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Communications in Mathematical Analysis
Communications in Mathematical Analysis
Bokalo, Mykola; Ilnytska, Olga
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.
Quintero, Jose R.; Montes, Alex M.
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.
Aramaki, Junichi
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.
Khan, M. Adil; Latif, N.; Pecaric, J.
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.
Yangari, Miguel
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.
Yakubovich, Semyon
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.
Zheng, Zhe-Ming; Ding, Hui-Sheng
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.
Blot, Joel; Cieutat, Philippe
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.
Mo, Huixia; Xue, Hongyang
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.
Shakkah, Ghada; Al-Salman, Ahmad
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
$B_{q}^{0,-1/2}(\mathbb{S}^{n-1}),$ $q>1$.
Kitada, Hitoshi
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.
Diagana, Toka
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.
Benmezai, Abdelhamid; Mechrouk, Salima; Henderson, Johnny
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.
Haluska, Jan; Hutnik, Ondrej
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.
Belloni, Marino; Marchi, Silvana
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.
McCalla, Peter; Ramaroson, Francois
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...
Asfaw, Teffera M.
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...
Hezzi, H.; Marzouk, A.; Saanouni, T.
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.
Shakhmurov, Veli.B.; Sahmurova, Aida
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.
Wang, Hongbin
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.