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In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of a solution of a fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators by utilizing a new version of Kransoselskii-Dhage type fixed-point theorem obtained in [13]. Moreover, we provide an example to support our result. [ABSTRACT FROM AUTHOR]

In this paper, the prime orbit theorem and Mertens' orbit theorem are proved for a shift dynamical system of infinite type called the Motzkin shift. Instead of using the zeta function approach to find the asymptotic formulas, different and more direct methods are used in the proof without any complicated theoretical discussion. [ABSTRACT FROM AUTHOR]

The intent of this manuscript is to present new rational symmetric ϖ − ξ -contractions and infer some fixed-points for such contractions in the setting of Θ -metric spaces. Furthermore, some related results such as Suzuki-type rational symmetric contractions, orbitally Υ -complete, and orbitally continuous mappings in Θ -metric spaces are introduced. Ultimately, the theoretical results are shared to study the existence of the solution to a fractional-order differential equation with one boundary stipulation. [ABSTRACT FROM AUTHOR]

In this work, the main concept of the homotopy perturbation method (HPM) was outlined and convergence theorems of the HPM for solving some classes of nonlinear integral, integro-differential and differential equations were proved. A theorem for estimating the error in the approximate solution was proved as well. The proposed HPM convergence theorems were confirmed and the efficiency of the technique was explored by applying the HPM for solving several classes of nonlinear integral/integro-differential equations. [ABSTRACT FROM AUTHOR]