scholarly journals Galois theories of q-difference equations: comparison theorems

2021 ◽  
Vol 12 (2) ◽  
pp. 11-35
Author(s):  
Lucia Di Vizio ◽  
Charlotte Hardouin
Keyword(s):  
Difference Equations ◽  
Comparison Theorems

scholarly journals Nonoscillation of Second-Order Dynamic Equations with Several Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-34 ◽  
Author(s):  
Elena Braverman ◽  
Başak Karpuz
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Comparison Theorems ◽  
Nonoscillatory Solutions ◽  
Several Delays ◽  
Delay Differential ◽  
Delay Difference Equations ◽  
Delay Difference

Existence of nonoscillatory solutions for the second-order dynamic equation(A0xΔ)Δ(t)+∑i∈[1,n]ℕAi(t)x(αi(t))=0fort∈[t0,∞)Tis investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the caseA0(t)≡1fort∈[t0,∞)Rand for second-order nondelay difference equations (αi(t)=t+1fort∈[t0,∞)N). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitraryA0and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.

scholarly journals A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

2020 ◽  
Vol 8 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed
Keyword(s):  
Fractional Calculus ◽  
Uniqueness Theorem ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Comparison Theorems ◽  
Existence And Uniqueness Theorem ◽  
Discrete Fractional Calculus ◽  
Uncertain Variables ◽  
Forward Difference ◽  
The Relationship

We consider the comparison theorems for the fractional forward h-difference equations in the context of discrete fractional calculus. Moreover, we consider the existence and uniqueness theorem for the uncertain fractional forward h-difference equations. After that the relations between the solutions for the uncertain fractional forward h-difference equations with symmetrical uncertain variables and their α-paths are established and verified using the comparison theorems and existence and uniqueness theorem. Finally, two examples are provided to illustrate the relationship between the solutions.

scholarly journals Oscillation of third-order delay difference equations with negative damping term

2018 ◽  
Vol 72 (1) ◽  
pp. 19
Author(s):  
Martin Bohner ◽  
Srinivasan Geetha ◽  
Srinivasan Selvarangam ◽  
Ethiraju Thandapani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Comparison Theorems ◽  
Asymptotic Behavior Of Solutions ◽  
Delay Difference Equation ◽  
Third Order ◽  
Negative Damping ◽  
The Difference ◽  
Delay Difference Equations ◽  
Delay Difference

The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.<br /><br />

scholarly journals OSCILLATION AND COMPARISON THEOREMS FOR NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (4) ◽  
pp. 343-352
Author(s):  
B. G. ZHANG ◽  
PENGXIANG YAN
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Comparison Theorems ◽  
Qualitative Properties ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Riccati Techniques ◽  
Qualitative Properties Of Solutions

In this paper we study qualitative properties of solutions of the neutral difference equation $$ \Delta(y_n-py_{n-k})+\sum_{i=1}^m q_n^i y_{n-k_i} =0 $$ $$ y_n=A_n \quad \text{ for } n=-M, \cdots, -1, 0$$ where $p \ge 1$, $M =\max\{k, k_1, \cdots, k_m\}$, and $k$, $k_i$, $i =1, \cdots, m$, are nonnegative integers. Riccati techniques are used.  

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