Class Of Second-order Differential Equations Research Articles

Enhanced criteria for detecting oscillations in neutral delay Emden-Fowler differential equations

In this article, we present new oscillation criteria for a specific class of second-order differential equations known as neutral delay Emden-Fowler equations. By utilizing advanced mathematical techniques, we are able to derive conditions that are more accurate and efficient than previous methods. Furthermore, our results also simplify the process of identifying oscillations in these types of equations. Our findings have important implications for various fields such as engineering, physics and mathematics, where these equations are widely used to model dynamic systems. Additionally, We have also highlighted some illustrative examples to further demonstrate the applicability and potential of our results.

Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method

The paper considers a simple and well-known method for reducing the differentiability order of an ordinary differential equation, defining the first derivative as a function that will become the new variable. Practically, we attach to the initial equation a supplementary one, very similar to the flow equation from the dynamical systems. This is why we name it as the “attached flow equation”. Despite its apparent simplicity, the approach asks for a closer investigation because the reduced equation in the flow variable could be difficult to integrate. To overcome this difficulty, the paper considers a class of second-order differential equations, proposing a decomposition of the free term in two parts and formulating rules, based on a specific balancing procedure, on how to choose the flow. These are the main novelties of the approach that will be illustrated by solving important equations from the theory of solitons as those arising in the Chafee–Infante, Fisher, or Benjamin–Bona–Mahony models.

Existence of Fixed Points for a Class of Decreasing Operators with Parameter and Applications

This paper studies the existence of fixed points for a class of decreasing operators with parameter in real Banach spaces. The existence theorems of fixed point are obtained as when the parameter is increasing, there will still be a large fixed point. These results have reduced the requirements of convexity, compactness, and lattice structure of spaces. By this new method, the existence of solutions for a class of second-order differential equations with parameter in infinite intervals is studied.

Periodic solutions for second-order difference equations with quadratic\u2013supquadratic condition

In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.

Asymptotic Representations of Rapidly Varying Solutions of Second-Order Differential Equations with Rapidly and Regularly Varying Nonlinearities

We establish necessary and sufficient conditions for the existence of Pω (Y0 ,Y1, λ0)-solutions of one class of second-order differential equations with rapidly and regularly varying nonlinearities. This result is a natural generalization of the results of previous investigations carried out by V. М. Evtukhov and his school. We also established the asymptotic representations of these solutions and their first-order derivatives.

К СУЩЕСТВОВАНИЮ ВПОЛНЕ НЕПРЕРЫВНОЙДИФФЕРЕНЦИАЛЬНОЙ РЕАЛИЗАЦИИ БИЛИНЕЙНОЙ СИСТЕМЫ ВТОРОГОПОРЯДКА

For a continuous nonlinear infinite-dimensional behavioristic system (dynamical system of J. Willems), a functional-geometric study of the necessary and sufficient conditions for the existence of six non-stationary coefficients-operators of the model of bilinear differential realization of this system in the class of second-order differential equations in a separable Hilbert space is carried out. The case is investigated when the simulated operators are burdened with a condition that ensures the complete continuity of the integral form of the equations of realization in the entropy setting.

Normalized Lucas wavelets: an application to Lane–Emden and pantograph differential equations

In this paper, a novel normalized Lucas wavelet scheme based on tau approach is proposed for the two classes of second-order differential equations, namely Lane–Emden and pantograph equations. The introduced scheme depends on shifted Lucas polynomials (SLPs) and their operational matrix of derivative (which are developed here). The weight function for the orthogonality of Lucas polynomials, and Rodrigues formula are proposed for the first time, which form the basis for the construction of SLPs. Normalized Lucas wavelets are constructed by utilizing SLPs and their novel properties. Literally, the present scheme transforms the given method to a set of nonlinear algebraic equations with undetermined coefficients which are here tackled by tau method. Meanwhile, new treatment of convergence and error analysis is provided for the established approach. Finally, the accuracy and applicability of present scheme is ensured by considering several examples.

Positive periodic solutions for a class of second-order differential equations with state-dependent delays

In this paper, we consider a class of second order differential equations with iterative source term. The main results are obtained by virtue of a Krasnoselskii fixed point theorem and some useful properties of a Green's function which allows us to prove the existence of positive periodic solutions. Finally, an example is included to illustrate the correctness of our results.

Asymptotic Representations for the Solutions of Second-Order Differential Equations with Rapidly and Regularly Varying Nonlinearities

We determine asymptotic representations for the solutions of one class of second-order differential equations with rapidly and regularly varying nonlinearities.

Periodic solutions for a class of conservative Liénard-type equations

In this paper, we study solvability of a class of second-order differential equations in a conservative Liénard form subject to periodic boundary conditions. Results on existence of non-trivial T-periodic solutions or positive T-periodic solutions are obtained respectively. Applications of the theorems are shown by examples. The results are proved by applying the coincidence degree theory for semilinear operator equations.

Asymptotic representations of solutions with slowly varying derivatives

Second-order differential equations with power and exponential nonlinearities on the right hand side play an important role in the development of a qualitative theory of differential equations. The authors of most works devoted to the establishment of asymptotic representations of solutions investigate equations with power and with regularly varying nonlinearities. Recently, the consideration of differential equations with exponential and a wider class than exponential functions - rapidly varying functions - has begun. In this paper, the asymptotic representations of solutions with slowly varying first-order derivatives of some new class of second-order differential equations with rapidly and regularly varying nonlinearities are established.

A Note on the Generalized and Universal Associated Legendre Equations**Supported of Natural Sciences and Engineering Research Council of Canada under Grant No. GP249507

A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated Legendre Equations are examined and established. The hypergeometric solutions, presented in this work, will promote future investigations of their mathematical properties and applications to problems in theoretical physics.

Study on the Existence of Sign-Changing Solutions of Case Theory Based a Class of Differential Equations Boundary-Value Problems

By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.

Periodic Solutions of a Class of Second-Order Differential Equation

We study the periodic solutions of the second-order differential equations of the form where the functions, , and are periodic of period in the variable t.

A variational approach of Sturm-Liouville problems with the nonlinearity depending on the derivative

In this paper, we are concerned with the existence of positive classical solutions for a class of second-order differential equations with the nonlinearity dependent on the derivative. We also provide a range of the parameter in order to obtain the existence of multiple solutions. The approach is based on variational methods. An example illustrates the abstract results of this paper.

Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

Abstract In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

Maximal regularity of second-order evolution equations with infinite delay in Banach spaces

By using Fourier multiplier theorems we characterize the existence and uniqueness of periodic solutions for a class of second-order differential equations with infinite delay. We also establish maximal regularity results for the equations in various space

Oscillation of second-order differential equations with a sublinear neutral term

This paper is concerned with oscillation of a certain class of second-order differential equations with a sublinear neutral term. Two oscillation criteria and two illustrative examples are included. In particular, the results obtained improve those reported in the literature.

Oscillation of second-order damped differential equations

AbstractWe study oscillatory behavior of a class of second-order differential equations with damping under the assumptions that allow applications to retarded and advanced differential equations. New theorems extend and improve the results in the literature. Illustrative examples are given.MSC:34C10, 34K11.

Boundedness of solutions for a class of second-order differential equation with singularity

In this paper, we study the following second-order periodic system: where has a singularity. Under some assumptions on the and by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.