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where $(p(n))$ is a sequence of non-negative real numbers, $(\tau (n))$ is a sequence of integers such that $\tau (n)< n$ for$\ n\in \mathbf {N},\,$and$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$ Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.
In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.
In this paper, we characterize jump phenomena of the $n$-th eigenvalue of self-adjoint discrete Sturm–Liouville problems in any dimension. For a fixed Sturm–Liouville equation, we completely characterize jump phenomena of the $n$-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in Hu et al. (2019, J. Differ. Equ.266, 4106–4136) and Kong et al. (1999, J. Differ. Equ.156, 328–354), the jump set here is involved with coefficients of the Sturm–Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing the jump areas. We study the singularity by partitioning and analysing the local coordinate systems, and provide a Hermitian matrix which can determine the areas’ division. To prove the asymptotic behaviour of the $n$-th eigenvalue, we generalize the method developed in Zhu and Shi (2016, J. Differ. Equ.260, 5987–6016) to any dimension. As an application, by transforming the continuous Sturm–Liouville problem of Atkinson type to a discrete one, we determine the number of eigenvalues and obtain complete characterization of jump phenomena of the $n$-th eigenvalue for the Atkinson type.
For an (irreducible) recurrence equation with coefficients from
$\mathbb Z[n]$
and its two linearly independent rational solutions
$u_n,v_n$
, the limit of
$u_n/v_n$
as
$n\to \infty $
, when it exists, is called the Apéry limit. We give a construction that realises certain quotients of L-values of elliptic curves as Apéry limits.
The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on
$\kappa _p$
symbols (here
$\kappa _p$
is the greatest common factor of k and
$p-1$
). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).
For
$R(z, w)\in \mathbb {C}(z, w)$
of degree at least 2 in w, we show that the number of rational functions
$f(z)\in \mathbb {C}(z)$
solving the difference equation
$f(z+1)=R(z, f(z))$
is finite and bounded just in terms of the degrees of R in the two variables. This complements a result of Yanagihara, who showed that any finite-order meromorphic solution to this sort of difference equation must be a rational function. We prove a similar result for the differential equation
$f'(z)=R(z, f(z))$
, building on a result of Eremenko.
We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in
$\mathbb {P}^{1}\!\times \mathbb {P}^{1}$
. These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.
Let P and Q be relatively prime integers greater than 1, and let f be a real valued discretely supported function on a finite dimensional real vector space V. We prove that if
$f_{P}(x)=f(Px)-f(x)$
and
$f_{Q}(x)=f(Qx)-f(x)$
are both
$\Lambda $
-periodic for some lattice
$\Lambda \subset V$
, then so is f (up to a modification at
$0$
). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.
We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:
This paper concerns the study of some bifurcation properties for the following class of Choquard-type equations:
(P)
$$\left\{ {\begin{array}{*{20}{l}}
{ - \Delta u = \lambda f(x)\left[ {u + \left( {{I_\alpha }*f( \cdot )H(u)} \right)h(u)} \right],{\rm{ in }} \ {{\mathbb{R}}^3},}\\
{{{\lim }_{|x| \to \infty }}u(x) = 0,\quad u(x) > 0,\quad x \in {{\mathbb{R}}^3},\quad u \in {D^{1,2}}({{\mathbb{R}}^3}),}
\end{array}} \right.$$
where ${I_\alpha }(x) = 1/|x{|^\alpha },\,\alpha \in (0,3),\,\lambda > 0,\,f:{{\mathbb{R}}^3} \to {\mathbb{R}}$ is a positive continuous function and h : ${\mathbb{R}} \to {\mathbb{R}}$ is a bounded Hölder continuous function. The main tools used are Leray–Schauder degree theory and a global bifurcation result due to Rabinowitz.
By using methods of subordinacy theory, we study packing continuity properties of spectral measures of discrete one-dimensional Schrödinger operators acting on the whole line. Then we apply these methods to Sturmian operators with rotation numbers of quasibounded density to show that they have purely $\unicode[STIX]{x1D6FC}$-packing continuous spectrum. A dimensional stability result is also mentioned.
We study a class of Schrödinger lattice systems with sublinear nonlinearities and perturbed terms. We get an interesting result that the systems do not have nontrivial homoclinic solutions if the perturbed terms are removed, but the systems have ground state homoclinic solutions if the perturbed terms are added. Besides, we also study the continuity of the homoclinic solutions in the perturbation terms at zero. To the best of our knowledge, there is no published result focusing on the perturbed Schrödinger lattice systems.
We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.
where $\mathbf{dt}=dt_{1}\,dt_{2}\cdots \,dt_{n}$ or $\mathbf{dt}=d_{q}t_{1}\,d_{q}t_{2}\cdots d_{q}t_{n}$ is the discrete measure in $q$-analysis. The sharp bounds for the multivariate Hausdorff operator on spaces $L^{p}$ with power weights are calculated, where $p\in \mathbb{R}\backslash \{0\}$.
The Toda equation and its variants are studied in the filed of integrable systems. One particularly generalized time discretisation of the Toda equation is known as the discrete hungry Toda (dhToda) equation, which has two main variants referred to as the dhTodaI equation and dhTodaII equation. The dhToda equations have both been shown to be applicable to the computation of eigenvalues of totally nonnegative (TN) matrices, which are matrices without negative minors. The dhTodaI equation has been investigated with respect to the properties of integrable systems, but the dhTodaII equation has not. Explicit solutions using determinants and matrix representations called Lax pairs are often considered as symbolic properties of discrete integrable systems. In this paper, we clarify the determinant solution and Lax pair of the dhTodaII equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhTodaII equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.
In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.
In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].