الفهرس 
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Abstract
To utilize mathematical techniques for solving practical or real-world problems, it is necessary to express the problem using mathematical language. This involves creating a mathematical representation, known as a model, for the problem. Mathematical models frequently incorporate equations that relate an unknown function and its derivatives, as derivatives mathematically represent rates of change. These equations, known as differential equations, find applications in diverse scientific fields such as physics (including dynamics, thermodynamics, heat, fluid mechanics, and electromagnetism), chemistry (involving rates of chemical reactions, physical chemistry, and radioactive decay), biology (including growth rates of bacteria, plants, and other organisms), and economics (encompassing economic growth rate and population growth rate), see [1]. A delay differential equation is an equation for a function of a single variable, usually called time, in which the function’s derivative at a certain time is given in terms of the values of the function at earlier times. Reliance on the past appears naturally in several applications in biology, electrical engineering, and physiology. A simple example in nature is reforestation. A cut forest, being replanted, will take at least 20 years before maturing. Hence, any mathematical forest harvesting and regeneration model must have time delays built into it [2]. Neutral delay differential equations are a type of functional differential equation in which the delayed argument appears in the state variable’s highest derivative. The qualitative analysis of such equations is quite beneficial in addition to its theoretical value. This form of the equation has fascinating applications in everyday life. As an example of some problems in networks with lossless transmission lines, as in high-speed computers, the vibration study of blocks connected to a flexible rod, and solving various problems with time delay, also in automated control theory and in aero-mechanical systems in which inertia plays an important role, we recommend the reader read Hale’s monograph [3] for additional science and technological applications. The main goals of oscillation theory are establishing conditions for the existence of oscillatory (non-oscillatory) solutions, researching zero distribution laws, estimating the number of zeros in a given interval and the distance between neighboring zeros, characterizing the relationship between oscillatory and other fundamental properties of solutions to different classes of differential equations, etc. An important numerical mathematical tool for many fields and advanced technologies is now oscillation theory. An application of oscillation theory is the model of blood cell production. The production of red and white blood cells in the bone marrow is regulated by the level of oxygen in the blood. A reduction in the number of cells in the blood, as a result of the loss of cells, causes the level of oxygen in the blood to decrease. When the level of oxygen in the blood decreases, a substance is released, this in turn leads to the release of blood elements from the bone marrow. Thus, the concentration c(t) of cells in the bloodstream, at any time t, changes according to the loss of cells and the release of new cells, from the bone marrow. But the bone marrow responds to a reduction in the number of blood cells and a decrease in the level of oxygen, with a delay that is in the order of 6 days. That means the release of new cells into the bloodstream at time t, depends on the cell concentration at an earlier time, namely, t-τ, where τ is the delay with which the bone marrow responds to a reduced level of oxygen in the blood. The simplest model of the concentration of cells in the bloodstream can be described by the delay differential equation c’(t)=λc(t-τ)-γc(t) where λ represents the flux of cells into the bloodstream, γ is the death rate, and τ is the delay. All of them are positive constants. The solutions of the above equation exhibit similar oscillations to the actual oscillatory pattern observed in the concentration of cells in the bloodstream. For more information, we refer the reader to [4]. Finding oscillation criteria for specific functional differential equations has been a very active field of study in recent decades, and a wealth of references and explanations of known results may be found in the monographs by Agarwal et al. [5], Gyori and Ladas [6], and Bainov and Mishev [7]. Compared to first- or second-order differential equations, we have a disproportionately small number of results for fourth-order equations. Therefore, the primary goal of this thesis is to shed light on numerous fourth-order equations by examining the asymptotic behavior of solutions to those equations using various techniques and contrasting the results. As evidenced by the examples and comments provided throughout the chapters, our discoveries in this thesis improve, generalize, and extend some of the prior results. Contributing to the significant advancement of the oscillation theory of higher-order delay differential equations is yet another primary goal of this thesis. The thesis is divided into five chapters. Chapter 1 is the introductory chapter. It includes some fundamental concepts, preliminary results that will be applied in the upcoming chapters, and a selection of the most important oscillation results for first-order differential equations that are available in the literature. In Chapter 2, we investigate the asymptotic behavior of solutions to the second-order half-linear differential equation. This chapter is divided into two parts. In the first part, we will present some oscillation results by using an integral average condition of the Philos-type. In the second part, depending on the sign of the derivatives, we will extend the results of previous studies and obtain one condition to guarantee that all solutions of the studied equation will oscillate. In Chapter 3, we discuss the oscillatory behavior of the delay and neutral delay differential equations under noncanonical case. The obtained results improve some of the known oscillation criteria in the literature. Some examples are provided to illustrate the significance of our main results. In Chapter 4, we investigate some qualitative properties of solutions to a class of functional differential equations with multi-delay. By using Riccati transformation technique and comparison theorems, we present new conditions for the oscillation of the studied equation. Further, some illustrative examples showing the applicability of the new results are included. In Chapter 5, we derive new inequalities that improve the asymptotic and oscillatory properties of solutions to the fourth-order neutral differential equations. We set new conditions that confirm the absence of positive solutions and thus confirm the oscillation of all solutions to the considered equation. Finally, we explain the importance of the new inequalities by applying our results to some special cases of the studied equation. The study of the oscillation of delay differential equations depends on three important issues: ∙The first of which is to classify the positive solutions according to the sign of their derivatives; The second is to find relationships between derivatives of different orders, which is also crucial, although doing so may impose further limitations on the study; Finally, estimating the relationship between the solution with delay and without delay. Our results can be summarized as follows :∙The most influential factor in the relationships between derivatives is the monotonic properties of the solutions to these equations;∙ Improving these properties or finding new properties of an iterative nature greatly affects the qualitative study of solutions to these equations;∙Obtaining oscillation conditions and criteria characterized by an iterative technique was a major goal because the iterative technique of the conditions allows them to be applied several times, even in the case that they fail in the application at the beginning;∙ As the conditions for oscillation are only sufficient (and not necessary) conditions, that is, if they are not satisfied, this does not mean that the equation is not oscillatory. Therefore, it is useful to find different formulas of conditions and to simplify them so that they are easy to apply; By using a different approach in the study, we got rid of some of the conditions that were previously imposed on the unknown functions in the equation, and that resulted in the application of the oscillation criteria we obtained on the equations for which the oscillation previous results failed;∙ The use of Riccati substitution with a nontraditional form led to unusual conditions and thus can be applied to different equations that cannot be covered by the known results; ∙ The use of a comparison technique with first-order delay differential equations allows us to obtain different conditions that ensure the application to a larger area of examples.