Functional Differential Equations Research Articles

Asymptotic Stepanov automorphic properties of solutions to stochastic functional differential equations

The primary objective of this study is to identify the criteria under which a unique μ-pseudo almost automorphic solution can exist for a stochastic functional differential equation with Stepanov forcing terms. To achieve this, spectral decomposition, Ito’s isometry property and semigroup theory are utilized. The Lipschitz condition and inequality techniques are then employed to establish the uniqueness of the μ-pseudo almost automorphic solution. The methodology is demonstrated through a representative example of a stochastic process.

Functional Differential Equations with an Advanced Neutral Term: New Monotonic Properties of Recursive Nature to Optimize Oscillation Criteria

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term.

The [formula omitted]th moment stability of stochastic functional differential equations with mixed time-varying delay and its applications

This paper investigates the pth moment stability with general decay rate for stochastic mixed time-varying delay functional differential equations with semi-Markov switching and Lévy noise (SMDFDEs-SMS-LN). By employing multiple degenerate Lyapunov functionals, the non-negative martingale convergence theorem, and functional Itô’s formula, a criterion for assessing the pth moment stability with general decay rate is derived. Auxiliary functions with multiple time-varying coefficients are introduced to regulate the diffusion operator, which enhances the system’s adaptability to high-order nonlinear coefficients. These conditions are applied to a stochastic mixed-delay neural network with semi-Markov switching and Lévy noise (SMDNN-SMS-LN) and a scalar non-autonomous stochastic mixed time-varying delay functional differential equations with semi-Markov switching and Lévy noise system with non-global Lipschitz conditions. Using vertex method and linear matrix inequalities, the pth moment stability criteria with general decay rate for the above systems are respectively derived. Finally, the effectiveness of the obtained results is validated through two numerical examples.

Infinitely many positive periodic solutions for second order functional differential equations

Infinitely many positive periodic solutions for second order functional differential equations

Abstract integro‐differential equations with state‐dependent integration intervals: Existence, uniqueness, and local well‐posedness

AbstractIn this work, we study a new class of integro‐differential equations with delay, where the informations from the past are represented as an average of the state over state‐dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro‐differential equations arising in the theory of population dynamics.

Boundary-Value Problem for an Elliptic Functional Differential Equation with Dilation and Rotation of Arguments

Boundary-Value Problem for an Elliptic Functional Differential Equation with Dilation and Rotation of Arguments

Pth Moment Exponential Stability of Impulsive Stochastic Functional Differential Equations

pth Moment Exponential Stability of Impulsive Stochastic Functional Differential Equations

New aspects of the development of Kantorovich’s one-product dynamic model of replacement of production funds

In the context of the development of scientific and technological progress and the growth of the capital stock, an important task is modeling and optimization of the terms of operation of production funds. The development of technologies and automation of production lead to a reduction in the workforce employed in production. In the article, for the previously developed single-product dynamic model of the replacement of production assets, taking into account the inertial properties of the funds being introduced, the case of a reduction in labor resources under the condition of an increase in output and capital investment is investigated. A solution was obtained that allows to determine the optimal strategy for the withdrawal of obsolete funds and the introduction of the new ones in case of decrease in labor resources. As the optimization criterion the principle of differential optimization is used. The theorem of equivalence of a trajectory satisfying the principle of differential optimization, the obtained solution is given. A system of functional differential equations is presented, which should be satisfied by variable models both in terms of growth and reduction of labor resources. Numerical methods are used to solve such a system. The variants of the system development are considered under various assumptions regarding changes in the total amount of labor resources. The results of numerical implementation of the model are presented.

New asymptotic study on the non-autonomous NFDEs involving Haddock conjecture

The classical Haddock conjecture is extended to a kind of non-autonomous neutral functional differential equations (NFDEs) incorporating time-varying delays in this paper. By using the Dini derivative theory and inequality analyses, without requiring the strictly monotonically increasing property of the delay feedback function, it is demonstrated that every solution of the considered NFDEs is bounded and converges to a constant, which fully refines and generalizes the existing findings.

Controllability of some nonlocal‐impulsive Volterra evolution systems via measures of noncompactness

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results.

Generalized Mean Square Exponential Stability for Stochastic Functional Differential Equations

This work focuses on a class of stochastic functional differential equations and neutral stochastic differential functional equations. By using a new approach, some sufficient conditions are obtained to guarantee the generalized mean square exponential stability for the equation under consideration. Certain existing results are refined and extended. Lastly, the validity of the main results is confirmed through several simulation examples.

Functional Differential Equations with Distributed Deviating Arguments: Oscillatory Features of Solutions

Functional Differential Equations with Distributed Deviating Arguments: Oscillatory Features of Solutions

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

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.

On the existence of a positive solution to a boundary value problem for a third-order nonlinear functional differential equation with an integral boundary condition at one of the ends of the segment

The article studies the existence of positive solutions on the segment [0,1] of a two-point boundary value problem for one nonlinear third-order functional differential equation with an integral boundary condition at one of the ends of the segment. Using the Go–Krasnoselsky fixed point theorem and some properties of the Green's function of the corresponding differential operator, sufficient conditions for the existence of at least one positive solution to the problem under consideration are obtained. An example is given to illustrate the results obtained.

Investigation of the Oscillatory Properties of Fourth-Order Delay Differential Equations Using a Comparison Approach with First- and Second-Order Equations

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method’s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results.

Numerical solutions of regime-switching functional diffusions with infinite delay

. We study a class of diffusion processes which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t). Under suitable conditions, we adopt Euler-Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we investigate convergence rates in the L 2-norm the stochastic functional differential equation with infinite memory and random switching under the global Lipschitz conditions. Then we also discuss L 2-convergence under the local Lipschitz conditions.

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.

On Uniqueness and Continuous Dependence of the Weak Solutions of Neutral Stochastic Functional Differential Equations in Infinite Dimensional Spaces

On Uniqueness and Continuous Dependence of the Weak Solutions of Neutral Stochastic Functional Differential Equations in Infinite Dimensional Spaces

A note on reflected BSDEs in infinite horizon with stochastic Lipschitz coefficients

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.