Noncanonical Operators Research Articles

Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients

Abstract In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.

On the monotonic properties and oscillatory behavior of solutions of neutral differential equations

Abstract In this work, we study new asymptotic properties of positive solutions of the even-order neutral differential equation with the noncanonical operator. The new properties are iterative, which means they can be used several times. We also use these properties to obtain new criteria for oscillation of the studied equation.

Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.

Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms

The study of the symmetric properties of differential equations is essential for identifying effective methods for solving them. In this paper, we examine the oscillatory behavior of solutions of Emden–Fowler-type mixed non-linear neutral differential equations with both canonical and non-canonical operators. By utilizing integral conditions and the integral averaging method, we present new sufficient conditions to ensure that all solutions are oscillatory. Our results enhance and extend previous findings in the literature and are illustrated with suitable examples to demonstrate their effectiveness.

Second-Order Dynamic Equations with Noncanonical Operator: Oscillatory Behavior

The present article aims to study the oscillatory properties of a class of second-order dynamic equations on time scales. We consider during this study the noncanonical case, which did not receive much attention compared to the canonical dynamic equations. The approach adopted depends on converting the noncanonical equation to a corresponding canonical equation. By using this transformation and based on several techniques, we create new, more effective, and sharp oscillation criteria. Finally, we explain the effectiveness and importance of the results by applying them to some special cases of the studied equation.

Oscillation of Second Order Nonlinear Neutral Differential Equations

The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential equations with noncanonical operators are established. The results extend and improve on those reported in the literature. Moreover, some examples are provided to show the significance of the results.

Even-Order Neutral Delay Differential Equations with Noncanonical Operator: New Oscillation Criteria

The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and reflection of developments in the study of second-order equations. We also derive criteria for improving conditions that exclude the decreasing positive solutions of the considered equation.

New Monotonic Properties of the Class of Positive Solutions of Even-Order Neutral Differential Equations

In this study, new asymptotic properties of positive solutions of the even-order neutral delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Using these properties, we obtain new criteria to exclude a class from the positive solutions of the studied equation, using the comparison principles.

New Sufficient Conditions for Oscillation of Second-Order Neutral Delay Differential Equations

In this work, new sufficient conditions for the oscillation of all solutions of the second-order neutral delay differential equations with the non-canonical operator are established. Using a generalized Riccati substitution, we obtained criteria that complement and extend some previous results in the literature.

On nonlinear impulsive differential systems with canonical and non-canonical operators

The investigation on the oscillatory and asymptotic properties of differential equations plays an important role in many applications, such as engineering, physics and medicine. In this work, we establish some new sufficient conditions for both the oscillatory and asymptotic behavior of the solutions to a second-order neutral impulsive differential system (IDS, for short) with deviating argument and fixed moments of impulse effect under both canonical and non-canonical operators with various values of the neutral coefficient. Also, we obtain the condition for the existence of a positive solution of the system under consideration. Finally, two examples are provided to illustrate the main results.

Delay differential equation of fourth-order: Asymptotic analysis and oscillatory behavior

The objective of this work is to offer sufficient conditions for the oscillation of all solutions of fourth-order delay differential equations with non-canonical operator. By establishing sufficient conditions for nonexistence of positive solutions, we attain oscillation of all solutions of the studied equations. Our results substantially extend and complement a number of existing ones.

New Asymptotic Properties of Positive Solutions of Delay Differential Equations and Their Application

In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison.

Oscillation and Asymptotic Properties of Differential Equations of Third-Order

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator (a(ι)[(b(ι)x(ι)+p(ι)x(ι−τ)′)′]β)′+∫cdq(ι,μ)xβ(σ(ι,μ))dμ=0, where ι≥ι0 and w(ι):=x(ι)+p(ι)x(ι−τ). New oscillation results are established by using the generalized Riccati technique under the assumption of ∫ι0ιa−1/β(s)ds<∫ι0ι1b(s)ds=∞asι→∞. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

Simplified and improved criteria for oscillation of delay differential equations of fourth order

An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form (a(ξ)((v(ξ)+b(ξ)v(ϑ(ξ)))′))′+c(ξ)G1(v(κ(ξ)))+d(ξ)G2(v(ς(ξ)))=0 under canonical and non-canonical operators, that is, ∫ξ0∞dξa(ξ)=∞ and ∫ξ0∞dξa(ξ)<∞. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results.

More Effective Results for Testing Oscillation of Non-Canonical Neutral Delay Differential Equations

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results.

On the Oscillatory Properties of Solutions of Second-Order Damped Delay Differential Equations

In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example is presented in order to illustrate the significance of the results.

Oscillation of a Class of Third-Order Neutral Differential Equations with Noncanonical Operators

The aim of this paper is to complement existing oscillation results for third-order neutral advanced differential equations under the condition of $$\gamma >0$$ ; in particular, the sufficient conditions are given in different way when $$\gamma =1$$ . Our main idea is by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Then, combining with the results which guarantee the equation almost oscillation, we establish sufficient condition for oscillation of all solutions.

Neutral differential equations with noncanonical operator: Oscillation behavior of solutions

The objective of this work is to study the oscillatory behavior of neutral differential equations with several delays. By using both Riccati substitution technique and comparison with delay equations of first-order, we establish new oscillation criteria. Our new criteria are simplifying and complementing some related results that have been published in the literature. Moreover, some examples are given to show the applicability of our results.

Oscillation criteria of third-order nonlinear neutral delay difference equations with noncanonical operators

In this paper, we present some new oscillation criteria for nonlinear neutral difference equations of the form ?(b(n)?(a(n)?z(n))) + q(n)x?(?(n)) = 0 where z(n) = x(n) + p(n)x(?(n)),? > 0, b(n) > 0, a(n) > 0, q(n) ? 0 and p(n) > 1. By summation averaging technique, we establish new criteria for the oscillation of all solutions of the studied difference equation above. We present four examples to show the strength of the new obtained results.