scholarly journals Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 918
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Ewa Schmeidel
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Solution ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Degree Of Approximation ◽  
The Other ◽  
Volterra Equations ◽  
Asymptotically Periodic ◽  
Monotone Type ◽  
Prescribed Asymptotic Behavior

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(un) as a measure of approximation.

scholarly journals Asymptotic behavior of solutions to difference equations in Banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Janusz Migda
Keyword(s):  
Asymptotic Behavior ◽  
Banach Spaces ◽  
Difference Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Degree Of Approximation ◽  
Asymptotic Behavior Of Solutions ◽  
Vector Version ◽  
A New Technique ◽  
Prescribed Asymptotic Behavior

We investigate the asymptotic properties of solutions to higher order nonlinear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L'Hospital's rule, remainder operator, and the regional topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.

scholarly journals Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Volterra Type ◽  
Order Difference ◽  
Prescribed Asymptotic Behavior

We consider the discrete Volterra type equation of the form Δ(rnΔxn)=bn+∑k=1nK(n,k)f(xk). We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use o(ns), for given nonpositive real s, as a measure of approximation.

scholarly journals Asymptotic properties of solutions of third order difference equations

2020 ◽  
Vol 14 (1) ◽  
pp. 1-19
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Asymptotic Properties ◽  
Harmonic Approximation ◽  
Sufficient Conditions ◽  
Degree Of Approximation ◽  
Third Order ◽  
Order Difference ◽  
Geometric Approximation ◽  
The Difference

We consider the difference equation of the form ?(rn?(pn?xn)) = anf (x?(n)) + bn. We present sufficient conditions under which, for a given solution y of the equation ?(rn?(pn?yn)) = 0, there exists a solution x of the nonlinear equation with the asymptotic behavior xn = yn + zn, where z is a sequence convergent to zero. Our approach allows us to control the degree of approximation, i.e., the rate of convergence of the sequence We examine two types of approximation: harmonic approximation when zn = o(ns), s ? 0, and geometric approximation when zn = o(?n), ? ? (0, 1).

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

scholarly journals Global Asymptotic Behavior of a Nonautonomous Competitor-Competitor-Mutualist Model

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Shengmao Fu ◽  
Fei Qu
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Time Dependent ◽  
Asymptotically Stable ◽  
Positive Periodic Solutions ◽  
Periodic Functions ◽  
Globally Asymptotically Stable ◽  
Periodic Model ◽  
Asymptotically Periodic

The global asymptotic behavior of a nonautonomous competitor-competitor-mutualist model is investigated, where all the coefficients are time-dependent and asymptotically approach periodic functions, respectively. Under certain conditions, it is shown that the limit periodic system of this asymptotically periodic model admits two positive periodic solutions(u1T,u2T,u3T),  (u1T,u2T,u3T)such thatuiT≤uiT  (i=1,2,3), and the sector{(u1,u2,u3):uiT≤ui≤uiT,  i=1,2,3}is a global attractor of the asymptotically periodic model. In particular, we derive sufficient conditions that guarantee the existence of a positive periodic solution which is globally asymptotically stable.

scholarly journals On Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Damped Neutral Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Ercan Tunç ◽  
Said R. Grace
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Second Order ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations

This paper discusses oscillatory and asymptotic properties of solutions of a class of second-order nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples.

Bounded solutions of nonlinear discrete volterra equations

2016 ◽  
Vol 66 (5) ◽  
Author(s):  
Małgorzata Migda ◽  
Janusz Migda
Keyword(s):  
Periodic Solutions ◽  
Volterra Equation ◽  
Sufficient Conditions ◽  
Volterra Equations ◽  
Real Constant ◽  
Bounded Solutions ◽  
All Solutions ◽  
Asymptotically Periodic ◽  
Asymptotically Periodic Solutions

AbstractWe give sufficient conditions, under which for every real constant, there exists a solution of the nonlinear discrete Volterra equationconvergent to this constant. We give also conditions under which all solutions are asymptotically constant. Sufficient conditions for the existence of asymptotically periodic solutions of the above equation are also derived.

scholarly journals A Construction of Meromorphic Functions with Prescribed Asymptotic Behavior

1971 ◽  
Vol 41 ◽  
pp. 75-87 ◽  
Author(s):  
J.L. Stebbins
Keyword(s):  
Asymptotic Behavior ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
The Other ◽  
Asymptotic Values ◽  
Other Hand ◽  
Extended Complex ◽  
The Given ◽  
Prescribed Asymptotic Behavior ◽  
Extended Complex Plane

Although there are several constructions of meromorphic functions with prescribed asymptotic sets [e.g., 5,6], it is usually difficult to determine or prescribe the nature of the asymptotic paths used in these constructions. On the other hand, there are several other constructions of meromorphic functions with prescribed asymptotic paths [e.g., 1, 10, 12], but the extent of the asymptotic values for these functions cannot always be restricted to the values approached along the given paths. Gross [3] has accomplished both results by prescribing paths for every value in the extended complex plane.

scholarly journals Existence and Global Asymptotic Behavior of S -Asymptotically ω -Periodic Solutions for Evolution Equation with Delay

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Hong Qiao ◽  
Qiang Li ◽  
Tianjiao Yuan
Keyword(s):  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Compact Semigroup ◽  
Existence Results ◽  
Asymptotically Periodic ◽  
Abstract Evolution Equation ◽  
Equation With Delay ◽  
Asymptotically Periodic Solutions

This paper is concerned with the abstract evolution equation with delay. Firstly, we establish some sufficient conditions to ensure the existence results for the S -asymptotically periodic solutions by means of the compact semigroup. Secondly, we consider the global asymptotic behavior of the delayed evolution equation by using the Gronwall-Bellman integral inequality involving delay. These results improve and generalize the recent conclusions on this topic. Finally, we give an example to exhibit the practicability of our abstract results.

scholarly journals Properties of Solutions of Generalized Sturm–Liouville Discrete Equations

2021 ◽  
Author(s):  
Janusz Migda ◽  
Magdalena Nockowska-Rosiak ◽  
Małgorzata Migda
Keyword(s):  
Periodic Solutions ◽  
Liouville Equation ◽  
Asymptotic Properties ◽  
Degree Of Approximation ◽  
Liouville Type ◽  
Discrete Equations ◽  
Nonincreasing Sequence ◽  
Asymptotically Periodic ◽  
Sturm Liouville ◽  
Asymptotically Periodic Solutions

AbstractWe consider discrete Sturm–Liouville-type equations of the form $$\begin{aligned} \varDelta (r_n\varDelta x_n)=a_nf(x_{\sigma (n)})+b_n. \end{aligned}$$ Δ ( r n Δ x n ) = a n f ( x σ ( n ) ) + b n . We present a theory of asymptotic properties of solutions which allows us to control the degree of approximation. Namely, we establish conditions under which for a given sequence y which solves the equation $$\varDelta (r_n\varDelta y_n)=b_n$$ Δ ( r n Δ y n ) = b n , the above equation possesses a solution x with the property $$x_n=y_n+\mathrm {o}(u_n)$$ x n = y n + o ( u n ) , where u is a given positive, nonincreasing sequence. The obtained results are applied to the study of asymptotically periodic solutions. Moreover, these results also allow us to obtain some nonoscillation criteria for the classical Sturm–Liouville equation.

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