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This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test whether the solutions of the studied equation exhibit oscillatory behavior. To verify the validity of the results we obtained and determine their applicability, we present some examples to confirm the strength and accuracy of our proposed criteria. [ABSTRACT FROM AUTHOR]

In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting criteria are easier to apply than other criteria that have appeared in the literature. We have obtained new oscillation criteria that hold up more robustly upon application. Some examples are presented to illustrate the significance of our main findings. [ABSTRACT FROM AUTHOR]

This work is concerned with necessary and sufficient conditions for oscillation of all solutions of 2-dim first order neutral delay difference system with constant coefficient of the form: Δ x (n) - p x (n - m) y (n) - p y (n - m) = a b c d x (n - α) y (n - β) , n ≥ n 0 by constructing several suitable characteristic equations. Also, an effort has been made to apply some results to nonlinear neutral systems aheading linearized oscillation theory. Some examples are in the discussion to illustrate our results. [ABSTRACT FROM AUTHOR]

In this study, we addressed the nonoscillation of the Sturm--Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm--Liouville type. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation and asymptotic behaviour of even order nonlinear differential equations with mixed nonlinear neutral terms. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first order advanced and/or delay differential equation. We provide novel oscillation criteria that improve, extend, and simplify previously published ones. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

This paper focuses on studying certain oscillatory properties of a new class of half-linear second-order differential equations with an advanced argument in a non-canonical case. By employing some new relations between the solution and its higher derivatives and taking into account the symmetry of positive and negative solutions, we have introduced new criteria to test whether all solutions of the studied equation exhibit oscillatory behavior. Our study aims to expand and enhance previous results, helping to understand these properties in the specified context. The results obtained are confirmed and clarified through an example involving Euler-type equations. [ABSTRACT FROM AUTHOR]

This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test whether the solutions of the studied equation exhibit oscillatory behavior. To verify the validity of the results we obtained and determine their applicability, we present some examples to confirm the strength and accuracy of our proposed criteria.

In the current paper, we aim to study the oscillatory behavior of a new class of third-order differential equations. In the present study, we are interested in a better understanding of the relationships between the solutions and their derivatives. The recursive nature of these relationships enables us to obtain new criteria that ensure the oscillation of all solutions of the studied equation. In comparison with previous studies, our results are more general and include models in a wider range of applications. Furthermore, our findings are also significant because no additional restrictive conditions are required. The presented examples illustrate the significance of the results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation and asymptotic behavior of solutions of third-order nonlinear neutral differential equations with a middle term and mixed nonlinear neutral terms in the case of the canonical operator. We establish several oscillation criteria that guarantee that all solutions are oscillatory or converge to zero. The given results are obtained by applying the comparison method, the Riccati transformation and the integral averaging technique. The results improve significantly and extend existing ones in the literature. Finally, illustrative examples are given. [ABSTRACT FROM AUTHOR]

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form \Delta\left[\matrix x(n)+p(n)x(n-m) y(n)+p(n)y(n-m) \right]= \left[\matrix a(n) & b(n) c(n) & d(n) \right]\left[\matrix x(n-\alpha) y(n-\beta) \right] are established, where $m>0$, $\alpha\geq0$, $\beta\geq0$ are integers and $a(n)$, $b(n)$, $c(n)$, $d(n)$, $p(n)$ are sequences of real numbers.

In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting criteria are easier to apply than other criteria that have appeared in the literature. We have obtained new oscillation criteria that hold up more robustly upon application. Some examples are presented to illustrate the significance of our main findings.

This paper focuses on studying certain oscillatory properties of a new class of half-linear second-order differential equations with an advanced argument in a non-canonical case. By employing some new relations between the solution and its higher derivatives and taking into account the symmetry of positive and negative solutions, we have introduced new criteria to test whether all solutions of the studied equation exhibit oscillatory behavior. Our study aims to expand and enhance previous results, helping to understand these properties in the specified context. The results obtained are confirmed and clarified through an example involving Euler-type equations.

In this work, we establish the necessary and sufficient conditions for oscillation of a class of functional differential equations of the form ((x(t)+p(t)x(to))' + q(t)(x(tr)) + v(t)(x( - )) = 0 of a neutral coefficient p(t), by using the Knaster-Tarski fixed point theorem and Banach's fixed point theorem. [ABSTRACT FROM AUTHOR]

In the present article, we set the if and only if conditions for the solutions of the class of neutral impulsive delay second-order differential equations. We consider two cases when it is non-increasing and non-decreasing for quotient of two positive odd integers. Our main tool is the Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form ... are established, where m > 0, α ≥ 0, β ≥ 0 are integers and a(n), b(n), c(n), d(n), p(n) are sequences of real numbers. [ABSTRACT FROM AUTHOR]

In the present article, we set the if and only if conditions for the solutions of the class of neutral impulsive delay second-order differential equations. We consider two cases when it is non-increasing and non-decreasing for quotient of two positive odd integers. Our main tool is the Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem. [ABSTRACT FROM AUTHOR]

In this paper we continue with our investigation of principal and antiprincipal solutions at infinity solutions of a dynamic symplectic system. The paper is a continuation of part I appeared in Differential Equations and Applications in 2022, where we have presenteded a theory of genera of conjoined bases for symplectic dynamic systems on time scales and its connections with principal solutions at infinity and antiprincipal solutions at infinity for these systems together with some basic properties of this new concept on time scales. Here we provide a characterization of all principal solutions of dynamic symplectic system at infinity in the given genus in terms of the initial conditions and a fixed principal solution at infinity from this genus. Further, we provide a characterization of all antiprincipal solutions of dynamic symplectic system at infinity in the given genus in terms of the initial conditions and a fixed principal solution at infinity from this genus. We also establish mutual limit properties of principal and antiprincipal solutions at infinity. [ABSTRACT FROM AUTHOR]

The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.

In this work, we present sufficient conditions for the oscillation of a class of forced second order neutral impulsive difference equations with deviating arguments and fixed moment of impulse effect. Furthermore, we prove the existence of a nonoscillatory solution to the aforementioned problem using Krasnosel'skii's fixed point theorem. To demonstrate our findings, we provide some illustrative examples. We point out that the results are new even for the neutral difference equations without impulse. [ABSTRACT FROM AUTHOR]

This work is concerned about the necessary and sufficient conditions for oscillation of solutions of 2-dimensional nonlinear neutral delay differential systems of the form: ... where m > 0, α ≥ 0, β ≥ 0 are real constants, aij(s), i, j = 1, 2 and p(s) are real-valued continuous functions, and f, g ∊ C(R,R). [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions are establish of the solutions to second-order delay differential equations of the form (r(t)(x′(t))λ)′ + ∑i=1m qi (t) fi (x(σi(t))) = 0 for t ≥ t0, We consider two cases when fi(u)/uβ is non-increasing for β < γ, and non-decreasing for β > γ where β and γ are the quotient of two positive odd integers. Our main tool is Lebesgue's Dominated Convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem. [ABSTRACT FROM AUTHOR]

In the current paper, we aim to study the oscillatory behavior of a new class of third-order differential equations. In the present study, we are interested in a better understanding of the relationships between the solutions and their derivatives. The recursive nature of these relationships enables us to obtain new criteria that ensure the oscillation of all solutions of the studied equation. In comparison with previous studies, our results are more general and include models in a wider range of applications. Furthermore, our findings are also significant because no additional restrictive conditions are required. The presented examples illustrate the significance of the results.

Abstract In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality’s weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero.

In this paper we present an existence result for conjoined bases of nonoscillatory linear Hamiltonian systems on an unbounded interval, which have prescribed numbers of left and right proper focal points. The result is based on a singular Sturmian separation theorem on an unbounded interval by the authors (2019) and it extends a similar property, which was recently derived for linear Hamiltonian systems on compact interval (2021). At the same time it is new even for completely controllable linear Hamiltonian systems, including higher order Sturm–Liouville differential equations. As the main tools we use the comparative index and properties of the minimal principal solution at infinity, which serves as the reference solution for calculating the numbers of proper focal points. We also provide several examples illustrating the presented theory. [ABSTRACT FROM AUTHOR]

This work is concerned about the necessary and sufficient conditions for oscillation of solutions of 2-dimensional nonlinear neutral delay difference systems of the form: Δ[x(n)+p(n)x(n−m)y(n)+p(n)y(n−m)]=[a(n)b(n)c(n)d(n)] [f(x(n−α))g(y(n−β))],\Delta \left[ {\matrix{ {x\left( n \right) + p\left( n \right)x\left( {n - m} \right)} \hfill \cr {y\left( n \right) + p\left( n \right)y\left( {n - m} \right)} \hfill \cr } } \right] = \left[ {\matrix{ {a\left( n \right)} \hfill & {b\left( n \right)} \hfill \cr {c\left( n \right)} \hfill & {d\left( n \right)} \hfill \cr } } \right]\,\,\left[ {\matrix{ {f\left( {x\left( {n - \alpha } \right)} \right)} \hfill \cr {g\left( {y\left( {n - \beta } \right)} \right)} \hfill \cr } } \right], where m > 0, α ≥ 0, β ≥ 0 are integers, a(n), b(n), c(n), d(n), p(n) are real sequences and f, g ∈ 𝒞(ℝ, ℝ).

Abstract The present work discusses the qualitative behaviour of solutions of third-order difference equations of the form: $$\begin{aligned} w(l+3)+a(l)w(l+2)+b(l)w(l+1)+c(l)w(l)=0,\,l\ne \theta _{k},\,l\ge l_{0} \end{aligned}$$ w ( l + 3 ) + a ( l ) w ( l + 2 ) + b ( l ) w ( l + 1 ) + c ( l ) w ( l ) = 0 , l ≠ θ k , l ≥ l 0 subject to the impulsive condition $$\begin{aligned} w(\theta _{k})=\alpha _{k} w(\theta _{k}-1),\, k\in {\mathbb {N}}. \end{aligned}$$ w ( θ k ) = α k w ( θ k - 1 ) , k ∈ N . Our state of the art is the inequality technique under the control of fixed moments of impulsive effect. We give some numerical examples to illustrate our findings.

In this study, we present an investigation of the asymptotic behavior of solutions of sum-difference equations. Based on some mathematical inequalities, we have obtained our results. The obtained results can apply to some fractional type difference equations as well.

The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker–Hill-type equations is also considered in this paper. The Whittaker–Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, we have discussed the oscillation properties of first order neutral impulsive difference equations with constant coefficients by using pulsatile constant. Also, we have made an effort to apply our constant coefficient results to nonlinear impulsive difference equations with variable coefficients. [ABSTRACT FROM AUTHOR]

This study is devoted to the investigation on the oscillation problem of nonlinear delay difference equations of the form x (n + 1) - a x (n) + f (n , x (n - k)) = 0 , n = 0 , 1 , 2 , ⋯ . Here, a is a positive number, k is a positive integer, and f(n, x) is a continuous function with respect to the second variable that satisfies suitable conditions. A pair of oscillation and nonoscillation theorems for the equation is established. This study also improves the lower bounded condition in an oscillation theorem and the upper bounded condition in a nonoscillation theorem for the equation when the nonlinear function f(n, x) is separable. The obtained theorems are proved by transforming the equation into a certain difference system and using a phase plane analysis. [ABSTRACT FROM AUTHOR]

Schauder's fixed point theorem and its applications to delay differential equations (DDEs) cannot be over emphasized. In this work, interesting results regarding the sufficient conditions for nonoscillation of more generalized forms of DDEs are established and provides improvements on the results obtained in the past. However, applying the theorem and the characteristic equation of DDE with constant coefficients helps to determine those conditions. Lastly, to ascertain a claim from an existing literature for a solution that is nonoscillatory and has such larger solutions, comparison theorem is used and the results are found to be true. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the oscillatory properties of two fourth order differential equations in dependence on boundary behavior of its coefficients at infinity. These properties are established based on two-sided estimates of the least constant of a certain weighted differential inequality.

In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality's weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero. [ABSTRACT FROM AUTHOR]

In this paper, we discuss mutual interrelations between the notions of weak disconjugacy and weak controllability for linear Hamiltonian differential systems. These notions have been used in connection with the study of exponential dichotomy, nonoscillation, and dissipative control processes for these systems [e.g. (Johnson et al., in: Nonautonomous linear Hamiltonian systems: oscillation, spectral theory and control developments in mathematics, Springer, Cham, 2016)]. As our main results, we derive characterizations of the weak controllability and weak disconjugacy in terms of properties of certain subspaces arising in the recently introduced theory of genera of conjoined bases for linear Hamiltonian systems (Šepitka in J Dyn Differ Equ 32(3):1139–1155, 2020). We also present new results regarding the zero value of the maximal order of abnormality of the system in terms of a weak controllability condition, or in terms of a weak disconjugacy condition when the system is nonoscillatory and satisfies the Legendre condition. In our accompanying comments, we highlight the connections of the theory of genera of conjoined bases with the existence of principal solutions at infinity, which arise in the study of weakly disconjugate linear Hamiltonian systems. The results in this paper may be regarded as a completion and clarification of the previous considerations in the literature about the weak disconjugacy and weak controllability conditions for linear Hamiltonian systems [e.g. (Fabbri et al. in: J Math Anal Appl 380(2):853–864, 2011), (Johnson et al., in Nonautonomous linear Hamiltonian systems: oscillation, spectral theory and control developments in mathematics, Springer, Cham, 2016)]. [ABSTRACT FROM AUTHOR]

The nonoscillation of Mathieu-type half-linear differential equations was investigated. The particular equation under consideration is an extension of the Mathieu equation, which has been widely applied in mechanical and electrical engineering. The investigation led to the main finding that all nontrivial solutions of the Mathieu-type half-linear differential equations are nonoscillatory under simple parametric conditions. Proving the finding requires a simple nonoscillation theorem to compare the two equations. As another application of the findings, by using a simple nonoscillation comparison theorem, we propose that all nontrivial solutions of the half-linear Whittaker–Hill-type equation do not oscillate. [ABSTRACT FROM AUTHOR]

In this paper, we present several sufficient conditions for the existence of nonoscillatory solutions to the following third order neutral type difference equation \Delta^3(x_n+a_n x_{n-l} +b_n x_{n+m})+p_n x_{n-k} - q_n x_{n+r}=0,\quad n\geq n_0 via Banach contraction principle. Examples are provided to illustrate the main results. The results obtained in this paper extend and complement some of the existing results.

Abstract The authors present necessary and sufficient conditions for the oscillation of a class of second order non-linear neutral dynamic equations with non-positive neutral coefficients by using Krasnosel’skii’s fixed point theorem on time scales. The nonlinear function may be strongly sublinear or strongly superlinear.

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

The present work discusses the qualitative behaviour of solutions of third-order difference equations of the form: w (l + 3) + a (l) w (l + 2) + b (l) w (l + 1) + c (l) w (l) = 0 , l ≠ θ k , l ≥ l 0 subject to the impulsive condition w (θ k) = α k w (θ k - 1) , k ∈ N. Our state of the art is the inequality technique under the control of fixed moments of impulsive effect. We give some numerical examples to illustrate our findings. [ABSTRACT FROM AUTHOR]

In this work, sufficient conditions are established for oscillation of all unbounded solutions of a class of fourth order neutral differential equations of the form (r (t) (y (t) + p (t) y (t − τ)) ″) ″ + q (t) G (y (t − α)) − h (t) H (y (t − σ)) = 0 under the non-canonical type assumption ∫ 0 ∞ t r (t) d t < ∞ for various ranges of p(t) with ∣p(t)∣ < ∞. [ABSTRACT FROM AUTHOR]

In this work, necessary and sufficient conditions for the oscillation of a class of second order neutral impulsive systems are established. [ABSTRACT FROM AUTHOR]

We have established sufficient conditions for oscillation of a class of first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect.