Riccati Transformation Research Articles

Oscillatory and Asymptotic Criteria for a Fifth-Order Fractional Difference Equation

In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms.

Riccati Transformation and Non-Oscillation Criterion for Half-Linear Difference Equations

Riccati Transformation and Non-Oscillation Criterion for Half-Linear Difference Equations

Oscillatory behavior of solutions of second-order non-linear differential equations with mixed non-linear neutral terms

This study primarily seeks to expand upon these developments by encompassing neutral differential equations of mixed type, incorporating both delay and advanced terms, particularly in the case of the canonical operator. The presented results are derived from the application of the comparison method, Riccati transformation, and integral averaging technique. These methodologies lead to substantial improvements and extensions of existing results found in the literature. Additionally, illustrative examples are provided to demonstrate the practical implications of the developed criteria.

Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm

Abstract We study linear differential equations whose coefficients consist of products of powers of natural logarithm and general continuous functions. We derive conditions that guarantee the non-oscillation of all non-trivial solutions of the treated type of equations. The conditions are formulated as a non-oscillation criterion, which is the counterpart of a previously obtained oscillation theorem. Therefore, from the presented main result, it follows that the analysed equations are conditionally oscillatory. The used method is based on averaging techniques for the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This article is finished by new corollaries and examples.

Some Oscillatory Criteria for Second-Order Emden–Fowler Neutral Delay Differential Equations

In this paper, by using the Riccati transformation and integral inequality technique, we establish several oscillation criteria for second-order Emden–Fowler neutral delay differential equations under the canonical case and non-canonical case, respectively. Compared with some recent results reported in the literature, we extend the range of the neutral coefficient. Therefore, our results generalize to some of the results presented in the literature. Furthermore, several examples are provided to illustrate our conclusions.

New Oscillation Criteria for Sturm–Liouville Dynamic Equations with Deviating Arguments

The aim of this study is to refine the known Riccati transformation technique to provide new oscillation criteria for solutions to second-order dynamic equations over time. It is important to note that the convergence or divergence of some improper integrals on time scales depends not only on the integration function but also on the integration time scale. Therefore, there has been a motivation to find new oscillation criteria that can be applicable regardless of whether ∫ζ0∞Δξa(ξ) is convergent or divergent, in contrast to what has been followed in most previous works in the literature. We have provided an example to illustrate the significance of the obtained results.

On the Oscillatory Behavior of Solutions of Second-Order Non-Linear Differential Equations with Damping Term

In this paper, we discuss the oscillatory behavior of solutions of two general classes of nonlinear second-order differential equations. New criteria are obtained using Riccati transformations and the integral averaging techniques. The obtained results improve and generalize some recent criteria in the literature. Moreover, a traditional condition is relaxed. Three examples are given to justify the results.

Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples.

Oscillatory and Property Conditions for Third Order Difference Equation with Negative Middle-Term

This paper presents a detailed analysis of the oscillation and Property A conditions for third-order advanced difference equations with negative middle terms. The authors employ the generalized Riccati transformation technique and summation by parts to establish sufficient criteria for the solutions to be oscillatory or have Property A. The main results and their proofs are presented clearly and concisely. The difference equations have been a topic of interest in various scientific fields such as economics, physics, and mathematical biology. The results presented in this paper can be used to inform the development of more accurate models for real-world problems in various scientific fields. Additionally, this paper explores the economic theory of business cycles and its mathematical foundations, specifically Samuelson's model, which outlines five potential trajectories or trends that economic activity can follow based on different combinations of the marginal propensity to consume multiplier parameter(α) and the accelerator coefficient (β).

A class of time-varying differential equations for vibration research and application

<p>As a pivotal branch within the realm of differential equations, the theory of oscillation holds a crucial position in the exploration of natural sciences and the construction of modern control theory frameworks. Despite the extensive research conducted globally, focusing on individual or combined analyses of key elements such as explicit damping terms, positive and negative coefficients, time-varying delays, and nonlinear neutral terms, systematic investigations into the oscillatory behavior of even-order differential equations that concurrently embody these four complex characteristics remain scarce. This paper, by establishing reasonable assumptions, innovatively presents two crucial criteria, aiming to preliminary delve into the oscillation patterns of even-order differential equations under specific complex settings. In the course of the study, a variety of mathematical techniques, such as Riccati transformation, calculus scaling methods, and partial integration, have been utilized by the researchers to perform the necessary derivations and confirmations.</p>

Fourth-order neutral dynamic equations oscillate on timescales with different arguments

<p>The theory of neutral dynamic equations on timescales was based to unify the study of differential and difference equations. The article described several oscillating criteria that will be developed for fourth-order-neutral dynamic equations in the presence of various types of arguments on timescales. The goal was to establish all necessary conditions for the solutions of these models to be oscillatory. To construct observation values, ideas from [Y. Sui and Z. Han, Oscillation of second order neutral dynamic equations with deviating arguments on time scales, <italic>Adv. Differ. Equ.</italic>, 10 (2018)] were used. The research seeked to provide sufficient criteria that ensured the oscillation of solutions to these complex dynamic equations using a technique Riccati transformations generalized, emphasizing their importance in the study of oscillatory processes within various scientific and engineering contexts.</p>

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.

Retraction notice to ‘‘Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales’’ [J. Amr. Uni. 3 (2023) 75 – 100

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamicequations. Applying suitable comparison theorems and by a Riccati transformation technique, weestablish some new sufficient conditions which insure that every solution of this equation either oscillatesor converges to zero. Our results not only unify the oscillation of third order nonlinear differential anddifference equations but also can be applied to different types of time scales with

New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new classes of equations that were not covered by the previous criteria. According to symmetry between the negative and positive solutions of the studied equation, we present some new criteria that ensure the oscillation of all solutions by using the Riccati transformation and comparison method. We provide an example to illustrate our main results.

Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamic equations. Applying suitable comparison theorems and by a Riccati transformation technique, we establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero. Our results not only unify the oscillation of third order nonlinear differential and difference equations but also can be applied to different types of time scales with

Oscillation of linear and half-linear difference equations via modified Riccati transformation

In this paper, we introduce a new oscillation criterion for Euler type half-linear difference equations with positive coefficients which are asymptotically periodic and bounded, respectively. As the mean of proving the criterion, the generalized adapted Riccati transformation is used. The presented main result is the oscillatory counterpart of a previously obtained non-oscillation criterion for the treated equations. Therefore, the result shows that the analyzed equations are conditionally oscillatory. Then we emphasize the novelty of the result in the linear case by several consequences.

Oscillation criteria for second-order delay dynamic equations with a sub-linear neutral term on time scales

This paper addresses the oscillatory behavior of the solutions of second-order dynamic equations with a sublinear neutral term. Using Riccati transformation and comparison principles, we obtain new oscillation criteria. The obtained results essentially improve, complement, and simplify some of the previous ones in the literature. Some examples have been provided herein to illustrate our main results.

Oscillation criteria for a class of half-linear neutral conformable differential equations

The main aim of this note is to obtain new oscillation criteria for a certain class of half-linear neutral conformable differential equations by the method of comparison and Riccati transformation technique. A suitable example is given to illustrate our new results.

Oscillation of Third-Order Nonlinear Generalized Difference Equation with Multiple Neutral Terms

In this paper, the authors discuss the oscillatory behaviour of a third order generalized difference equation with multiple neutral terms. We have also applied Riccati transformation and Philo type technique to derive new oscillation criteria for the difference equation in question. Suitable examples are provided to validate our main results.

On the oscillation and non-oscillation of solutions of forced second order differential equations

The oscillatory behavior of solutions of a class of second order forced non-linear differential equations is discussed. Several oscillation and non-oscillation criteria are established using Riccati transformations technique. Four examples are given to illustrate our results.