Fourth-order Delay Differential Equations Research Articles

Investigation of the Oscillatory Properties of Fourth-Order Delay Differential Equations Using a Comparison Approach with First- and Second-Order Equations

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method’s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results.

Oscillation results for nonlinear weakly canonical fourth-order delay differential equations via canonical transform

By using canonical transformation technique, we convert the nonlinear weakly canonical fourth-order delay differential equations into strongly canonical equations. Then we apply oscillation results for canonical equations to get similar results for the studied equation. The significance and novelty of our results are illustrated via numerical examples.

Optimizing the Monotonic Properties of Fourth-Order Neutral Differential Equations and Their Applications

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields.

Non-Canonical Functional Differential Equation of Fourth-Order: New Monotonic Properties and Their Applications in Oscillation Theory

In the present article, we iteratively deduce new monotonic properties of a class from the positive solutions of fourth-order delay differential equations. We discuss the non-canonical case in which there are possible decreasing positive solutions. Then, we find iterative criteria that exclude the existence of these positive decreasing solutions. Using these new criteria and based on the comparison and Riccati substitution methods, we create sufficient conditions to ensure that all solutions of the studied equation oscillate. In addition to having many applications in various scientific domains, the study of the oscillatory and non-oscillatory features of differential equation solutions is a theoretically rich field with many intriguing issues. Finally, we show the importance of the results by applying them to special cases of the studied equation.

New Criteria for Oscillation of Half-Linear Differential Equations with p-Laplacian-like Operators

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our 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 Oscillation Criteria for Neutral Delay Differential Equations of Fourth-Order

New oscillatory properties for the oscillation of solutions to a class of fourth-order delay differential equations with several deviating arguments are established, which extend and generalize related results in previous studies. Some oscillation results are established by using the Riccati technique under the case of canonical coefficients. The symmetry plays an important and fundamental role in the study of the oscillation of solutions of the equations. Examples are given to prove the significance of the new theorems.

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.

On the oscillation of certain fourth-order differential equations with p-Laplacian like operator

• Oscillation of certain fourth order differential equations. • To study the oscillatory properties of fourth order delay differential equations. • Comparison with first and second order differential equations and Riccati transformations. The aim of this work is to study the oscillatory properties of the solutions of fourth-order delay differential equations with a p -Laplacian like operator of the form ( r ( ν ) | w ″ ′ ( ν ) | p 1 − 2 w ″ ′ ( ν ) ) ′ + q ( ν ) | w ( η ( ν ) ) | p 2 − 2 w ( η ( ν ) ) = 0 . The results obtained are based on the comparison with first and second-order differential equations and the use of a Riccati transformation. Some examples are provided to illustrate the main results.

Improved Conditions for Oscillation of Functional Nonlinear Differential Equations

The aim of this work is to study oscillatory properties of a class of fourth-order delay differential equations. New oscillation criteria are obtained by using generalized Riccati transformations. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.

Oscillation of solutions to fourth-order delay differential equations with middle term

Oscillation of solutions to fourth-order delay differential equations with middle term

Oscillation of Fourth-Order Functional Differential Equations with Distributed Delay

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples.

Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations

In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained.

On the oscillation of fourth-order delay differential equations

In the paper, fourth-order delay differential equations of the form \t\t\t(r3(r2(r1y′)′)′)′(t)+q(t)y(τ(t))=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(r_{3} \\bigl(r_{2} \\bigl(r_{1}y' \\bigr)' \\bigr)' \\bigr)'(t) + q(t) y \\bigl( \\tau (t) \\bigr) = 0 $$\\end{document} under the assumption \t\t\t∫t0∞dtri(t)<∞,i=1,2,3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int _{t_{0}}^{\\infty }\\frac{\\mathrm {d}t}{r_{i}(t)} < \\infty , \\quad i = 1,2,3, $$\\end{document} are investigated. Our newly proposed approach allows us to greatly reduce a number of conditions ensuring that all solutions of the studied equation oscillate. An example is also presented to test the strength and applicability of the results obtained.

On the asymptotic behavior of fourth-order functional differential equations

The aim of this work is to study asymptotic properties of a class of fourth-order delay differential equations. Our results in this paper not only generalize some previous results, but also improve the earlier ones. Examples are considered to elucidate the main results.

New oscillation results to fourth order delay differential equations with damping

New oscillation results to fourth order delay differential equations with damping

A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form:X(4)+F(X˙,X¨)X⃛+Φ(X¨)+G(X˙(t-r))+H(X(t-r))=0.By constructing a Lyapunov functional, we obtained the result of stability.

Oscillation of Fourth-Order Delay Differential Equations

This article deals with the oscillation of a certain class of fourth-order delay differential equations. Some new oscillation criteria (including Hille- and Nehari-type criteria) are presented. The results obtained in the paper improve some results from [C. Zhang, T. Li, B. Sun, and E. Thandapani, Appl. Math. Lett., 24, 1618 (2011)]. Two examples are presented to illustrate our main results.

ON STABILITY AND BOUNDEDNESS OF SOLUTIONS OF CERTAIN FOURTH ORDER DELAY DIFFERENTIAL EQUATION

In this paper, sufficient criteria which guarantee the existence of uniform asymptotic stability and boundedness of solution of a scalar real fourth-order delay differential equation were established with the aid of a suitable Lyapunov function. With the Lyapunov function, conditions on the nonlinear terms to guarantee stability and boundedness of the solution and its derivatives were given. Our results include and improve on the existing results in the literature.

Bifurcation Analysis in a Kind of Fourth-Order Delay Differential Equation

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the analytic results.