Moroccan Journal of Pure and Applied Analysis Feed

On a New Parabolic Sobolev Embedding Map

Fri, 29 Sep 2023 00:00:00 GMT

The purpose of the present article is to provide a new parabolic Sobolev embedding map between a parabolic weighted Sobolev space and the space of square-integrable functions on a cylinder. Furthermore, the embedding constant is furnished explicitly.

On best proximity point theorems for relatively nonexpansive mappings in locally convex spaces

Fri, 29 Sep 2023 00:00:00 GMT

This paper explores the concept of normal structure by introducing a generalized form known as P-proximal normal structure. Focusing on the framework of Hausdorff locally convex spaces, the study establishes optimal proximity outcomes for both cyclic and noncyclic relatively P-nonexpansive mappings. Furthermore, the paper presents in the realm of probabilistic normed spaces, considered as instances of Hausdorff locally convex spaces, some theorems that address the existence of best proximity points within this context.

On Darbo’s fixed point principle

Fri, 29 Sep 2023 00:00:00 GMT

In this paper, we prove the following generalization of the classical Darbo fixed point principle : Let X be a Banach space and µ be a montone measure of noncompactness on X which satisfies the generalized Cantor intersection property. Let C be a nonempty bounded closed convex subset of X and T : C → C be a continuous mapping such that for any countable set Ω ⊂ C, we have µ(T(Ω)) ≤ kµ(Ω), where k is a constant, 0 ≤ k < 1. Then T has at least one fixed point in C. The proof is based on a combined use of topological methods and partial ordering techniques and relies on the Schauder and the Knaster-Tarski fixed point principles.

Nonlinear elliptic boundary value problems with convection term and Hardy potential

Fri, 29 Sep 2023 00:00:00 GMT

In this paper, we deal with a nonlinear elliptic problems that incorporate a Hardy potential and a nonlinear convection term. We establish the existence and regularity of solutions under various assumptions concerning the summability of the source term f.

Solvability of Parametric Elliptic Systems with Variable Exponents

Fri, 29 Sep 2023 00:00:00 GMT

In this paper, we study the solvability to the left of the positive infimum of all eigenvalues for some non-resonant quasilinear elliptic problems involving variable exponents. We first prove the existence of at least a weak solution for some non-variational systems by using a surjectivity result for pseudomonotone operators. Furthermore, under additional conditions, we show that the solution is unique and provide examples. Second, we deal with non-resonant gradient-type systems and obtain existence by using a variational approach.

Basins of attraction of a one-parameter family of root-finding techniques

Fri, 29 Sep 2023 00:00:00 GMT

Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter A within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (A = 0), Halley’s (A = 1) and BSC (A = 2) techniques. According to the results, the one-parameter family provides the best performance for values near A = 1, which equals to the Halley’s method.

On some generalization of order cancellation law for subsets of topological vector space

Fri, 29 Sep 2023 00:00:00 GMT

In this paper we generalize a result of R. Urbański from paper [7] which states that for subsets A, B, C of topological vector space X the following implication holds A+B⊂B+C⇒A⊂C A + B \subset B + C \Rightarrow A \subset C provided that B is bounded and C is closed and convex. The generalization is given in Theorem 5 where we prove this result for k- convex subsets of a topological vector space. Also we introduce a notion of some abstract like-closure operation for subsets of linear space and we study its connections to order cancellation law.

An existence result for two-dimensional parabolic integro-differential equations involving CEV model

Fri, 29 Sep 2023 00:00:00 GMT

In this paper, we present an existence result of weak solutions for some parabolic equations involving the so-called CEV model with jumps.

p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators

Fri, 29 Sep 2023 00:00:00 GMT

We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators { M(σ)(Δp(x)2u-Δp(x)u)=λϑ(x)| u |q(x)-2u(∫Ωϑ(x)q(x)| u |q(x)dx)r in Ω,u∈W2,p(.)(Ω)∩W01,p(.)(Ω), \left\{ {\matrix{ {M\left( \sigma \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill \cr {u \in {W^{2,p\left( . \right)}}\left( \Omega \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right),} \hfill \cr } } \right. under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where σ:=∫Ω| Δu |p(x)p(x)+| ∇u |p(x)p(x)dx. \sigma : = \int_\Omega {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.} Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.

Mohamed Zarrabi 1964-2021

Wed, 07 Jun 2023 00:00:00 GMT

This note is dedicated to recalling the virtues and the important contributions in mathematics of mohamed zarabi who passed a way on mid december 2021.

Sums and products of periodic functions

Wed, 07 Jun 2023 00:00:00 GMT

There exist two real valued periodic functions on the real line such that, for every x ∈ ℝ, f1(x) + f2(x) = x, but it is impossible to find two real valued periodic functions on the real line such that, for every x ∈ ℝ, f1(x) + f2(x) = x2. The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.

Cyclicity in de Branges–Rovnyak spaces

Wed, 07 Jun 2023 00:00:00 GMT

In this paper, we study the cyclicity problem with respect to the forward shift operator Sb acting on the de Branges–Rovnyak space ℋ (b) associated to a function b in the closed unit ball of H∞ and satisfying log(1− |b| ∈ L1(𝕋). We present a characterisation of cyclic vectors for Sb when b is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of Sb in this case, but we provide here an elementary proof. We also study the situation where b has the form b = (1+ I)/2, where I is a non-constant inner function such that the associated model space KI = ℋ (I) has an orthonormal basis of reproducing kernels.

Carleson’s formula for some weighted Dirichlet spaces

Wed, 07 Jun 2023 00:00:00 GMT

We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

Wed, 07 Jun 2023 00:00:00 GMT

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice X, including weighted ℓp spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 𝔻X∞ \mathbb{D}_X^\infty . Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 𝔻X∞,|ℂN=𝔻N \mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N} and the ℓp-unit balls 𝔻X∞,|ℂN=𝔹pN \mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as X=ℓp(ℤ+N,(1+α)s) X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right)) , s = (s1, s2, … ) and X=ℓp(ℤ+N, (α!| α |!)t(1+| α |)s) X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right) , s,t ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.

The Maximum Locus of the Bloch Norm

Wed, 07 Jun 2023 00:00:00 GMT

For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.

H∞ interpolation constrained by Beurling–Sobolev norms

Wed, 07 Jun 2023 00:00:00 GMT

We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.

Negative Powers of Contractions Having a Strong AA+ Spectrum

Wed, 07 Jun 2023 00:00:00 GMT

Zarrabi proved in 1993 that if the spectrum of a contraction T on a Banach space is a countable subset of the unit circle 𝕋, and if limn→+∞log(‖ T−n ‖)n=0 {\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0 , then T is an isometry, so that ‖Tn‖ = 1 for every n ∈ ℤ. It is also known that if C is the usual triadic Cantor set then every contraction T on a Banach space such that Spec(T ) ⊂ 𝒞 satisfying lim supn→+∞log(‖ T−n ‖)nα<+∞ \lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} < + \infty for some α0 ‖T−n ‖ ≤ K, where K < + ∞ denotes the “AA+-constant” of E (closed countanble subsets of 𝕋 and the triadic Cantor set are strong AA+-sets of constant 1).

Volterra operator norms : a brief survey

Wed, 07 Jun 2023 00:00:00 GMT

In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.

Numerical comparison of three a posteriori error estimators for nonconforming finite element method

Wed, 01 Feb 2023 00:00:00 GMT

In this paper, we propose to compare three a posteriori error estimators namely equilibrated, star-based and residual based for the Poisson problem and the Stokes problem with lowest-order Crouzeix-Raviart finite element discretization. The numerical results are presented to compare the performance of the three estimators in an adaptive refinement strategy.

Existence result of the global attractor for a triply nonlinear thermistor problem

Wed, 01 Feb 2023 00:00:00 GMT

We study the existence and uniqueness of a bounded weak solution for a triply nonlinear thermistor problem in Sobolev spaces. Furthermore, we prove the existence of an absorbing set and, consequently, the universal attractor.