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 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 Oscillation of nonlinear third order perturbed functional difference equations

2019 ◽  
Vol 6 (1) ◽  
pp. 57-64 ◽  
Author(s):  
P. Dinakar ◽  
S. Selvarangam ◽  
E. Thandapani
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Third Order ◽  
All Solutions ◽  
Functional Difference Equation ◽  
Functional Difference Equations

AbstractThis paper deals with oscillatory and asymptotic behavior of all solutions of perturbed nonlinear third order functional difference equation\Delta {\left( {{b_n}\Delta ({a_n}(\Delta {x_n}} \right)^\alpha })) + {p_n}f\left( {{x_{\sigma \left( n \right)}}} \right) = g\left( {n,{x_n},{x_{\sigma (n)}},\Delta {x_n}} \right),\,\,\,n \ge {n_0}.By using comparison techniques we present some new sufficient conditions for the oscillation of all solutions of the studied equation. Examples illustrating the main results are included.

scholarly journals The Asymptotic Behavior of Solutions of Second order Difference Equations With Damping Term

2018 ◽  
Vol 14 (2) ◽  
pp. 7806-7811
Author(s):  
Jai Kumar S ◽  
K. Alagesan
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Damping Term ◽  
Order Difference ◽  
Keywords And Phrases ◽  
Second Order Difference Equation ◽  
Order Difference Equation

  The author presents some sufficient conditions for second order difference equation with damping term of the form                                                                             ^(an ^(xn + cxn-k)) + pn^xn + qnf(xn+1-l) = 0 An example is given to illustrate the main results. 2010 AMS Subject Classification: 39A11 Keywords and Phrases: Second order, difference equation, damping term.

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 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 third order delay difference equations with a negative middle term

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. H. Saker ◽  
S. Selvarangam ◽  
S. Geetha ◽  
E. Thandapani ◽  
J. Alzabut
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Delay Difference Equation ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Middle Term ◽  
The Third ◽  
Delay Difference Equations ◽  
Delay Difference

AbstractIn this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term $$ \Delta \bigl(a_{n}\Delta (\Delta w_{n})^{\alpha } \bigr)-p_{n}(\Delta w_{n+1})^{ \alpha }-q_{n}h(w_{n-l})=0,\quad n\geq n_{0}, $$ Δ ( a n Δ ( Δ w n ) α ) − p n ( Δ w n + 1 ) α − q n h ( w n − l ) = 0 , n ≥ n 0 , are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration.

scholarly journals Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2751-2770
Author(s):  
Aleksandra Kapesic ◽  
Jelena Manojlovic
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Order Difference ◽  
Varying Coefficients ◽  
Regularly Varying ◽  
Necessary And Sufficient

Positive decreasing solutions of the nonlinear difference equation ?(pn|?xn|?-1?xn)=qn|xn+1|?-1xn+1, n ? 1, ? > ? > 0, are studied under the assumption that p; q are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.

scholarly journals Global behavior of a third order difference equation with quadratic term

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
R. Abo-Zeid ◽  
H. Kamal
Keyword(s):  
Difference Equation ◽  
Global Behavior ◽  
Quadratic Term ◽  
Real Numbers ◽  
Third Order ◽  
Order Difference ◽  
Initial Values ◽  
Order Difference Equation ◽  
Admissible Solutions ◽  
The Difference

AbstractIn this paper, we solve and study the global behavior of the admissible solutions of the difference equation $$\begin{aligned} x_{n+1}=\frac{x_{n}x_{n-2}}{-ax_{n-1}+bx_{n-2}}, \quad n=0,1,\ldots , \end{aligned}$$ x n + 1 = x n x n - 2 - a x n - 1 + b x n - 2 , n = 0 , 1 , … , where $$a, b>0$$ a , b > 0 and the initial values $$x_{-2}$$ x - 2 , $$x_{-1}$$ x - 1 , $$x_{0}$$ x 0 are real numbers.

scholarly journals Global behavior of a third order difference equation

2012 ◽  
Vol 43 (3) ◽  
pp. 375-384
Author(s):  
Raafat Abo-zeid
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Global Behavior ◽  
Real Numbers ◽  
Third Order ◽  
Order Difference ◽  
Order Difference Equation ◽  
The Difference

The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation\[x_{n+1}=\frac{A+Bx_{n-1}}{C+Dx_{n}x_{n-2}},\qquad n=0,1,2,\ldots\] where $A,B$ are nonnegative real numbers and$C, D>0$.

scholarly journals Quasi-adjoint third order difference equations: oscillatory and asymptotic behavior

1986 ◽  
Vol 9 (4) ◽  
pp. 781-784 ◽  
Author(s):  
B. Smith
Keyword(s):  
Asymptotic Behavior ◽  
Adjoint Equation ◽  
Asymptotic Properties ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Oscillatory Solutions ◽  
Order Difference ◽  
Necessary And Sufficient

In this paper, asymptotic properties of solutions ofΔ3Vn+Pn−1Vn+1=0          (E+)are investigated via the quasi-adjoint equationΔ3Un+PnUn+2=0.             (E−)A necessary and sufficient condition for the existence of oscillatory solutions of(E+)is given. An example showing that it is possible for(E+)to have only nonoscillatory solutions is also given.

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