Yasar Bolat

Institute of Mathematics of the Czech Academy of Sciences

In this study, we examine the trichotomy of non-oscillatory solution for the nonlinear first-order neutral difference equation ∆_a (x_n-a^(n+1) x_(n-1) )+∆_a (x_(n-1)/b_(n-1) )+q_n f(x_(n-τ) )=0,n∈N_max{1,τ} , where ∆_a x_n=x_(n+1)-ax_n,τ∈N,a∈R^+ with ∆_a^m Δ=∆_a^(m-1) (∆_a ), aⁿ is a general term of exponential sequence, (q_n) is real valued sequences. The accuracy of the primary findings is demonstrated by examples.

In this manuscript, we studied stability of fractional order delayed differential equations with two fractional terms and gave some criteria on the stability of equation (1.1) - (1.2). The type of equation that we have considered in this article is the type of equation that has more general and more complex application areas than the equations discussed so far, and the examination of such an equation type colud not been seen in the literature until now. We use some well known inequalities for the proof of our results. We also provide some examples to check the accuracy of our results.

In this manuscript, new oscillation criteria are given for solutions of h‐order delay difference equation in the equation (1.1). There, four cases of are considered: for , as oscillatory periodic sequences with periods and and a quickly oscillating sequence such that . It is well‐known that coefficient functions have a very important role in the behavior of the solutions of an equation. The coefficient sequence has been considered as positive sequences in all oscillation criteria so far. But it is considered as oscillatory periodic sequences and quickly oscillatory sequences for the first time in this study. We use a new technique to prove our results. Also, some examples are provided to demonstrate the effectiveness of the main results.

In this manuscript new oscillation criteria are given for solutions of h-order delay difference equation in the form ∆ E n + p n E n − α = 0 . Here, four cases of p are considered; for α ∈N , as oscillatory periodic sequences with periods 2 α, (2 α +1) and ( α +1) , and a quickly oscillating sequence such that p n p n + 1 < 0 . It is well known that coefficient functions has a very important role in the behavior of the solutions of an equation. The coefficients sequence p has been considered as positive sequences in all oscillation criteria so far. But It is considered as a oscillatory periodic sequences and quickly oscillatory sequences for the first time in this study. We use a new technique to prove our results. Also some examples are provided to demonstrate the effectiveness of the main results.

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results.

In this paper, we derive new interval oscillation criteria for impulsive conformable fractional differential equations having fixed moments of impulse actions. The results are extended to a more general class of nonlinear impulsive conformable fractional differential equations. Examples are also given to illustrate the relevance of the result.