Sufficient Conditions For Boundedness Research Articles

Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis

In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population dynamics’ models is applied and fixed moments impulsive perturbations are considered. The combined concept of practical stability with respect to manifolds is adapted to the introduced model. Sufficient conditions for boundedness and generalized practical stability of the solutions are obtained by using an analogue of the Lyapunov function method. The uncertain case is also studied. Examples are given to demonstrate the effectiveness of the established results.

Volterra-type operator on the subclasses of univalent functions

UDC 517.5 In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order and spirallike functions of type with the complex order We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes. In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces.

Weighted composition operators from Bloch-type into Bers-type spaces

In this paper we consider the weighted composition operator uC_{\varphi} from Bloch-type space B^{\alpha} into Bers-type space H_{\beta}^{\infty}, in three cases, \alpha>1, \alpha=1 and \alpha<1. We give the necessary and sufficient conditions for boundedness and compactness of the above operator.

SUFFICIENT CONDITIONS OF BOUNDEDNESS OF SOLUTIONS OF ONE NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL SECOND-ORDER EQUATION ON THE HALF-AXIS

Sufficient conditions of boundedness on the half-axle of all solutions and their first derivatives of a single nonlinear integro-differential equation of the second-order Volterra type are established. For this purpose, a partial cutting method is developed. The illustrative example is given.

Substitution vector-valued integral type operators on Orlicz spaces

In this paper, we consider some fundamental properties of a substitution vector-valued integral operator Tu? from Orlicz space L?(?) to Hilbert space H by the language of conditional expectation operators. First, we present necessary and sufficient conditions for boundedness and compactness Tu? from L?(?) to H. Next, we investigate the problem of conditions on the generating Young functions, the functions u, ? and h = d(? ? ?-1)/d?, under which operator Tu? is of closed range or finite rank. Finally, we determine the lower and upper estimates for the essential norm of Tu? on Orlicz spaces under certain conditions.

Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

Abstract In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball B 2 = { z ∈ C 2 : | z | = | z 1 | 2 + | z 2 | 2 < 1 } , $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l 1, l 2): 𝔹2 → R + 2 $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function. Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

Fast enclosing the solution set of the parametric Sylvester matrix equations

ABSTRACT In this work, we study the parametric Sylvester matrix equation whose elements are linear functions of uncertain parameters varying within some intervals. We first explore some properties of its solution set and then present some sufficient conditions for boundedness of the solution set. The main work is developing a modified variant of the parametric Krawczyk operator which reduces the computational complexity of enclosing the solution set significantly, compared to the parametric Krawczyk operator on the Kronecker product form. Some numerical experiments are given to illustrate the effectiveness of the proposed method.

Об интегральных операторах с периодическими ядрами в пространствах суммируемых функций

We consider integral operators with periodic kernels acting from Lp(ℝn) to Lq(ℝn). We obtain sufficient conditions for boundedness of that operators. Moreover we obtain compactness conditions for the product of the integral operator with periodic kernel and the operator of multiplication by an essentially bounded function.

Analogs of Fricke's theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables.

In this paper, we present necessary and sufficient conditions of boundedness of $\mathbb{L}$-index in joint variables for vector-functions analytic in the unit ball, where $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ Particularly, we deduce analog of Fricke's theorems for this function class, give estimate of maximum modulus on the skeleton of bidisc. The first theorem concerns sufficient conditions. In this theorem we assume existence of some radii, for which the maximum of norm of vector-function on the skeleton of bidisc with larger radius does not exceed maximum of norm of vector-function on the skeleton of bidisc with lesser radius multiplied by some costant depending only on these radii. In the second theorem we show that boundedness of $\mathbf{L}$-index in joint variables implies validity of the mentioned estimate for all radii.

Analysis of leukaemic cells dynamics with multi‐stage maturation process using a new non‐linear positive model with distributed time‐delay

This study proposes a new non-linear positive model with distributed time-delay describing the dynamics of leukaemic cells lineages and their levels of maturity. The dynamic model is first reformulated into coupled non-linear equations with distributed time-delay using an appropriate transformation of the age-structured transport partial differential equations that govern the dynamics of leukaemic cell populations. After assessing the positiveness of the proposed model, the conditions for the existence of positive steady-states are established. Furthermore, the necessary and sufficient conditions for boundedness of the solutions are derived. Finally, the convergence of the proposed model to the origin is studied as this has an important biological interpretation which is the extinction of all generations of leukaemic cells following anti-leukaemia treatments. To investigate the global asymptotic stability of this important equilibrium point, both model properties (i.e. positiveness and boundedness) are used to construct a suitable Lyapunov function for the characterisation of leukaemic cells dynamics. The results demonstrate a significant improvement for the treatment of leukaemia. Some simulations are presented to illustrate the effectiveness of the proposed model.

Weighted composition operators between Fock spaces ℱ∞(ℂ) and ℱp(ℂ)

In this paper, we establish necessary and sufficient conditions for boundedness and compactness of weighted composition operators acting between Fock spaces [Formula: see text] and [Formula: see text]. We also give complete descriptions of path connected components for the space of composition operators and the space of nonzero weighted composition operators in this context.

A nonstandard Volterra integral equation on time scales

Abstract This paper introduces the more general result on existence, uniqueness and boundedness for solutions of nonstandard Volterra type integral equation on an arbitrary time scales. We use Lipschitz type function and the Banach’s fixed point theorem at functional space endowed with a suitable Bielecki type norm. Furthermore, it allows to get new sufficient conditions for boundedness and continuous dependence of solutions.

Kernel Theorems for Modulation Spaces

We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces $M^p$ for every $1\leq p\leq\infty$, by the membership of its kernel to (mixed) modulation spaces. Whereas Feichtinger's kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz' kernel theorem for temperate distributions, our results do not have a couterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism upon distribution theory, as already emphasized in Feichtinger's manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on $M^{p,q}$ a similar characterization is not expected, but sufficient conditions for boundedness can be sated in the same spirit.

Properties of power series of analytic in a bidisc functions of bounded $\mathbf{L}$-index in joint variables

We generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in a bidisc functions, where $\mathbf{L}(z)=(l_1(z_1,z_2),$ $l_{2}(z_1,z_2)),$ $l_j:\mathbb{D}^2\to \mathbb{R}_+$ is a continuous function, $j\in\{1,2\},$ $\mathbb{D}^2$ is a bidisc $\{(z_1,z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1\}.$ The propositions describe a behaviour of power series expansion on a skeleton of a bidisc. We estimated power series expansion by a dominating homogeneous polynomial with the degree that does not exceed some number depending only from radii of bidisc. Replacing universal quantifier by existential quantifier for radii of bidisc, we also proved sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for analytic functions which are weaker than necessary conditions.

$$L^{p}$$ L p -boundedness, compactness of pseudo-differential operators on compact Lie groups

In this paper we investigate boundedness and compactness of pseudo-differential operators on compact Lie group \( \mathbb {G} \) and offer a new sufficient condition for boundedness and compactness of these operators on \(L^{p}(\mathbb {G}),\ \ p\ge 1\).

Deciding the boundedness and dead-beat stability of constrained switching systems

We study computational questions related with the stability of discrete-time linear switching systems with switching sequences constrained by an automaton.We first present a decidable sufficient condition for their boundedness when the maximal exponential growth rate equals one. The condition generalizes the notion of the irreducibility of a matrix set, which is a well known sufficient condition for boundedness in the arbitrary switching (i.e. unconstrained) case.Second, we provide a polynomial time algorithm for deciding the dead-beat stability of a system, i.e. that all trajectories vanish to the origin in finite time. The algorithm generalizes one proposed by Gurvits for arbitrary switching systems, and is illustrated with a real-world case study.

A study on pseudo-differential operators on $$\mathbb {S}^{1} $$ S 1 and $$\mathbb {Z}$$ Z

In this paper we offer a new sufficient condition for boundedness of pseudo-differential operators on \(L^{p}(\mathbb {S}^{1})\) and \(L^{P}(\mathbb {Z}),\ p\ge 1. \) Also a new sufficient condition for \(L^{2}\)-compactness of these operators is obtained.

Necessary and sufficient conditions for boundedness of multilinear fractional integrals with rough kernels on Morrey type spaces

In this article, we study necessary and sufficient conditions on the parameters of the boundedness on Morrey spaces and modified Morrey spaces for $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$ , which are a multilinear fractional integral and a multilinear fractional maximal operator with rough kernel, respectively. Our results extend some known results significantly.

Synchronization of Identical Linear Systems and Diffusive Time-Delayed Couplings

We study the problem of controlled network synchronization for a class of identical linear systems. The systems are interconnected through static and dynamic diffusive couplings with time-delays. In particular, we derive conditions on the systems, on the couplings, on the time-delay, and on the network topology that guarantee global synchronization of the systems. Diffusive time-delayed dynamic couplings are constructed by combining linear observers and output feedback controllers. Using passivity properties, sufficient conditions for boundedness of the interconnected systems are derived. Moreover, predictor-based dynamic couplings are proposed in order to enhance robustness against time-delays in the network. The results are illustrated by numerical simulations.

Boundedness of solutions of second order nonlinear dynamic equations

This paper deals with the boundedness of the solutions of the following dynamic equations $$(r(t)x^\Delta (t))^\Delta + a(t)f(x^\sigma (t)) + b(t)g(x^\sigma (t)) = 0$$ and $$(r(t)x^\Delta (t))^\Delta + a(t)x^\sigma (t) + b(t)f(x(t - \tau (t))) = e(t)$$ on a time scale \(\mathbb{T}\). By using the Bellman integral inequality, we establish some sufficient conditions for boundedness of solutions of the above equations. Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.