Second-order Functional Differential Equations Research Articles

Stability and boundedness criteria for certain second-order nonlinear neutral stochastic functional differential equations

This paper presents stochastic stability and stochastic boundedness to certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is weakened to a neutral stochastic system of first-order equations and used together with a second-order quadratic function to obtain perfect Lyapunov-Krasovskii functional. This functional is adapted and applied to obtain criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, fresh results on stochastic boundedness and uniform stochastic boundedness of solutions are obtained. The results of this paper are original, new, essentially improving, complementing, and simplifying several related ones in the literature. Two special cases of the theoretical results are supplied to demonstrate the applicability of the hypothetical results.

Existence and Approximate Controllability of Random Impulsive Neutral Functional Differential Equation with Finite Delay

This study investigates second-order neutral functional differential equations with delays, prevalent in various scientific and engineering fields. These equations, characterized by their neutral nature and delays, present unique challenges within Banach spaces. The research focuses on the existence and approximate controllability of solutions, using advanced mathematical tools like cosine family theory and the Leray-Schauder theorem to establish rigorous solution conditions. These theoretical results are empirically validated through practical examples, enhancing understanding of real-life behavior and bridging theory with practice. The study’s findings advance the understanding of delayed feedback systems, facilitating effective control strategies and practical engineering solutions, thereby contributing significantly to dynamical systems and control theory.

Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients

Abstract In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

Oscillation of second-order functional differential equations with mixed argument

Oscillation of second-order functional differential equations with mixed argument

The Dirichlet Problem for an Elliptic Functional Differential Equation with the Compressed, Expanded, and Rotated Argument

The paper is devoted to the Dirichlet problem in a plain bounded domain for a linear divergent-form second-order functional differential equation with the compressed (expanded) and rotated argument of the highest derivatives of the unknown function. Necessary and sufficient conditions for the Gårding-type inequality are obtained in algebraic form. The result may depend not only on the absolute value of the coefficients but also on their signature. Under some restrictions on the structure of the operator and the geometry of the domain, the questions of existence, uniqueness, and smoothness of generalized solutions are studied for all possible values of the coefficients and parameters of transformations in the equation, even when the equation is not strongly elliptic.

Boundary-value problem for an elliptic functional differential equation with dilation and rotation of arguments

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation.

On the existence of a positive solution to a boundary value problem for one nonlinear functional-differential equation of the second order

This article considers a boundary value problem for one non-linear second-order functional differential equation on the segment [0, 1] with an integral boundary condition at one of the ends of the segment. Using the well-known Go-Krasnoselsky theorem, sufficient conditions for the existence of at least one positive solution to the problem under consideration are established. A non-trivial example is given, illustrating the fulfillment of the conditions for the unique solvability of the problem posed.

Oscillation Criteria for Advanced Half-Linear Differential Equations of Second Order

In this paper, we find new oscillation criteria for second-order advanced functional half-linear differential equations. Our results extend and improve recent criteria for the same equations established previously by several authors and cover the existing classical criteria for related ordinary differential equations. We give some examples to illustrate the significance of the obtained results.

Improved results for testing the oscillation of functional differential equations with multiple delays

<abstract><p>In this article, we test whether solutions of second-order delay functional differential equations oscillate. The considered equation is a general case of several important equations, such as the linear, half-linear, and Emden-Fowler equations. We can construct strict criteria by inferring new qualities from the positive solutions to the problem under study. Furthermore, we can incrementally enhance these characteristics. We can use the criteria more than once if they are unsuccessful the first time thanks to their iterative nature. Sharp criteria were obtained with only one condition that guarantees the oscillation of the equation in the canonical and noncanonical forms. Our oscillation results effectively extend, complete, and simplify several related ones in the literature. An example was given to show the significance of the main results.</p></abstract>

On the existence of a positive solution to a periodic boundary value problem for one non-linear second-order functional differential equation

On the existence of a positive solution to a periodic boundary value problem for one non-linear second-order functional differential equation

On the existence of a positive solution to a boundary-value problem for a nonlinear second-order functional differential equation

On the existence of a positive solution to a boundary-value problem for a nonlinear second-order functional differential equation

Criteria for Oscillation of Half-Linear Functional Differential Equations of Second-Order

The present article aims to establish more effective criteria for testing the oscillation of a class of functional differential equations with delay arguments. In the non-canonical case, we deduce some improved monotonic and asymptotic properties of the class of decreasing positive solutions of the studied equation. Depending on both the new properties and the linear representation of the studied equation, we obtain new oscillation criteria. Moreover, we test the effectiveness of the new criteria by applying them to some special cases of the studied equation.

Oscillation of Noncanonical Second-Order Functional Differential Equations via Canonical Transformation

Oscillation of Noncanonical Second-Order Functional Differential Equations via Canonical Transformation

An Effective Approximation Algorithm for Second-Order Singular Functional Differential Equations

In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane–Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system of algebraic equations and by solving this system the unknown Bessel coefficients are determined and the solution will be known. The method is tested on several test examples and proves to provide accurate results as compared to other existing methods from the literature. The simplicity and robustness of the proposed technique drive us to investigate more of their applications to several similar problems in the future.

A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

Oscillation of Second-Order Functional Differential Equations with Superlinear Neutral Terms

Oscillation of Second-Order Functional Differential Equations with Superlinear Neutral Terms

Oscillation Results for Second-Order Neutral Functional Differential Equations

In this article, we obtain several comparison theorems for the second-order neutral differential equation , t ⩾ t 0, where γ, λ, β are ratios of positive odd integers. We compare above equation with the first-order differential inequalities in the sense that the absence of the eventually positive solutions of these first-order inequalities implies the oscillation of the studied equation.

Existence of positive periodic solutions for a class of second-order neutral functional differential equations

Abstract In this paper, under some ordered conditions, we investigate the existence of positive ω-periodic solutions for a class of second-order neutral functional differential equations with delayed derivative in nonlinearity of the form (x(t) − cx(t − δ))″ + a(t)g(x(t))x(t) = λb(t)f(t, x(t), x(t − τ 1(t)), x′(t − τ 2(t))). By utilizing the perturbation method of a positive operator and the fixed point index theory in cones, some sufficient conditions are established for the existence as well as the non-existence of positive ω-periodic solutions.

Oscillation theorems of solution of second-order neutral differential equations

<abstract><p>In this paper, we aim to explore the oscillation of solutions for a class of second-order neutral functional differential equations. We propose new criteria to ensure that all obtained solutions are oscillatory. The obtained results can be used to develop and provide theoretical support for and further develop the oscillation study for a class of second-order neutral differential equations. Finally, an illustrated example is given to demonstrate the effectiveness of our new criteria.</p></abstract>