Communications in Mathematical Analysis Articles (Project Euclid)
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The latest articles from Communications in Mathematical Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTFri, 25 Feb 2011 16:39 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On The Uniqueness Of Meromorphic Functions Sharing Two Sets
http://projecteuclid.org/euclid.cma/1275586728
<strong>Abhijit Banerjee</strong><p><strong>Source: </strong>Commun. Math. Anal., Volume 9, Number 2, 1--11.</p><p><strong>Abstract:</strong><br/>
In the paper we employ the notion of weighted sharing of sets to deal with the
well known question of Gross and obtain a uniqueness result on meromorphic
functions sharing two sets which will improve an earlier result of Lahiri [14]
.
</p>projecteuclid.org/euclid.cma/1275586728_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTSingular and Fractional Integral Operators with Variable Kernels on the Weak
Hardy Spaceshttp://projecteuclid.org/euclid.cma/1439384427<strong>Hua Wang</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 1, 48--63.</p><p><strong>Abstract:</strong><br/>
In this paper, by using the decomposition theorem for weak Hardy spaces, we will
obtain the boundedness properties of some integral operators with variable
kernels on these spaces, under some Dini type conditions imposed on the variable
kernel $\Omega(x,z)$.
</p>projecteuclid.org/euclid.cma/1439384427_20150812090030Wed, 12 Aug 2015 09:00 EDTExistence of Multiple Limit Cycles in a Predator-Prey Model with $\arctan(ax)$ as
Functional Responsehttp://projecteuclid.org/euclid.cma/1439384428<strong>Gunog Seo</strong>, <strong>Gail S. K. Wolkowicz</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 1, 64--68.</p><p><strong>Abstract:</strong><br/>
We consider a Gause type predator-prey system with functional response given by
$θ(x)=\arctan(ax), where $a \gt 0$, and give a counterexample to the criterion
given in Attili and Mallak [Commun. Math. Anal. 1:33-40(2006)] for the
nonexistence of limit cycles. When this criterion is satisfied, instead this
system can have a locally asymptotically stable coexistence equilibrium
surrounded by at least two limit cycles.
</p>projecteuclid.org/euclid.cma/1439384428_20150812090030Wed, 12 Aug 2015 09:00 EDT1D-Solitons for a Generalized Dispersive Equationhttp://projecteuclid.org/euclid.cma/1439384429<strong>Alex M. Montes</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 1, 69--82.</p><p><strong>Abstract:</strong><br/>
In this paper we show the existence of one-dimensional solitons (travelling waves
of finite energy) for a generalized nonlinear dispersive equation modeling the
deformations of a hyperelastic compressible plate. From the Hamiltonian
structure for such equation we find the natural space for the travelling wave
solutions and characterize travelling waves variationally as minimizers of an
energy functional under a suitable constraint. Our approach involves the Lions's
Concentration-Compactness Lemma.
</p>projecteuclid.org/euclid.cma/1439384429_20150812090030Wed, 12 Aug 2015 09:00 EDT$L^p$ Quantitative Uncertainty Principles for the Generalized Fourier Transform Associated with the Spherical Mean Operatorhttp://projecteuclid.org/euclid.cma/1439384430<strong>Hatem Mejjaoli</strong>, <strong>Youssef Othmani</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 1, 83--99.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to prove new quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator. The first of these results is an extension of the Donoho and Stark's uncertainty principle. The second result extends the Heisenberg-Pauli-Weyl uncertainty principle. From these two results we deduce a continuous-time principle for the $L^p$ theory, when $1 \lt p \le 2$.
</p>projecteuclid.org/euclid.cma/1439384430_20150812090030Wed, 12 Aug 2015 09:00 EDTAnisotropic Herz Spaces with Variable Exponentshttp://projecteuclid.org/euclid.cma/1439384456<strong>Hongbin Wang</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 1--14.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the anisotropic Herz spaces with two variable
exponents and establish their block decomposition. Using this decomposition, we
obtain some boundedness on the anisotropic Herz spaces with two variable
exponents for a class of sublinear operators.
</p>projecteuclid.org/euclid.cma/1439384456_20150812090058Wed, 12 Aug 2015 09:00 EDTDegenerate Abstract Parabolic Equations and Applicationshttp://projecteuclid.org/euclid.cma/1446210171<strong>Veli.B. Shakhmurov</strong>, <strong>Aida Sahmurova</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 15--33.</p><p><strong>Abstract:</strong><br/>
Linear and nonlinear degenerate abstract parabolic equations with variable
coefficients are studied. Here the equations and boundary conditions are
degenerated on all boundary and contain some parameters. The linear problem is
considered on the moving domain. The separability properties of elliptic and
parabolic problems and Strichartz type estimates in mixed $L_{\mathbf{p}} $
spaces are obtained. Moreover, the existence and uniqueness of optimal regular
solution of mixed problem for nonlinear parabolic equation is established. Note
that, these problems arise in fluid mechanics and environmental engineering.
</p>projecteuclid.org/euclid.cma/1446210171_20151030090254Fri, 30 Oct 2015 09:02 EDTA Note on the Inhomogeneous Schrödinger Equation with Mixed Power
Nonlinearityhttp://projecteuclid.org/euclid.cma/1446210172<strong>H. Hezzi</strong>, <strong>A. Marzouk</strong>, <strong>T. Saanouni</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 34--53.</p><p><strong>Abstract:</strong><br/>
We investigate the initial value problem for an inhomogeneous nonlinear
Schrödinger equation with a combined power nonlinearity. We prove global
well-posedness in the defocusing case. In the focusing case, we prove existence
of ground state and nonlinear instability of standing waves.
</p>projecteuclid.org/euclid.cma/1446210172_20151030090254Fri, 30 Oct 2015 09:02 EDTNew Developments on Nirenberg's Problem for Compact Perturbations of
Quasimonotone Expansive Mappings in Reflexive Banach Spaceshttp://projecteuclid.org/euclid.cma/1446210173<strong>Teffera M. Asfaw</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 54--75.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a real locally uniformly convex reflexive Banach space with locally
uniformly convex dual space $X^*$. Let $T:X\to X^*$ be demicontinuous,
quasimonotone and $\alpha$-expansive, and $C: X\to X^*$ be compact such that
either (i) $\langle Tx+Cx, x\rangle \geq -d\|x\|$ for all $x\in X$ or (ii)
$\langle Tx+Cx, x\rangle \geq-d\|x\|^2$ for all $x\in X$ and some suitable
positive constants $\alpha$ and $d.$ New surjectivity results are given for the
operator $T+C.$ The results are new even for $C=\{0\}$, which gives a partial
positive answer for Nirenberg's problem for demicontinuous, quasimonotone and
$\alpha$-expansive mapping. Existence result on the surjectivity of
quasimonotone perturbations of multivalued maximal monotone operator is
included. The theory is applied to prove existence of generalized solution in
$H^{1}_{0}(\Omega)$ of nonlinear elliptic equation of the type \begin{align*}
\begin{split} \left\{\begin{array}{cc}
-\sum\limits_{i=1}^{N}{\frac{\partial}{\partial x_i} a_i(x, u(x), \nabla
u(x))})+G_{\lambda}(x, u(x))=f(x) &\textrm{in $\Omega$}\\
u(x)=0&\textrm{$x\in\partial \Omega$},\\ \end{array}\right. \end{split}
\end{align*} where $f\in L^{2}(\Omega)$, $\Omega$ is a nonempty, bounded and
open subset of $\mathbb{R}^{N}$ with smooth boundary, $\lambda>0$, $
G_{\lambda}(x, u)=-div (\beta (\nabla u(x)))+\lambda u(x)+a_0(x, u(x), \nabla
u(x))+g(x, u(x))$, $\beta: \mathbb{R}^{N}\to\mathbb{R}^{N}$, $a_i: \Omega\times
\mathbb{R}\times \mathbb{R}^{N}\to\mathbb{R}$ ($i=0, 1, 2, ..., N$) and
$g:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}$ satisfy certain
conditions.
</p>projecteuclid.org/euclid.cma/1446210173_20151030090254Fri, 30 Oct 2015 09:02 EDTOn the Maximality of Certain Hyperellptic Curves with an Application to Character
Sums Peter McCalla and Francois Ramarosonhttp://projecteuclid.org/euclid.cma/1446210174<strong>Peter McCalla</strong>, <strong>Francois Ramaroson</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 76--85.</p><p><strong>Abstract:</strong><br/>
In [4], Kodama, Top, Washio studied the maximality of a family of elliptic
curves, mostly of genus 3, over a finite field. They used the Jacobians of the
curves and differential forms to obtain their results. In this note, in order to
prove the maximality of the curves under study, we use analytical tools, namely
character and Jacobsthal sums, together with an important result which says that
if a curve is the image of a maximal curve under a rational map, then it is
itself maximal. Character sums are suitable for counting the number of points on
a curve over a finite field, and their use makes the proofs natural and rather
elementary. The norm and trace curves are utilized to construct rational maps to
the hyperelliptic curves. As an application, the maximality of a certain
hyperelliptic curve is used to find the explicit value of a character sum.
</p>projecteuclid.org/euclid.cma/1446210174_20151030090254Fri, 30 Oct 2015 09:02 EDTExtremal Viscosity Solutions of Almost Periodic Hamilton-Jacobi Equationshttp://projecteuclid.org/euclid.cma/1446210175<strong>Marino Belloni</strong>, <strong>Silvana Marchi</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 86--105.</p><p><strong>Abstract:</strong><br/>
This paper deals with viscosity solutions of Hamilton-Jacobi equations in which
the Hamiltonian $H$ is weakly monotone with respect to the zero order term: this
leads to non-uniqueness of solutions, even in the class of periodic or
almost periodic (briefly a.p.) functions. The lack of uniqueness of
a.p. solutions leads to introduce the notion of minimal (maximal) a.p. solution
and to study its properties. The classes of asymptotically almost
periodic (briefly a.a.p.) and pseudo almost periodic
(briefly p.a.p.) functions are also considered.
</p>projecteuclid.org/euclid.cma/1446210175_20151030090254Fri, 30 Oct 2015 09:02 EDTThe Egoroff Theorem for Operator-Valued Measures in Locally Convex Spaceshttp://projecteuclid.org/euclid.cma/1449496831<strong>Jan Haluska</strong>, <strong>Ondrej Hutnik</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 106--111.</p><p><strong>Abstract:</strong><br/>
The Egoroff theorem for measurable ${\mathbb X}$-valued functions and
operator-valued measures ${\mathbb m}: \Sigma \to L({\mathbb X}, {\mathbb Y})$
is proved, where $\Sigma$ is a $\sigma$-algebra of subsets of $T \neq \emptyset$
and ${\mathbb X}$, ${\mathbb Y}$ are both locally convex spaces.
</p>projecteuclid.org/euclid.cma/1449496831_20151207090036Mon, 07 Dec 2015 09:00 ESTFixed Point Theorems for Positive Maps and Applicationshttp://projecteuclid.org/euclid.cma/1449496832<strong>Abdelhamid Benmezai</strong>, <strong>Salima Mechrouk</strong>, <strong>Johnny Henderson</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 18, Number 2, 112--126.</p><p><strong>Abstract:</strong><br/>
We prove in this article new fixed point theorems for positive maps having
approximative minorant and majorant at $0$ and $\infty$ in specific classes of
operators. Then, the new fixed point theorems are used to obtain existence
results for positive solutions to boundary value problems involving a
generalized $p(t)$-Laplacian operator.
</p>projecteuclid.org/euclid.cma/1449496832_20151207090036Mon, 07 Dec 2015 09:00 EST<link>http://projecteuclid.org/euclid.cma/1455715852</link><description><strong>Toka Diagana</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 1, 1--5.</p><p><strong>Abstract:</strong><br/>
This Interview is a part of the Special Issue of Communications in Mathematical
Analysis dedicated to late Prof. Tosio Kato on his 100th birthday. We extend our
deepest thanks to Prof. Hitoshi Kitada for dedicating his paper "Wave Operators
and Similarity for Long Range N-body Schrodinger Operators" to Prof. Tosio Kato.
Further, we thank him for accepting to answer to our questions.
</p></description><guid isPermaLink="false">projecteuclid.org/euclid.cma/1455715852_20160217083054</guid><pubDate>Wed, 17 Feb 2016 08:30 EST</pubDate></item><item><title>Wave Operators and Similarity for Long Range $N$-body Schrödinger
Operatorshttp://projecteuclid.org/euclid.cma/1455715853<strong>Hitoshi Kitada</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 1, 6--66.</p><p><strong>Abstract:</strong><br/>
We consider asymptotic behavior of $e^{-itH}f$ for $N$-body Schrödigner operator
$H=H_{0}+\sum_{1 \leq i < j \leq N } V_{ij}(x)$ with long- and short-range
pair potentials $V_{ij}(x)=V_{ij}^L(x)+V_{ij}^S(x)$ $(x\in {\mathbb R}^\nu)$
such that $\partial_x^\alpha V_{ij}^L(x)=O(|x|^{-\delta |\alpha|})$ and
$V_{ij}^S(x)=O(|x|^{-1-\delta})$ $(|x|\to\infty)$ with $\delta>0$.
Introducing the concept of scattering spaces which classify the initial states
$f$ according to the asymptotic behavior of the evolution $e^{-itH}f$, we give a
generalized decomposition theorem of the continuous spectral subspace ${\mathcal
H}_c(H)$ of $H$. The asymptotic completeness of wave operators is proved for
some long-range pair potentials with $\delta>1/2$ by using this decomposition
theorem under some assumption on subsystem eigenfunctions.
</p>projecteuclid.org/euclid.cma/1455715853_20160217083054Wed, 17 Feb 2016 08:30 ESTA Class of Parabolic Maximal Functionshttp://projecteuclid.org/euclid.cma/1455715866<strong>Ghada Shakkah</strong>, <strong>Ahmad Al-Salman</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 1--31.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove $L^{p}$ estimates of a class of parabolic maximal
functions provided that their kernels are in $L^{q}$. Using the obtained
estimates, we prove the boundedness of the maximal functions under very weak
conditions on the kernel. In particular, we prove the$\ L^{p}$-boundedness of
our maximal functions when their kernels are in $L\log
L^{\frac{1}{2}}(\mathbb{S}^{n-1})$ or in the block space
$B_{q}^{0,-1/2}(\mathbb{S}^{n-1}),$ $q>1$.
</p>projecteuclid.org/euclid.cma/1455715866_20160217083106Wed, 17 Feb 2016 08:31 ESTCommutators Generated by Singular Integral Operators with Variable Kernels and
Local Campanato Functions on Generalized Local Morrey Spaceshttp://projecteuclid.org/euclid.cma/1465475796<strong>Huixia Mo</strong>, <strong>Hongyang Xue</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 32--42.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain the boundedness for the singular integral operator with
rough variable kernel $T_\Omega$ on the generalized local Morrey spaces, as well
as the boundedness for the multilinear commutators generated by $T_\Omega$ and
local Campanato functions.
</p>projecteuclid.org/euclid.cma/1465475796_20160609083641Thu, 09 Jun 2016 08:36 EDTCompleteness of Sums of Subspaces of Bounded Functions and Applicationshttp://projecteuclid.org/euclid.cma/1465475797<strong>Joel Blot</strong>, <strong>Philippe Cieutat</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 43--61.</p><p><strong>Abstract:</strong><br/>
We establish the completeness of spaces of $\mu$-pseudo almost periodic functions (or sequences) and $\mu$-pseudo almost automorphic functions (or sequences) by establishing a new result on the closedness of the sum of closed vector subspaces of the Banach space of bounded functions. To obtain this result we use abstract tools on the closedness of the image of linear operators and the sum of closed vector subspaces of a Banach space.
</p>projecteuclid.org/euclid.cma/1465475797_20160609083641Thu, 09 Jun 2016 08:36 EDTA Note on Closedness of the Sum of Two Closed Subspaces in a Banach Spacehttp://projecteuclid.org/euclid.cma/1473854217<strong>Zhe-Ming Zheng</strong>, <strong>Hui-Sheng Ding</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 62--67.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We collect
several necessary and sufficient conditions for the closedness of $M+N$ ($M+N$
is not necessarily direct sum), and show that a necessary condition in a
classical textbook is also sufficient.
</p>projecteuclid.org/euclid.cma/1473854217_20160914075700Wed, 14 Sep 2016 07:57 EDTLebedev's Type Index Transforms with the Modified Bessel Functionshttp://projecteuclid.org/euclid.cma/1473854218<strong>Semyon Yakubovich</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 68--81.</p><p><strong>Abstract:</strong><br/>
New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. Boundedness properties are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the modulus of Macdonald's function, recently considered by the author.
</p>projecteuclid.org/euclid.cma/1473854218_20160914075700Wed, 14 Sep 2016 07:57 EDTPropagation Speed for Fractional Cooperative Systems with Slowly Decaying Initial Conditionshttp://projecteuclid.org/euclid.cma/1486782020<strong>Miguel Yangari</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 82--100.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to study the time asymptotic propagation for mild solutions to the fractional reaction diffusion cooperative systems when at least one entry of the initial condition decays slower than a power. We state that the solution spreads at least exponentially fast with an exponent depending on the diffusion term and on the smallest index of fractional Laplacians.
</p>projecteuclid.org/euclid.cma/1486782020_20170210220032Fri, 10 Feb 2017 22:00 ESTGeneralizations of Majorization Inequality via Lidstone's Polynomial and Their Applicationshttp://projecteuclid.org/euclid.cma/1486782021<strong>M. Adil Khan</strong>, <strong>N. Latif</strong>, <strong>J. Pecaric</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 19, Number 2, 101--122.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain the generalizations of majorization inequalities by using Lidstone's interpolating polynomials and conditions on Green's functions. We give bounds for identities related to the generalizations of majorization inequalities by using Čebyšev functionals. We also give Grüss type inequalities and Ostrowski-type inequalities for these functionals. We present mean value theorems and $n$-exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity.
</p>projecteuclid.org/euclid.cma/1486782021_20170210220032Fri, 10 Feb 2017 22:00 ESTVariational Inequality with Evolutional Curl Constraint in a Multi-Connected
Domainhttp://projecteuclid.org/euclid.cma/1494986414<strong>Junichi Aramaki</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 1, 1--26.</p><p><strong>Abstract:</strong><br/>
We consider a system of quasilinear parabolic type equations involving operator
curl associated with the Maxwell equations in a multi-connected domain. The
paper is a continuation of the author's previous paper. We deal with a
variational inequality with curl constraint. It is an extension of the results
of Miranda et al. for $p$-curl system.
</p>projecteuclid.org/euclid.cma/1494986414_20170516220020Tue, 16 May 2017 22:00 EDTPeriodic Travelling Waves and its Inter-relation with Solitons for the 2D
abc-Boussinesq Systemhttp://projecteuclid.org/euclid.cma/1494986415<strong>Jose R. Quintero</strong>, <strong>Alex M. Montes</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 1, 27--49.</p><p><strong>Abstract:</strong><br/>
Via a variational approach involving Concentration-Compactness principle, we show
the existence of $x$-periodic travelling wave solutions for a general
2D-Boussinesq system that arises in the study of the evolution of long water
waves with small amplitude in the presence of surface tension. We also establish
that $x$-periodic travelling waves have almost the same shape of solitons as the
period tends to infinity, by showing that a special sequence of $x$-periodic
travelling wave solutions parameterized by the period converges to a solitary
wave in a appropriate sense.
</p>projecteuclid.org/euclid.cma/1494986415_20170516220020Tue, 16 May 2017 22:00 EDTBoundary Value Problems for Degenerate Coupled Systems with Variable Time
Delayhttp://projecteuclid.org/euclid.cma/1494986416<strong>Mykola Bokalo</strong>, <strong>Olga Ilnytska</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 1, 50--68.</p><p><strong>Abstract:</strong><br/>
The boundary value problems for coupled systems of parabolic and ordinary
differential equations, where all equations contain time depended delay and
degenerate at initial moment, are considered. Existence and uniqueness of
classical solutions of these problems are proved. A priori estimates are
obtained.
</p>projecteuclid.org/euclid.cma/1494986416_20170516220020Tue, 16 May 2017 22:00 EDTNonlinear Eigenvalue Problem for the p-Laplacianhttp://projecteuclid.org/euclid.cma/1500084077<strong>Najib Tsouli</strong>, <strong>Omar Chakrone</strong>, <strong>Omar Darhouche</strong>, <strong>Mostafa Rahmani</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 1, 69--82.</p><p><strong>Abstract:</strong><br/>
This article is devoted to the study of the nonlinear eigenvalue problem $$-\Delta_{p} u
\quad=\quad \lambda |u|^{p-2}u \;\mbox{in}\; \Omega,\\ |\nabla u|^{p-2}\frac{\partial
u}{\partial \nu}\quad+\quad\beta |u|^{p-2}u=\lambda |u|^{p-2}u
\;\mbox{on}\quad\partial\Omega,$$ where $ν$ denotes the unit exterior normal, $1 \lt p \lt
∞ \,\mathrm {and} ∆_{p}u = div(|∇u|^{p−2}∇u)$ denotes the p-laplacian. $Ω ⊂
\mathbb{R}^{N}$ is a bounded domain with smooth boundary where $N ≥ 2$ and $β \in
L^{∞}(∂Ω) \,\mathrm{with}\, β^{−} := \mathrm{inf}_{x∈∂Ω}β(x) > 0$. Using
Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of
positive eigenvalues and the first eigenvalue is simple and isolated. Moreover, we will
prove that the second eigenvalue coincides with the second variational eigenvalue obtained
via the Ljusternik-Schnirelman theory.
</p>projecteuclid.org/euclid.cma/1500084077_20170714220122Fri, 14 Jul 2017 22:01 EDTA Poincaré Inequality for Functions with Locally Bounded Variation in $\mathbb{R}^{d}$http://projecteuclid.org/euclid.cma/1500084078<strong>Bacary Savadogo</strong>, <strong>Ibrahim Fofana</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 1, 83--106.</p><p><strong>Abstract:</strong><br/>
We prove a weighted Poincaré inequality in a subspace of $BV_\text{loc}$ whose elements
have variation measure in a Wiener amalgam space of Radon measures.
</p>projecteuclid.org/euclid.cma/1500084078_20170714220122Fri, 14 Jul 2017 22:01 EDTNorm Estimates for Powers of Products of Operators in a Banach Spacehttps://projecteuclid.org/euclid.cma/1509674425<strong>Michael Gil’</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 2, 1--7.</p><p><strong>Abstract:</strong><br/>
Let $A$ and $B$ be bounded linear operators in a Banach space. We consider the following problem: if $\Sigma_{k=0}^{\infty} || A^{k} |||| B^{k} || \lt\infty$, under what conditions $\Sigma_{k=0}^{\infty} || (AB)^{k} || \lt \infty$?
</p>projecteuclid.org/euclid.cma/1509674425_20171102220029Thu, 02 Nov 2017 22:00 EDTVector Inequalities For Two Projections in Hilbert Spaces and Applicationshttps://projecteuclid.org/euclid.cma/1509674426<strong>Silvestru Sever Dragomir</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 2, 8--30.</p><p><strong>Abstract:</strong><br/> In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$ Applications for norm and numerical radius inequalities of two bounded operators are given as well. </p>projecteuclid.org/euclid.cma/1509674426_20171102220029Thu, 02 Nov 2017 22:00 EDTGeneral Adjoint on a Banach Spacehttps://projecteuclid.org/euclid.cma/1512442821<strong>Tepper L. Gill</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 2, 31--47.</p><p><strong>Abstract:</strong><br/>
In this paper, we show that the continuous dense embedding of a separable Banach space $\mathcal B$ into a Hilbert space $\mathcal H$ offers a new tool for studying the structure of operators on a Banach space. We use this embedding to demonstrate that the dual of a Banach space is not unique. As a application, we consider this non-uniqueness within the $\mathbb C[0,1] \subset L^2[0,1]$ setting. We then extend our theory every separable Banach space $\mathcal B$. In particular, we show that every closed densely defined linear operator $A$ on $\mathcal B$ has a unique adjoint $A^*$ defined on $\mathcal B$ and that $\mathcal L[\mathcal B]$, the bounded linear operators on $\mathcal B$, are continuously embedded in $\mathcal L[\mathcal H]$. This allows us to define the Schatten classes for $\mathcal L[\mathcal B]$ as the restriction of a subset of $\mathcal L[\mathcal H]$. Thus, the structure of $\mathcal L[\mathcal B]$, particularly the structure of the compact operators $\mathbb K[\mathcal B]$, is unrelated to the basis or approximation problems for compact operators. We conclude that for the Enflo space $\mathcal B_e$, we can provide a representation for compact operators that is very close to the same representation for a Hilbert space, but the norm limit of the partial sums may not converge, which is the only missing property.
</p>projecteuclid.org/euclid.cma/1512442821_20171204220026Mon, 04 Dec 2017 22:00 ESTWeak Solutions for Implicit Hilfer Fractional Differential Equations With Not Instantaneous Impulseshttps://projecteuclid.org/euclid.cma/1515467126<strong>S. Abbas</strong>, <strong>M. Benchohra</strong>, <strong>J. Henderson</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 2, 48--61.</p><p><strong>Abstract:</strong><br/>
In this paper, we present results concerning the existence of weak solutions for some functional implicit Hilfer fractional differential equations with not instantaneous impulses in Banach spaces. The main results are proved by applying Mönch's fixed point theorem associated with the technique of measure of weak non compactness, and we present an illustrative example.
</p>projecteuclid.org/euclid.cma/1515467126_20180108220530Mon, 08 Jan 2018 22:05 ESTOn the Oscillation of Solutions of First-Order Difference Equations with Delayhttps://projecteuclid.org/euclid.cma/1515467127<strong>Y. Shoukaku</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 20, Number 2, 62--67.</p><p><strong>Abstract:</strong><br/>
Consider the first order delay difference equation $$\Delta x_{n} + p_{n} x_{\sigma(n)} = 0, \quad n \in {\mathbb N}_0,$$ where $\{p_{n}\}_{n \in {\mathbb N}_0}$ is a sequence of nonnegative real numbers, and $\{\sigma(n)\}_{n \in {\mathbb N}_0}$ is a sequence of integers such that $\sigma(n) \le n-1$, and $\displaystyle \lim_{n \to \infty}\sigma(n) = +\infty$. We obtain similar oscillation criteria of delay differential equations. This criterion is used by more simple method until now.
</p>projecteuclid.org/euclid.cma/1515467127_20180108220530Mon, 08 Jan 2018 22:05 ESTDerivatives on Function Spaces Generated By the Dirichlet Laplacian and the Neumann Laplacian in One Dimensionhttps://projecteuclid.org/euclid.cma/1523498574<strong>Tsukasa Iwabuchi</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 1--8.</p><p><strong>Abstract:</strong><br/>
We investigate the relation between Besov spaces generated by the Dirichlet Laplacian and the Neumann Laplacian in one space dimension from the view point of the boundary value of functions. Derivatives on spaces with such boundary conditions are defined, and it is proved that the derivative operator is isomorphic from one to the other.
</p>projecteuclid.org/euclid.cma/1523498574_20180411220256Wed, 11 Apr 2018 22:02 EDTHardy Classes and Symbols of Toeplitz Operatorshttps://projecteuclid.org/euclid.cma/1523498575<strong>Marco López-García</strong>, <strong>Salvador Pérez-Esteva</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 9--22.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to study functions in the unit disk $\mathbb D$ through the family of Toeplitz operators $\{T_{φdσ_{t}}\}_{t∈[0,1)}$, where $T_{φdσ_{t}}$ is the Toeplitz operator acting the Bergman space of $\mathbb D$ and where $dσ_t$ is the Lebesgue measure in the circle $tS^1$. In particular for $1\le p \lt \infty$ we characterize the harmonic functions $φ$ in the Hardy space $h^{p}(\mathbb D)$ by the growth in $t$ of the $p$-Schatten norms of $T_{φdσ_{t}}$. We also study the dependence in $t$ of the norm operator of $T_{adσ_{t}}$ when $a∈H^p_{at}$, the atomic Hardy space in the unit circle with $1/2 \lt p \le 1$.
</p>projecteuclid.org/euclid.cma/1523498575_20180411220256Wed, 11 Apr 2018 22:02 EDTA Compactness Result For An Equation with Hölderian Conditionhttps://projecteuclid.org/euclid.cma/1525831213<strong>Samy Skander Bahoura</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 23--34.</p><p><strong>Abstract:</strong><br/>
We give blow-up behavior for a Brezis and Merle's problem with Dirichlet and Hölderian conditions. Also we derive a compactness criterion as in the work of Brezis and Merle.
</p>projecteuclid.org/euclid.cma/1525831213_20180508220017Tue, 08 May 2018 22:00 EDTA Simple Estimate of the Bloch Constanthttps://projecteuclid.org/euclid.cma/1525831214<strong>Mitsuo Izuki</strong>, <strong>Takeshi Koyama</strong>, <strong>Takahiro Noi</strong>, <strong>Tatsuki Takeuchi</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 35--41.</p><p><strong>Abstract:</strong><br/>
Our aim is to give a simple estimate of the Bloch constant applying some fundamental facts on complex analysis. Our method is based on the Cauchy estimate, the maximum modulus principle, the Schwarz lemma and the Rouche theorem.
</p>projecteuclid.org/euclid.cma/1525831214_20180508220017Tue, 08 May 2018 22:00 EDTExistence and Global Stability Results for Volterra Type Fractional Hadamard Partial Integral Equationshttps://projecteuclid.org/euclid.cma/1530842637<strong>S. Abbas</strong>, <strong>W. Albarakati</strong>, <strong>M. Benchohra</strong>, <strong>J.J. Nieto</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 42--53.</p><p><strong>Abstract:</strong><br/>
This paper deals with the global existence and stability of solutions of a new class of partial integral equations of Hadamard fractional order.
</p>projecteuclid.org/euclid.cma/1530842637_20180705220406Thu, 05 Jul 2018 22:04 EDTExistence and Regularity for the Neumann Problem to the Poisson Equation and an Application to the Maxwell-Stokes Type Equationhttps://projecteuclid.org/euclid.cma/1536717642<strong>Junichi Aramaki</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 54--66.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the Neumann problem for the Laplace operator with a given data containing a divergence of a vector field. We demonstrate the existence and regularity of a weak solution. As an application, we consider the existence and regularity of a weak solution in regard to the Maxwell-Stokes type equation.
</p>projecteuclid.org/euclid.cma/1536717642_20180911220059Tue, 11 Sep 2018 22:00 EDTAnalysis of the Small Oscillations of a Heavy Viscous Liquid in a Container Supported by an Elastic Structurehttps://projecteuclid.org/euclid.cma/1536717643<strong>Hilal Essaouini</strong>, <strong>Pierre Capodanno</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 1, 67--82.</p><p><strong>Abstract:</strong><br/>
We propose here a mathematical study of the small oscillations of a heavy viscous liquid in an arbitrary open rigid container supported by an elastic structure, i.e a spring-mass-damper system. From the equations of motion of the system, we deduce a variational formulation of the problem and after, an operatorial equation in a suitable Hilbert space. Then, we can study the spectrum of the problem. At first, we prove that it is formed by eigenvalues that are located in the right half-plane, so that the equilibrium position is stable. Besides, we show that the operator pencil of the problem is a well-known pencil, whose we prove by a simple method that it has two branches of real eigenvalues having as points of accumulation zero and the infinity and a number at most finite of complex eigenvalues. Finally, we prove the existence and the unicity of the solution of the associated evolution problem by means of Lions method. Afterwards, we consider the case where the damper is removed, that is very different. We prove in this case that the equilibrium position is stable, but the problem is reduced to the study of a Krein-Langer pencil, so that in particular, there exist always non oscillatory eigenmotions.
</p>projecteuclid.org/euclid.cma/1536717643_20180911220059Tue, 11 Sep 2018 22:00 EDTA Uniform Ergodic Theorem for Some Nörlund Meanshttps://projecteuclid.org/euclid.cma/1538704836<strong>Laura Burlando</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 2, 1--34.</p><p><strong>Abstract:</strong><br/>
We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums, $T$ is a bounded linear operator on a Banach space and $s$ is a divergent nondecreasing sequence of strictly positive real numbers, such that $\lim_{n\rightarrow+\infty} s(n+1)/s(n)=1$ and $\varDelta^qs\in\ell_1$ for some positive integer $q$. Indeed, we prove that if $T^{n}/s(n$) converges to zero in the uniform operator topology, then the sequence of averages above converges in the same topology if and only if $1$ is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$.
</p>projecteuclid.org/euclid.cma/1538704836_20181004220041Thu, 04 Oct 2018 22:00 EDT$(\omega,c)$-Periodic Solutions for Impulsive Differential Systemshttps://projecteuclid.org/euclid.cma/1545361382<strong>Mengmeng Li</strong>, <strong>JinRong Wang</strong>, <strong>Michal Feckan</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 2, 35--45.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the existence of $(\omega,c)$-periodic solutions for a nonhomogeneous linear impulsive system by constructing Green functions and adjoint systems, respectively. In addition, we study the existence and uniqueness of $(\omega,c)$-periodic solutions for a semilinear impulsive system via fixed point approach. Two examples are provided to illustrate our results.
</p>projecteuclid.org/euclid.cma/1545361382_20181220220323Thu, 20 Dec 2018 22:03 ESTJensen-Type Inequalities on Time Scales For $n$-Convex Functionshttps://projecteuclid.org/euclid.cma/1545361386<strong>Rozarija Mikic</strong>, <strong>Josip Pecaric</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 2, 46--67.</p><p><strong>Abstract:</strong><br/>
By utilizing some scalar inequalities obtained via Hermite's interpolating polynomial, we will obtain lower and upper bounds for the difference in Jensen's inequality and in the Edmundson-Lah-Ribaric inequality in time scale calculus that hold for the class of $n$-convex functions. Main results are then applied to generalized means, with a particular emphasis to power means, and in that way some new reverse relations for generalized and power means that correspond to $n$-convex functions are obtained.
</p>projecteuclid.org/euclid.cma/1545361386_20181220220323Thu, 20 Dec 2018 22:03 ESTKarakostas Fixed Point Theorem and the Existence of Solutions for Impulsive Semilinear Evolution Equations with Delays and Nonlocal Conditionshttps://projecteuclid.org/euclid.cma/1547262053<strong>Hugo Leiva</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 21, Number 2, 68--91.</p><p><strong>Abstract:</strong><br/>
We prove the existence and uniqueness of the solutions for the following impulsive semilinear evolution equations with delays and nonlocal conditions: $$\left\{ \begin{array}{lr} \acute{z} =-Az +F(t,z_{t}), & z\in Z, \quad t \in (0, \tau], t \neq t_k, \\ z(s)+(g(z_{\tau_1},z_{\tau_2},\dots, z_{\tau_q}))(s) = \phi(s), & s \in [-r,0],\\ z(t_{k}^{+}) = z(t_{k}^{-})+J_{k}(z(t_{k})), & k=1,2,3, \dots, p. \end{array} \right.$$ where $0 \lt t_1 \lt t_2 \lt t_3 \lt ··· \lt t_p \lt \tau, 0 \lt \tau_{1} \lt \tau_{2} \lt ··· \lt \tau_{q} \lt r \lt \tau, Z$ is a Banach space $Z$, $z_t$ defined as a function from $[−r, 0]$ to $Z^ \alpha$ by $z_{t}(s) = z(t + s),−r \le s \le 0, g : C([−r, 0];Z^{\alpha}_{q}) \rightarrow C([−r, 0];Z^{\alpha})$ and $J_{k} : Z^{\alpha} \rightarrow Z^{\alpha}, F : [0,\tau] ×C(−r,0;Z^{\alpha}) \rightarrow Z$. In the above problem, $A : D(A)\subset Z \rightarrow Z$ is a sectorial operator in $Z$ with $−A$ being the generator of a strongly continuous compact semigroup $\{T(t)\}_{t \ge 0}$, and $Z^{\alpha}= D(A^{\alpha})$. The novelty of this work lies in the fact that the evolution equation studied here can contain non-linear terms that involve spatial derivatives and the system is subjected to the influence of impulses, delays and nonlocal conditions, which generalizes many works on the existence of solutions for semilinear evolution equations in Banch spaces. Our framework includes several important partial differential equations such as the Burgers equation and the Benjamin-Bona-Mohany equation with impulses, delays and nonlocal conditions.
</p>projecteuclid.org/euclid.cma/1547262053_20190111220058Fri, 11 Jan 2019 22:00 ESTDeterministic Homogenization of Variational Inequalities with Unilateral Constrainthttps://projecteuclid.org/euclid.cma/1566266424<strong>Hermann Douanla</strong>, <strong>Cyrille Kenne</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 1, 1--13.</p><p><strong>Abstract:</strong><br/>
The article studies the reiterated homogenization of linear elliptic variational inequalities arising in problems with unilateral constraints. We assume that the coefficients of the equations satisfy and abstract hypothesis covering on each scale a large set of concrete deterministic behavior such as the periodic, the almost periodic and the convergence at infinity. Using the multiscale convergence method, we derive a homogenization result whose limit problem is of the same type as the problem with rapidly oscillating coefficients.
</p>projecteuclid.org/euclid.cma/1566266424_20190819220042Mon, 19 Aug 2019 22:00 EDTLaplace Transform, Gronwall Inequality and Delay Differential Equations for General Conformable Fractional Derivativehttps://projecteuclid.org/euclid.cma/1566266425<strong>Michal Pospíšil</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 1, 14--33.</p><p><strong>Abstract:</strong><br/>
In the present paper, the general conformable fractional derivative (GCFD) is considered and a corresponding Laplace transform is defined. Gronwall inequality is proved to show the exponential boundedness of a solution and using the Laplace transform the solution is found for certain classes of delay differential equations with GCFD.
</p>projecteuclid.org/euclid.cma/1566266425_20190819220042Mon, 19 Aug 2019 22:00 EDTExistence and Regularity of a Weak Solution to the Maxwell-Stokes Type System Containing $p$-curlcurl Equationhttps://projecteuclid.org/euclid.cma/1566266426<strong>Junichi Aramaki</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 1, 34--50.</p><p><strong>Abstract:</strong><br/>
We consider the existence and regularity of a weak solution to the Maxwell-Stokes type system containing a $p$-curlcurl equation in a multiply-connected domain with holes. In this paper, we shows that the compatibility condition is necessary and sufficient for the existence of a weak solution to the Maxwell-Stokes type system, and that the unique solution has the $C^{1,\beta}$-regularity. The $C^{1,\beta}$-regularity is optimal.
</p>projecteuclid.org/euclid.cma/1566266426_20190819220042Mon, 19 Aug 2019 22:00 EDTConvergence to Attractors of Nonexpansive Set-Valued Mappingshttps://projecteuclid.org/euclid.cma/1566266427<strong>Simeon Reich</strong>, <strong>Alexander J. Zaslavski</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 1, 51--60.</p><p><strong>Abstract:</strong><br/>
In our previous work we have shown that if for any initial point there exists a trajectory of a nonexpansive set-valued mapping attracted by a given set, then this property is stable under small perturbations of the mapping. In the present paper we obtain several extensions of this result.
</p>projecteuclid.org/euclid.cma/1566266427_20190819220042Mon, 19 Aug 2019 22:00 EDTMultiple Solutions for Semilinear $\Delta_{\gamma}-$differential Equations in $\mathbb R^N$ with Sign-changing Potentialhttps://projecteuclid.org/euclid.cma/1566266428<strong>Duong Trong Luyen</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 1, 61--75.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the existence of infinitely many nontrivial solutions of the semilinear $\Delta_{\gamma}$ differential equations in $\mathbb{R}^N$ $$ - \Delta_{\gamma}u+ b(x)u=f(x,u)\quad \mbox{ in }\; \mathbb{R}^N, \quad u \in S^2_{\gamma}(\mathbb{R}^N), $$ where $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, ..., \gamma_N),$$ and the potential $b$ is allowed to be sign-changing, and the primitive of the nonlinearity $f$ is of superquadratic growth near infinity in $u$ and allowed to be sign-changing.
</p>projecteuclid.org/euclid.cma/1566266428_20190819220042Mon, 19 Aug 2019 22:00 EDTSome New Stability, Boundedness, and Square Integrability Conditions for Third-Order Neutral Delay Differential Equationshttps://projecteuclid.org/euclid.cma/1566266429<strong>John R. Graef</strong>, <strong>Djamila Beldjerd</strong>, <strong>Moussadek Remili</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 1, 76--89.</p><p><strong>Abstract:</strong><br/>
In this paper, the authors establish some new sufficient conditions under which all solutions of a third order nonlinear neutral delay differential equation are stable, bounded, and square integrable. An example is also given to illustrate the results.
</p>projecteuclid.org/euclid.cma/1566266429_20190819220042Mon, 19 Aug 2019 22:00 EDTPractical Stability of Differential Equations with State Dependent Delay and Non-instantaneous Impulseshttps://projecteuclid.org/euclid.cma/1575428420<strong>Michal Feckan</strong>, <strong>Snezhana Hristova</strong>, <strong>Krasimira Ivanova</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 2, 1--17.</p><p><strong>Abstract:</strong><br/>
In this paper some practical stability results for nonlinear differential equations with non-instantaneous impulses and state dependent delays are presented. The impulses start abruptly at some points and their action continue on given finite intervals. The delay depends on both the time and the state variable which is a generalization of time variable delay. Some sufficient conditions for practical stability and strong practical stability are obtained by the help with the appropriate modification of Razumikhin method and an appropriate definition of the derivative of the Lyapunov function. Examples are given to illustrate the results.
</p>projecteuclid.org/euclid.cma/1575428420_20191203220034Tue, 03 Dec 2019 22:00 ESTExistence of Standing Waves in DNLS with Saturable Nonlinearity on 2D-Latticehttps://projecteuclid.org/euclid.cma/1575428421<strong>Sergiy Bak</strong>, <strong>Galyna Kovtonyuk</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 2, 18--34.</p><p><strong>Abstract:</strong><br/>
In this paper we obtain results on existence of standing waves in Discrete Nonlinear Shrödinger equation (DNLS) with saturable nonlinearity on a two-dimensional lattice. We consider two types of solutions: with periodic amplitude and vanishing at infinity (localized solution). Sufficient conditions for the existence of such solutions are obtained with the aid of Nehari manifold and periodic approximations.
</p>projecteuclid.org/euclid.cma/1575428421_20191203220034Tue, 03 Dec 2019 22:00 ESTTwo Results Relating an $L^p$ Regularity Condition and the $L^q$ Dirichlet Problem for Parabolic Equationshttps://projecteuclid.org/euclid.cma/1575428422<strong>Luis San Martin</strong>, <strong>Jorge Rivera-Noriega</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 2, 35--60.</p><p><strong>Abstract:</strong><br/>
We consider variations and generalizations of the initial Dirichlet problem for linear second order divergence form equations of parabolic type, with vanishing initial values and non-continuous lateral data, in the setting of Lipschitz cylinders. More precisely, lateral data in adequations of the Lebesgue classes $L^p$, and a family of Sobolev-type classes are considered. We also establish some basic connections between estimates related to solvability of each of these problems. This generalizes some of the well-known works for Laplace's equation, heat equation and some linear elliptic-type equations of second order.
</p>projecteuclid.org/euclid.cma/1575428422_20191203220034Tue, 03 Dec 2019 22:00 ESTSufficient Conditions for the Lebesgue Integrability of Fourier Transforms in Amalgam Spaceshttps://projecteuclid.org/euclid.cma/1575428423<strong>Sekou Coulibaly</strong>, <strong>Moumine Sanogo</strong>, <strong>Ibrahim Fofana</strong>. <p><strong>Source: </strong>Communications in Mathematical Analysis, Volume 22, Number 2, 61--77.</p><p><strong>Abstract:</strong><br/>
Let $f$ be an element of the subspace $(L^{p},l^{q})^{\alpha}(\mathbb{R}^d)$ $(1\leq p \leq \alpha \leq q \leq 2)$ of the Wiener amalgam space $(L^{p},l^{q})(\mathbb{R}^{d})$. We give sufficient conditions for Lebesgue integrability of the Fourier transform of $f$. These conditions are in terms of the $(L^{p},l^{q})^{\alpha}(\mathbb{R}^{d})$ integral modulus of continuity of $f$. As an application, we obtain that if $1\leq\alpha\leq q\le 2$ with $\frac{1}{\alpha}-\frac{1}{q}\lt\frac{1}{d}$ and $N= [\frac{d}{\alpha}] + 1$, then the Fourier inversion theorem can be applied to the elements of the Sobolev space $W^{N}((L^{1},l^{q})^{\alpha}(\mathbb{R}^d))$.
</p>projecteuclid.org/euclid.cma/1575428423_20191203220034Tue, 03 Dec 2019 22:00 EST