Partial Neutral Functional Differential Equations Research Articles

Existence and Regularity of solutions in alpha norm for some second order partial neutral functional differential equations in Banach Spaces

The purpose of this paper is to study the existence and regularity of solutions in the $\alpha$-norm for some second order partial neutral functional differential equations with finite delay in Banach spaces. To do this, we use the cosine family theory and Schauder's fixed point Theorem to establish the local existence of solutions and we give some sufficients conditions to enssure the regularity of solutions. Finally, we give an example to illustrate the theoretical results.

Pseudo almost automorphy of stochastic neutral partial functional differential equations with Lévy noise

ABSTRACT The paper introduces and studies the concepts of (Poisson) -pseudo almost automorphy of class r for stochastic processes. Under suitable conditions on the coefficients, we establish the existence, and uniqueness of mild solutions, which are -pseudo almost automorphic of class r in distribution for stochastic neutral partial functional differential equations with Lévy noise.

Trotter-Kato approximations of stochastic neutral partial functional differential equations

Trotter-Kato approximations of stochastic neutral partial functional differential equations

Regularity results for some class of nonautonomous partial neutral functional differential equations with finite delay

Regularity results for some class of nonautonomous partial neutral functional differential equations with finite delay

Pseudo almost periodic solutions of infinite class under the light of measure theory and applications

The aim of this work is to present a new approach to study weighted pseudo almost periodic functions with infinite delay using the measure theory. We study the existence and uniqueness of (μ, ν)‐pseudo almost periodic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed on ] − ∞, 0].

Existence, Uniqueness and Stability of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays Driven by a Fractional Brownian Motion

This article presents the result on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations driven by a fractional Brownian motion. The results are obtained by using the method of successive approximation and Bihari’s inequality.

Existence and Stability Results of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays and Poisson Jumps

In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations with infinite delays and poisson jumps. The results are obtained by using the method of successive approximation and Bihari’s inequality.

Periodic Solutions to Partial Neutral Functional Differential Equations in Admissible Spaces on a Half-Line

We prove the existence and uniqueness of periodic solutions to partial neutral function differential equation of the form $\frac {\partial Fu_{t}}{\partial t}= A(t)Fu_{t} +g(t,u_{t}),~t\in [0,\infty )$, on a Banach space X, where the operator-valued function t↦A(t) and the nonlinear delay operator g(t,v) are both T-periodic with respect to t, whereas g is φ-Lipschitz with respect to v for φ belonging to an admissible space. Then, in the case that the family (A(t))t≥ 0 generates an evolution family having an exponential dichotomy, we apply our abstract results to study the existence, uniqueness of periodic solutions. We also prove the conditional stability, and the existence of local stable manifolds around such periodic solutions.

Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line

A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is studied.

New Results for Some Neutral Partial Functional Differential Equations

In this research work, we study some new results for the existence and uniqueness of doubly measures pseudo almost automorphic and pseudo almost periodic solutions for some neutral partial functional differential equations in Hilbert spaces. The main results of this paper are based on the compositional theorem and the Banach fixed point theorem. Some examples are given to illustrate the theory.

Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line

For the following class of partial neutral functional differential equations \[ \begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases} \] we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.

Stable manifolds for some partial neutral functional differential equations with non-dense domain

We investigate the asymptotic solution beha\-vior for the partial neutral functional differential equation$$ \begin{cases} & \frac {d}{dt}\mathcal {D}u_t=(A+B(t))\mathcal {D}u_t+f(t,u_t), \quad \hbox {$t\geq s\geq 0$,} \\ & u_s=\phi \in \mathcal {C}:=C([-r,0],X), \end{cases} $$ where the linear operator $A$ is not necessarily densely defined and satisfies the Hille-Yosida condition, and the delayed part $f$ is assumed to satisfy the $\varphi $-Lipschitz condition, i.e., $\|f(t,\phi )-f(t,\psi )\|\leq \varphi (t)\|\phi -\psi \|_{\mathcal {C}}$. Here $\varphi $ belongs to some admissible spaces and $\phi ,\ \psi \in \mathcal {C}:=C([-r,0],X)$. More precisely, we prove the existence of stable (respectively, center stable) manifolds when the linear part generates an evolution family having an exponential dichotomy (respectively, trichotomy) on the half positive line. Furthermore, we show that such stable manifold attracts all mild solutions of the considered neutral equation. An example is given to assimilate our theory.

Periodic solutions and their conditional stability for partial neutral functional differential equations

We prove the existence and uniqueness of the periodic solution as well as its conditional stability for the semilinear partial neutral function differential equation of the form $$\begin{aligned} \frac{\partial Fu_t}{\partial t}= A(t)Fu_t +g(t,u_t), t\in (0,\infty );\;u_0=\phi . \end{aligned}$$Here the operator-valued function $$t\mapsto A(t)$$ is T-periodic; the nonlinear delay operator $$g(t,\psi )$$ is T-periodic with respect to t and Lipschitz continuous with respect to $$\psi $$. In our strategy, on the one hand, we prove a Massera-type theorem for the linearized neutral equation, then pass to the semilinear equation using fixed-point arguments and Neumann series. On the other hand, our abstract results fit perfectly with the case that the family $$(A(t))_{t\ge 0}$$ generates an evolution family having exponential dichotomy. In such a case, we can apply our abstract results to show the existence, uniqueness, and conditional stability of periodic solution to the above equation.

Pseudo almost periodic in distribution solutions to impulsive partial stochastic functional differential equations

ABSTRACTIn this paper, we study the piecewise pseudo almost periodicity in distribution for a stochastic process. Using the analytic semigroup theory and fixed point strategy with stochastic analysis theory, we obtain the existence and the exponential stability of piecewise pseudo almost periodic in distribution mild solutions for impulsive partial neutral stochastic functional differential equations under non-Lipschitz conditions. Moreover, an example is given to illustrate the general theorems.

Weighted Pseudo Almost Periodic and Pseudo Almost Automorphic Solutions of Class r for Some Partial Differential Equations

The aim of this work is to present new approach to study weighted pseudo almost periodic and automorphic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We study the existence and uniqueness of $$\mu $$ -pseudo almost periodic and automorphic solutions of class r for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed in Adimy and co-authors. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part are assumed to be pseudo almost periodic or pseudo almost automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.

Solutions to neutral partial functional differential equations with functional delay

An approximate solution to a neutral partial functional differential equation with functional delay is established by using method of semidiscretization. Sufficient conditions for the existence and uniqueness of a strong solution are also obtained.

On the oscillation of impulsive vector partial differential equations with distributed deviating arguments

AbstractIn this paper, we consider a class of nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. For this class, we establish sufficient conditions for theH-oscillation of the solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. We provide an example to illustrate the main result.

The asymptotic behavior for neutral stochastic partial functional differential equations

ABSTRACTIn this article, we consider the existence and uniqueness, and the asymptotic behavior of the strong solution for the following neutral stochastic partial functional differential equations.where A(t): V → V* is a linear bounded operator. , and are some appropriate measurable functions. By establishing the variational framework, the existence and uniqueness for the strong solution of such equations is shown under the coercivity condition. Then, the asymptotic behavior of the strong solution is investigated and some well-known results are extended. As a by-product, the exponential stability in mean square and almost sure exponential stability for a strong solution of neutral stochastic partial differential equations with delays are also discussed by utilizing the integral inequality. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained results.

Oscillation of Even Order Impulsive Neutral Partial Differential Equations with Distributed Deviating Arguments

In this paper, we will consider a class of boundary value problems associated with even order nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. Sufficient conditions are obtained for the oscillation of solutions using impulsive differential inequalities and integral averaging scheme with pair of boundary conditions. Examples are specified to point up our important results.

Inertial Manifolds for Partial Neutral Functional Differential Equations in Admissible Spaces

We prove the existence of an inertial manifold for the partial neutral functional differential equations of the form \(\frac {\partial }{\partial t}(Fu_{t}) +A (Fu_{t}) = {\Phi }(t,u_{t})\), where the partial differential operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large gap; the difference operator \(F: \mathcal {C}_{\beta } \rightarrow X\) is a bounded linear operator, and the nonlinear delay operator Φ satisfies the φ-Lipschitz condition, i.e., \(\|{\Phi }(t,\phi )\| \leq \varphi (t)(1+|\phi |_{\mathcal {C}_{\beta }})\) and \(\|{\Phi }(t,\phi )-{\Phi }(t,\psi )\| \leq \varphi (t)|\phi - \psi |_{\mathcal {C}_{\beta }}\), where φ belongs to an admissible function space defined on \(\mathbb {R}\). Our main method is based on Lyapunov–Perron’s equations combined with the admissibility of function spaces and the technique of choosing F-induced trajectories.