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 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 Half-Linearq-Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Pavel Řehák
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Special Limit ◽  
Bounded Functions ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linearq-difference equationDq(Φ(Dqy(t)))+p(t)Φ(y(qt))=0,t∈{qk:k∈N0}withq>1,Φ(u)=|u|α−1sgn⁡uwithα>1, to behave likeq-regularly varying orq-rapidly varying orq-regularly bounded functions (that is, the functionsy, for which a special limit behavior ofy(qt)/y(t)ast→∞is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

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 Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
E. Messina ◽  
Y. Muroya ◽  
E. Russo ◽  
A. Vecchio
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Methods ◽  
Integral Equations ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Difference Equations ◽  
Discrete Equations

We consider nonlinear difference equations of unbounded order of the formxi=bi−∑j=0iai,jfi−j(xj),  i=0,1,2,…,wherefj(x)  (j=0,…,i)are suitable functions. We establish sufficient conditions for the boundedness and the convergence ofxiasi→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.

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 Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

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