# Get e-book Introduction to Functional Differential Equations

Differential difference equations are functional differential equations in which the argument values are discrete. Differential difference equations are also referred to as retarded , neutral , advanced , and mixed functional differential equations.

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This classification depends on whether the rate of change of the current state of the system depends on past values, future values, or both. In other words, this class of functional differential equations depends on the past and present values of the function with delays.

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Neutral differential equations depend on past and present values of the function, similarly to retarded differential equations, except it also depends on derivatives with delays. In other words, retarded differential equations do not involve the given function's derivative with delays while neutral differential equations do. Integro-differential equations of Volterra type are functional differential equations with continuous argument values. Functional differential equations have been used in models that determine future behavior of a certain phenomenon determined by the present and the past.

Future behavior of phenomena, described by the solutions of ODEs, assumes that behavior is independent of the past.

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FDEs are applicable for models in multiple fields, such as medicine, mechanics, biology, and economics. FDEs have been used in research for heat-transfer, signal processing, evolution of a species, traffic flow and study of epidemics.

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From Wikipedia, the free encyclopedia. Main article: Delay differential equation. Main article: Integro-differential equation. Applied Theory of Functional Differential Equations. The Netherlands: Kluwer Academic Publishers.

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Functional Differential Equations. United States: Springer-Verlag. Introduction to Functional Differential Equations. Recently, Ahmed et al. A primary purpose of this paper is to further study the oscillation of solutions of 1. Our results extend and generalize some of the relevant results in [ 1 — 19 ]. Furthermore, z t is a differentiable solution, while w t is twice differentiable. As usual, a solution of 1 is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is either eventually positive or eventually negative.

Equation 1 is said to be oscillatory if all its solutions are oscillatory. In the sequel, unless otherwise specified, when we write a functional inequality, we assume that it holds for all sufficiently large t. To specify the proofs of our main results, we need the following essential lemmas.

## Introduction to Functional Differential Equations

Assume that 3 holds. Let x t be an eventually positive solution of 7.

In this section, we establish some infinite integral conditions for all solutions of 1 to oscillate. We assume that condition 3 holds. Set z t to be defined as in 5. Then by Lemma 1 , it follows that. Substituting in 22 yields. So by Lemma 4 , we have that the delay differential equation.

## Stability of Functional Differential Equations

Applying the inequality cf. Erbe et al. From 35 and 38 , we find that. Therefore, from 40 in 39 , we get. This is a contradiction with The proof is complete. All conditions of Theorem 5 are satisfied. Then all solutions of 45 oscillate. Let z t and w t be defined as in 5 and 6. It is easily seen, by direct substituting, that z t and w t are also solutions of 1 when p and r are constants; that is.

Also, we have indeed that.

## Stability of Functional Differential Equations - 1st Edition

Using 54 in 52 implies that. As w t is positive solution, so by Lemma 4 , the delay differential equation. Applying the inequality 32 to 64 , we have. From 67 and 70 , we find that.

Therefore, from 72 in 71 , we get. Then all conditions of Theorem 7 are satisfied and therefore all solutions of 77 oscillate. Theorems 5 and 7 generalize and extend Theorems 3. See also the results of Ahmed et al. The authors declare that there is no conflict of interests regarding the publication of this paper.

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Journal List ScientificWorldJournal v. Published online Aug Fatima N. Ahmad , Ummul K. Din , and Mohd S. Rokiah R. Ummul K. Mohd S. Author information Article notes Copyright and License information Disclaimer. Ahmed: moc. Ahmed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract We will consider a class of neutral functional differential equations. Introduction During the past few decades, neutral differential equations have been studied extensively and the oscillatory theory for these equations is well developed; see [ 1 — 19 ] and the references cited therein. Auxiliary Lemmas To specify the proofs of our main results, we need the following essential lemmas. Lemma 1 see [ 12 ]. Then a. Lemma 2 see [ 16 ].