Fourth Order Functional 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.

More Effective Conditions for Testing the Oscillatory Behavior of Solutions to a Class of Fourth-Order Functional Differential Equations

This paper presents an investigation into the qualitative behavior of solutions for a specific class of fourth-order half-linear neutral differential equations. The main objective of this study is to improve the relationship between the solution and its corresponding function. By developing improved relationships, a novel criterion is proposed to determine the oscillatory behavior of the studied equation. The exclusion of positive solutions is achieved through a comparative approach in which the examined equation is compared to second-order equations. Additionally, the significance of the obtained results is demonstrated by applying them to various illustrative examples.

Stability, boundedness and existence of unique periodic solutions to a class of fourth order functional differential equations

In this paper a novel class of fourth order functional differential equations is discussed. By reducing the fourth order functional differential equation to system of first order, a suitable complete Lyapunov functional is constructed and employed to obtain sufficient conditions that guarantee existence of a unique periodic solution, asymptotic and uniform asymptotic stability of the zero solutions, uniform boundedness and uniform ultimate boundedness of solutions. The obtained results are new and include many prominent results in literature. Finally, two examples are given to show the feasibility and reliability of the theoretical results.

Unbounded oscillation of fourth order functional differential equations

Abstract In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay 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 $$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0 \end{array}$$ are discussed under the assumption ∫ 0 ∞ t r ( t ) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}~~ t=\infty \end{array}$$

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.

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.

ON SOME QUALITATIVE BEHAVIORS OF CERTAIN DIFFERENTIAL EQUATIONS OF FOURTH ORDER WITH MULTIPLE RETARDATIONS

In this paper, we give sufficient conditions to guarantee the asymptotic stability and boundedness of solutions to a kind of fourth-order functional differential equations with multiple retardations. By using the LyapunovKrasovskii functional approach, we establish two new results on the stability and boundedness of solutions, which include and improve some related results in the literature.

A new result on the existence of periodic solutions for fourth-order functional differential equations with multiple deviating arguments

A new result on the existence of periodic solutions for fourth-order functional differential equations with multiple deviating arguments

Oscillation criteria for fourth-order functional differential equations

Abstract Some new criteria for the oscillation of all solutions of certain fourth-order functional differential equations are established.

Oscillation theorems for fourth order functional differential equations

The authors investigate the oscillatory behavior of all solutions of the fourth order functional differential equations $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})+q(t)f(x[g(t)])=0$ and $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})=q(t)f(x[g(t)])+p(t)h(x[\sigma(t)])$ in the case where ∫ ∞ a −1/α (s)ds<∞. The results are illustrated with examples.

On the oscillation of certain fourth order functional differential equations

No AccessOn the oscillation of certain fourth order functional differential equationsSaid R. Grace, Ravi P. Agarwal, Raffaella Pavani and Jelena V. ManojlovicSaid R. Grace Faculty of Engineering, Department of Engineering Mathematics, Cairo University, Orman, Giza 12221, Egypt. , Ravi P. Agarwal Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA. , Raffaella Pavani Department of Mathematics, Politecnico diMilano, 20133 Milano, Italy. and Jelena V. Manojlovic Faculty of Science and Mathematics, Department of Mathematics and Computer Science, University of Nis, Visegradska 33, 18000 Nis, Serbia Published Online:July 21, 2008pp 155-163https://doi.org/10.1504/IJDSDE.2008.019677PDF ToolsAdd to FavouritesDownload CitationsTrack Citations Share this article on social mediaShareShare onFacebookTwitterLinkedInReddit AboutAbstractSome new criteria for the oscillation of the fourth order functional differential equation (1/a(t)(x″(t))α)″ = q(t)f(x[g(t)]) + p(t)h(x[σ(t)]) are established.Keywordscomparison, functional differential equations, fourth order FDE, nonoscillation, oscillation Previous article Next article FiguresReferencesRelatedDetails Volume 1Issue 32008 ISSN: 1752-3583eISSN: 1752-3591 HistoryPublished onlineJuly 21, 2008 Keywordscomparisonfunctional differential equationsfourth order FDEnonoscillationoscillationAuthors and AffiliationsSaid R. Grace1 Ravi P. Agarwal2 Raffaella Pavani3 Jelena V. Manojlovic4 1. Faculty of Engineering, Department of Engineering Mathematics, Cairo University, Orman, Giza 12221, Egypt. 2. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA. 3. Department of Mathematics, Politecnico diMilano, 20133 Milano, Italy. 4. Faculty of Science and Mathematics, Department of Mathematics and Computer Science, University of Nis, Visegradska 33, 18000 Nis, Serbia PDF download

Oscillation of certain fourth-order functional differential equations

Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the form $$\frac{{d^2 }}{{dt^2 }}\left( {a(t)\left( {\frac{{d^2 x(t)}}{{dt^2 }}} \right)^\alpha } \right) + q(t)f(x[g(t)]) = 0, \alpha > 0,$$ are established.

Characterization of oscillation of fourth order functional differential equations with deviating arguments

This paper is devoted to the study of oscillation of all solutions of the fourth order functional differential equation with general deviating arguments $$L_4 x(t) + f(t,x(g_1 (t)),...,x(g_n (t))) = 0$$ where $$L_4 = \frac{1}{{p_4 }}\frac{d}{{dt}}\frac{1}{{p_3 }}\frac{d}{{dt}}\frac{1}{{p_2 }}\frac{d}{{dt}}\frac{1}{{p_1 }}\frac{d}{{dt}}\frac{ \cdot }{{p_0 }}$$ 2 necessary and sufficient conditions are established for all solutions of equation (1)to be oscillatory when L 4 is in canonical form, that is, $$\mathop \smallint \limits^\infty p_i (t)dt = \infty ,1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 3$$ , 1≦i≦3.In Section 3,on the basis of Trench's theory, a characterization of oscillation of equation (1)is obtained when L 4 is not necessarily in canonical form.

Nonlinear oscillation of fourth order functional differential equations

In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) 2 ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations.