointwise approximations. Stability and convergence are studied in weighted uniform spaces and some numerical tests are provided to asses the performance of the procedure.