Functional Differential Equations Of Mixed Type Research Articles

Solutions of mixed-type functional differential equations with state-dependence

In this paper we study existence and asymptotic behaviors of solutions for mixed-type functional differential equation with state-dependence x˙(t)=f(t,x(T1(t,x(t))),...,x(Tn(t,x(t)))). Because of complexity of state-dependent terms, it is so difficult to discuss its stability by Lyapunov direct method. Through an order-preserving transformation, we generalize Gronwall inequality to obtain boundedness and asymptotics of solutions. It can be regarded as an extension to Lyapunov indirect method. Moreover, we indicate that Krasnoselskii fixed point theorem also can be used to study existence of sign-changing solutions. Finally, we apply our method to iterative functional differential equations and provide two concrete examples.

Lower and upper estimates of semi-global and global solutions to mixed-type functional differential equations

AbstractIn the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered and the results are illustrated by selected examples.

Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

Micropterons, nanopterons and solitary wave solutions to the diatomic Fermi–Pasta–Ulam–Tsingou problem

We use a specialized boundary-value problem solver for mixed-type functional differential equations to numerically examine the landscape of traveling wave solutions to the diatomic Fermi–Pasta–Ulam–Tsingou (FPUT) problem. By using a continuation approach, we are able to uncover the relationship between the branches of micropterons and nanopterons that have been rigorously constructed recently in various limiting regimes. We show that the associated surfaces are connected together in a nontrivial fashion and illustrate the key role that solitary waves play in the branch points. Finally, we numerically show that the diatomic solitary waves are stable under the full dynamics of the FPUT system.

A continuous kernel functions method for mixed-type functional differential equations

A continuous kernel functions method for mixed-type functional differential equations

Existence of global solutions to nonlinear mixed-type functional differential equations

The paper considers systems of nonlinear mixed-type functional differential equations. It aims to find conditions for the existence of global solutions with graphs staying in a prescribed domain. A relevant result is proved by Schauder–Tychonoff fixed-point technique with illustrative examples shown. A linear variant of the derived result is formulated, comparisons with known results are discussed and some open problems are formulated.

Mixed-Type Functional Differential Equations: A $C_{0}$-Semigroup Approach

In this paper we study certain systems of mixed-type functional differential equations, from the point of view of the $C_{0}$-semigroup theory. In general, this type of equations are not well-posed as initial value problems. But there are also cases where a unique differentiable solution exists. For these cases and in order to achieve our goal, we first rewrite the system as a classical Cauchy problem in a suitable Banach space. Second, we introduce the associated semigroup and its infinitesimal generator and prove important properties of these operators. As an application, we use the results to characterize the null controllability for those systems, where the control $u$ is constrained to lie in a non-empty compact convex subset $\Om{}$ of $\R^n$.

Computing Transitional Cycles for a Deterministic Time-to-Build Growth Model

Capital investment requires a gestation lag for being productive. This time-to-build feature can lead to an autonomous system of mixed-type functional differential equations (FDEs), causing aggregate fluctuations for a deterministic economy. We present a continuation method to solve a system of mixed FDEs by solving a sequence of boundary value problems for systems of ordinary differential equations (ODEs). Unlike other iteration techniques, this method can avoid the need to predetermine both forward-looking and backward-looking difference terms. The strategy is to form a homotopy that can deform a simpler ODE system into a target FDE system, while treating the deformation process as a sequence of ODEs. We can thereby compute oscillatory cycles for a time-to-build economy while it is in transition to the steady state.

R&D-based Calibrated Growth Models with Finite-Length Patents: A Novel Relaxation Algorithm for Solving an Autonomous FDE System of Mixed Type

The statutory patent length is 20 years in most countries. R&D-based growth models, however, often presume an infinite patent length. In this paper, finite-length patents are embedded in a non-scale R&D-based growth model, while allowing any patent's effective life to be terminated prematurely, subject to two idiosyncratic hazards from imitation and creative destruction. This gives rise to an autonomous system of mixed-type functional differential equations (FDEs) that had never been encountered in the growth literature. Its dynamics are driven by current, delayed and advanced states. We present a relaxation algorithm to solve these FDEs by solving a sequence of standard boundary value problems for systems of ordinary differential equations. We use this algorithm to simulate a calibrated U.S. economy's transitional dynamics by making discrete changes from the baseline 20 years patent length. We find that if transitional impacts are taken into account, the switch to the long-run optimal patent length can incur a welfare loss, albeit rather small.

Neutral Mixed Type Functional Differential Equations

We extend the existing Fredholm theory for mixed type functional differential equations developed by Mallet-Paret (J Dyn Differ Equ 11:1–47, 1999) to the case of implicitly defined mixed type functional differential equations. We present analogous results for the Fredholm alternative theorem, the cocycle property, and spectral flow property. In particular, we apply the theory to examples, one of which arises from modeling signal propagation in nerve fibers, and show the existence of traveling wave solutions via local continuation.

Optimal pollution control with distributed delays

We present a model of optimal stock pollution control with general distributed delays in the stock accumulation dynamics. Using generic functional forms and a distribution structure covering a wide range of distributions, we solve analytically the complex dynamic system that arises from the introduction of these distributed delays. From a theoretical standpoint, our contribution extends the dynamic optimization literature that focused on single discrete delays and develops an original method to address control problems written as mixed type functional differential equations with general kernels. Our results show the qualitative impact of acknowledging these distributed delays on the optimal pollution paths dynamics. We study analytically the properties of the dynamics and we identify the conditions for the occurrence of limit cycles. This theoretical work contributes to the design of efficient environmental policies in the presence of complex delays.

Stability and determinacy conditions for mixed-type functional differential equations

This paper analyzes the solution of linear mixed-type functional differential equations with either predetermined or non-predetermined variables. Conditions characterizing the existence and uniqueness of a solution are given and related to the local stability and determinacy properties of the steady state. In particular, it is shown that the relationship between the uniqueness of the solution and the stability of the steady-state is more subtle than the one that holds for ordinary differential equations, and gives rise to new dynamic configurations.

Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a monotone solution of the equation defined in the whole real axis, which tends to given values at ±∞. We introduce new numerical methods for the solution of the equation, analyse their performance, and present and discuss the results of the numerical simulations.

Almost homoclinic solutions for a certain class of mixed type functional differential equations

We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: $\ddot{q}(t)+V_{q}(t,q(t))+u(t,q(t),q(t-T),q(t+T))=f(t)$, where $t\in\mathbb R$, $q\in\mathbb R^{n}$ and $

Finite element solution of a linear mixed-type functional differential equation

This paper is devoted to the approximate solution of a linear first-order functional differential equation which involves delayed and advanced arguments. We seek a solution x, defined for t???(0, k???1],(k???IN ), which takes given values on the intervals [???1, 0] and (k???1, k]. Continuing the work started in previous articles on this subject, we introduce and analyse a computational algorithm based on the finite element method for the solution of this problem which is applicable both in the case of constant and variable coefficients. Numerical results are presented and compared with the results obtained by other methods.

The numerical solution of forward–backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x ′ ( t ) = a x ( t ) + b x ( t − 1 ) + c x ( t + 1 ) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.

Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x ( t ) , defined for t ∈ [ − 1 , k ] , ( k ∈ N ), that satisfies this equation almost everywhere on [ 0 , k − 1 ] and assumes specified values on the intervals [ − 1 , 0 ] and ( k − 1 , k ] . We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.

Mixed-type functional differential equations: A numerical approach

The equations considered in this paper are mixed-type functional equations (sometimes known as forward–backward differential equations) that take the form u ′ ( t ) = a u ( t ) + b u ( t − 1 ) + c u ( t + 1 ) , t ∈ I ⊂ R . We consider basic existence and uniqueness properties when I = [ t 1 , t 2 ] and we seek solutions u ∈ C [ t 1 − 1 , t 2 + 1 ] that satisfy u ( t ) = w 1 ( t ) , t ∈ [ t 1 − 1 , t 1 ] , u ( t ) = w 2 ( t ) , t ∈ [ t 2 , t 2 + 1 ] , for prescribed functions w 1 , w 2 absolutely continuous, respectively, on [ t 1 − 1 , t 1 ] , [ t 2 , t 2 + 1 ] . With arbitrary boundary conditions specified in this way, the problem turns out to be ill-posed and so existence and uniqueness questions have an important role to play in developing numerical schemes. We consider, with t 1 , t 2 ∈ N , numerical approximations of a solution when it exists. The numerical methods that we consider are linear θ -methods and we investigate computationally their effectiveness through some illustrative examples.

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations. It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.

Positive periodic solutions of impulsive functional differential equations

We study the existence and nonexistence of positive periodic solutions of a non-autonomous functional differential equation with impulses. The equations we study may be of delay, advance or mixed type functional differential equations and the impulses may cause the existence of positive periodic solutions. The methods employed are fixed-point index theorem, Leray-Schauder degree, and upper and lower solutions. The results obtained are new, and some examples are given to illustrate our main results.