scholarly journals Ulam stability results for boundary value problem of fractional difference equations

2020 ◽  
pp. 219-230
Author(s):  
A. George Maria Selvam ◽  
R. Dhineshbabu
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Boundary Value ◽  
Ulam Stability ◽  
Fractional Difference ◽  
Stability Results

Boundary value problems have wide applications in science and technology. This paper is concerned with various kinds of Ulam stability analysis for the nonlinear discrete boundary value problem of fractional order $\sigma\in(2,3]$ with Riemann-Liouville fractional difference operator. Finally, some examples are presented to illustrate the main results.

scholarly journals Hyers - Ulam Stability Results for Discrete Antiperiodic Boundary Value Problem with Fractional Order 2 <  3

2019 ◽  
Vol 9 (1) ◽  
pp. 4997-5003 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Fractional Order ◽  
Difference Operator ◽  
Boundary Value ◽  
Ulam Stability ◽  
Fractional Difference ◽  
Stability Results

In this present work, we investigate Ulam stability for the following nonlinear discrete antiperiodic boundary value problem with fractional order of the form 0 ( ) = 1, ( 1) ,   C k v k k v k          for   0 k L L    [0, 2] = 0,1,..., 2  , with boundary conditions v v L ( 3) = ( )      ,     v v L ( 3) = ( )   , 2 2     v v L ( 3) = ( )   , where 2 :[ 2, ] L             is a continuous and 0 C k   is the Caputo fractional difference operator with order 2 < 3   . Finally, the main results are illustrated by some examples.

scholarly journals Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Fulai Chen ◽  
Yong Zhou
Keyword(s):  
Boundary Value Problem ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Ulam Stability ◽  
Stability Of Solutions ◽  
Fractional Boundary Value Problem ◽  
Fractional Difference ◽  
Fractional Difference Equations

We discuss the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations. Two examples are also provided to illustrate our main results.

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 204-217
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko ◽  
Nikita Popov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Oscillations ◽  
Bounded Operator ◽  
Boundary Value ◽  
Homogeneous Part ◽  
Solvability Conditions ◽  
Engineering Control ◽  
Generalized Green Operator

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

scholarly journals Discrete fractional order two-point boundary value problem with some relevant physical applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
R. Dhineshbabu ◽  
S. Rashid ◽  
M. Rehman
Keyword(s):  
Boundary Value Problem ◽  
Fractional Order ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Difference Operators ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Rassias Stability

Abstract The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

Sign Properties of Green's Functions For Two Classes of Boundary Value Problems

1987 ◽  
Vol 30 (1) ◽  
pp. 28-35 ◽  
Author(s):  
P. W. Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Difference Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Value ◽  
Partial Derivatives

AbstractLet G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

scholarly journals Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Fractional Operators ◽  
Damping Term ◽  
Fractional Difference ◽  
Riccati Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Positive Integers

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

scholarly journals Boundary-Value Problems for Differential-Difference Equations with Incommeasurable Shifts of Arguments Reducible to Nonlocal Problems

2019 ◽  
Vol 65 (4) ◽  
pp. 613-622
Author(s):  
E. P. Ivanova
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Original Problem ◽  
Difference Operator ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlocal Problems ◽  
Higher Order Terms ◽  
The Difference

We consider boundary-value problems for differential-difference equations containing incommeasurable shifts of arguments in higher-order terms. We prove that in the case of finite orbits of boundary points generated by the set of shifts of the difference operator, the original problem is reduced to the boundary-value problem for differential equation with nonlocal boundary conditions.

scholarly journals Impulsive Antiperiodic Boundary Value Problems for Nonlinearqk-Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Lihong Zhang ◽  
Bashir Ahmad ◽  
Guotao Wang
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Uniqueness Of Solutions

We show the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsiveqk-difference equations by applying some well-known fixed point theorems. An example is presented to illustrate the main results.

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