Functional Differential Equations Research Articles

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.

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.

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.

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.

A Study of Positive Solutions for Semilinear Fractional Measure Driven Functional Differential Equations in Banach Spaces

In this paper, we deal with the delayed measure differential equations with nonlocal conditions via measure of noncompactness in ordered Banach spaces. Combining (β,γk)-resolvent family, regulated functions and fixed point theorem with respect to convex-power condensing operator and measure of noncompactness, we investigate the existence of positive mild solutions for the mentioned system under the situation that the nonlinear function satisfies measure conditions and order conditions. In addition, we provide an example to verify the rationality of our conclusion.

Analysis of a parametric delay functional differential equation with nonlocal integral condition

Analysis of a parametric delay functional differential equation with nonlocal integral condition

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

FuNVol: multi-asset implied volatility market simulator using functional principal components and neural SDEs

We introduce a new approach for generating sequences of implied volatility (IV) surfaces across multiple assets that are faithful to historical prices. We do so using a combination of functional data analysis and neural stochastic differential equations (SDEs) combined with a probability integral transform penalty to reduce model misspecification. We demonstrate that learning the joint dynamics of IV surfaces and prices produces market scenarios that are consistent with historical features and lie within the sub-manifold of surfaces that are essentially free of static arbitrage. Finally, we demonstrate that delta hedging using the simulated surfaces generates profit and loss (P&L) distributions that are consistent with realized P&Ls.

On the Existence of a Positive Solution of a Boundary Value Problem for a Nonlinear Second-Order Functional-Differential Equation with Integral Boundary Conditions

On the Existence of a Positive Solution of a Boundary Value Problem for a Nonlinear Second-Order Functional-Differential Equation with Integral Boundary Conditions

Partial Stability in Probability of Nonlinear Stochastic Discrete-Time Systems with Delay

A system of nonlinear stochastic functional-difference equations with finite delay is considered. By assumption, this system has a “partial” trivial equilibrium (with respect to part of the state variables). The problem under study is to analyze partial stability in probability of this equilibrium: stability is considered with respect to part of the variables determining it. The problem is solved using a discrete-stochastic modification of the method of Lyapunov–Krasovskii functionals. Conditions for partial stability in probability are established. An example is provided to illustrate the features of the proposed approach and the rationale for introducing a one-parameter family of functionals.

Over-order multiplicities and their application in controlling delay dynamics. On zeros’ distribution of linear combinations of Kummer hypergeometric functions

A series of recent works have shown that, for a system of linear functional differential equations, a spectral value having a multiplicity exceeding the order of the system tends to correspond to the spectral abscissa of the system, a property called MID for multiplicity-induced-dominancy. In particular, when this multiplicity coincides with the degree of the characteristic quasipolynomial, this property is called generic MID (GMID), in opposition to the intermediate MID (IMID), which corresponds to a multiplicity strictly smaller than the degree. The GMID has been fully characterized for single-delay retarded as well as neutral delay-differential equations thanks to the representation of the corresponding quasipolynomial in terms of a Kummer hypergeometric function. However, apart from partial results, in full generality, no result of the literature enables the characterization of the dominance of a spectral value having an intermediate multiplicity, which is essentially due to the lack of existing results among the open literature pertaining to linear combinations of Kummer functions’ zeros distribution. In this work, we overcome this difficulty and we further investigate the MID to cover the so-called over-order MID, that is, the cases where the multiplicity is larger than the order of the corresponding differential equation.

Tensor approximation of functional differential equations.

Functional differential equations(FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations(PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance (parallel) tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equationevolving from a random initial state.

Stability of a generalised difference functional equation

In this paper, we discuss the generalised Ulam Hyers stability of a generalized dierence functional equation