Functional Differential Equations Research Articles

Global Hopf bifurcation and positive periodic solutions of a delayed diffusive predator–prey model with weak Allee effect for predator

In this paper, we investigate a delayed diffusive predator–prey model with weak Allee effect for predator. First, we discuss the existence and uniqueness of positive steady state and the local Hopf bifurcation. Next, we obtain the permanence and uniform boundedness of positive periodic solutions of the delayed reaction–diffusion system. The existence range of positive periodic solutions is further compressed by using iterative approach. For the system without delay, we prove the global attractiveness of unique positive steady state by using Lyapunov function, which ensures that the periods of positive periodic solutions are uniformly bounded. Then we obtain the global Hopf bifurcation and extended existence of positive periodic solutions by using the global Hopf bifurcation theorem of partial functional differential equation. Finally, the validity of the conclusion is verified through numerical simulation.

Normal Form Formulae of Turing–Turing Bifurcation for Partial Functional Differential Equations With Nonlinear Diffusion

ABSTRACTFor most systems, the appearance of Turing bifurcation means that it is possible to excite Turing–Turing bifurcation, thus inducing superimposed spatial patterns, multistable spatial patterns co‐existing, and others.It is undoubtedly of interest to qualitatively analyze the structures and stability of these spatially heterogeneous solutions generated by Turing–Turing bifurcation. Therefore, in this paper, with the aid of the center manifold theory and the normal form method, for general partial functional differential equations with nonlinear diffusion, the third‐order normal form is firstly derived. It is locally topologically equivalent to the primitive partial functional differential equations at the Turing–Turing bifurcation point. And then the explicit formulae for the coefficients in the normal form associated with three different spatial modes are presented. As an application, a predator–prey model with predator–taxis is considered. It is theoretically revealed that the system admits the coexistence of a pair of stable steady states with a single characteristic wavelength and the coexistence of four stable steady states with different single characteristic wavelengths. Further, the parameter regions in which these phenomena will arise are quantitatively given. It is illustrated that the self‐diffusion of prey and predator–taxis complement each other to promote the formation for the spatially homogeneous distributions of populations.

On the existence-uniqueness and exponential estimate for solutions to stochastic functional differential equations driven by G-Lévy process

The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received tremendous attention. This article aims to study the theory for stochastic functional differential equations (SFDEs) driven by the G-Lévy process. It derives the existence-uniqueness theorem for solutions to SFDEs driven by the G-Lévy process. Moreover, it shows the error estimation between the exact solution x(t) and Picard approximate solutions xn(t),n≥1. Ultimately, the exponential estimate has been derived.

Averaging principle for two-time-scale stochastic functional differential equations with past-dependent switching

Averaging principle for two-time-scale stochastic functional differential equations with past-dependent switching

Asymptotic almost periodicity of bounded solutions for some partial functional differential equations with delay

Asymptotic almost periodicity of bounded solutions for some partial functional differential equations with delay

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.

Hamiltonian Boundary Value Methods (HBVMs) for functional differential equations with piecewise continuous arguments

Hamiltonian Boundary Value Methods (HBVMs) for functional differential equations with piecewise continuous arguments

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.

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.

Infinitely many positive periodic solutions for second order functional differential equations

Infinitely many positive periodic solutions for second order functional differential equations

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.

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.

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.

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

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.

On spectral data for (2,2) Berry connections, difference equations and equivariant quantum cohomology

In this paper, we study supersymmetric Berry connections of 2d [Formula: see text] gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin–Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arise naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a quantization of the Cherkis–Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target X, we show that the difference modules are related to the equivariant quantum cohomology of X.

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