Search

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept. [ABSTRACT FROM AUTHOR]

In this work, we first propose a concept of Weyl almost anti-periodic functions. Then, we make use of the contraction mapping principle and analysis techniques to research the existence of a unique Weyl almost anti-periodic solution to a neutral functional semilinear abstract differential equation. Finally, we give an example of a neutral functional partial differential equation to show the validity of the obtained results.

In this work, we first propose a concept of Weyl almost anti-periodic functions. Then, we make use of the contraction mapping principle and analysis techniques to research the existence of a unique Weyl almost anti-periodic solution to a neutral functional semilinear abstract differential equation. Finally, we give an example of a neutral functional partial differential equation to show the validity of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, under some ordered conditions, we investigate the existence of positive ω-periodic solutions for a class of second-order neutral functional differential equations with delayed derivative in nonlinearity of the form (x(t) − cx(t − δ))″ + a(t)g(x(t))x(t) = λb(t)f(t, x(t), x(t − τ1(t)), x′(t − τ2(t))). By utilizing the perturbation method of a positive operator and the fixed point index theory in cones, some sufficient conditions are established for the existence as well as the non-existence of positive ω-periodic solutions. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity ( x ( t ) − c x ( t − δ ) ) ″ + a ( t ) g ( x ( t ) ) x ( t ) = λ b ( t ) f ( t , x ( t ) , x ( t − τ 1 ( t ) ) , x ′ ( t − τ 2 ( t ) ) ) $(x(t)-cx(t-\delta)){''}+a(t)g(x(t))x(t)=\lambda b(t)f(t,x(t),x(t-\tau_{1}(t)),x'(t-\tau_{2}(t)))$ . By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.

This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity (x (t) − c x (t − δ)) ″ + a (t) g (x (t)) x (t) = λ b (t) f (t , x (t) , x (t − τ 1 (t)) , x ′ (t − τ 2 (t))) . By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established. [ABSTRACT FROM AUTHOR]

One fundamental step towards grasping the global dynamic structure of a population system involves characterizing the convergence behavior (specifically, how to characterize the convergence behavior). This paper focuses on the neutral functional differential equations arising from population dynamics. With the help of monotonicity techniques and functional methods, we analyze the subtle relations of both the ω-limited set and special point. Meanwhile, we prove that every bounded solution converges to a constant vector, as t tends to positive infinity. Our results correlate with the findings from earlier publications, and our proof yields an improved Haddock conjecture.

In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) (x(t) - cx(t - σ))(2m) + f (x(t))x'(t) + g(t, x(t - δ)) = e(t). The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new criteria on the boundedness and asymptotic constancy of solutions for a class of scalar neutral functional differential equations with time-varying delays via differential inequality techniques. Our results are an improvement of existing ones and generalization of the Haddock conjecture.

In this article, we introduced and explore the properties of two sets of functions: weighted pseudo periodic functions of class r, and weighted Stepanov-like pseudo periodic functions of class r. We show the existence and uniqueness of weighted pseudo periodic solution of class r that are solutions to neutral functional differential equations. Other applications to partial differential equations and scalar reaction-diffusion equations with delay are also given.

In this paper, we establish some new criteria on the boundedness and asymptotic constancy of solutions for a class of scalar neutral functional differential equations with time-varying delays via differential inequality techniques. Our results are an improvement of existing ones and generalization of the Haddock conjecture. [ABSTRACT FROM PUBLISHER]

Motivated by recent stability results on one-step methods, especially Runge–Kutta methods, for the generalized pantograph equation (GPE), in this paper we study the stability of one-leg multistep methods for these equations since the one-leg methods have less computational cost than Runge–Kutta methods. To do this, a new stability concept, G q ( q ¯ ) -stability defined for variable stepsizes one-leg methods with the stepsize ratio q which is an extension of G -stability defined for constant stepsizes one-leg methods, is introduced. The Lyapunov functional of linear system is obtained and numerically approximated. It is proved that a G q ( q ¯ ) -stable fully-geometric mesh one-leg method can preserve the decay property of the Lyapunov functional for any q ∈ [ 1 , q ¯ ] . The asymptotic contractivity, a new stability concept at vanishing initial interval, is introduced for investigating the effect of the initial interval approximation on the stability of numerical solutions. This property and the bounded stability of G q ( q ¯ ) -stable one-leg methods for linear and nonlinear problems are analyzed. A numerical example which further illustrates our theoretical results is provided. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of an infinite number of subharmonic periodic solutions to a class of second-order neutral nonlinear functional differential equations. Subdifferentiability of lower semicontinuous convex functions $varphi(x(t),x(t-au))$ and the corresponding conjugate functions are constructed. By combining the critical point theory, Z2-group index theory and operator equation theory, we obtain the infinite number of subharmonic periodic solutions to such system.

In this article, by using a fixed point theorem, we study the existence and regularity of mild solutions for a class of abstract neutral functional differential equations with infinite delay. The fraction power theory and alpha-norm is used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. A stability result for the autonomous case is also established. We conclude with an example that illustrates the applications of the results obtained.

In this paper, the authors consider the problem of existence of periodic solutions for a second order neutral functional differential system with nonlinear difference D-operator. For such a system, since the possible periodic solutions may not be differentiable, our method is based on topological degree theory of condensing field, not based on Leray Schauder topological degree theory associated to completely continuous field. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of non-autonomous neutral functional differential equations with multi-proportional delays. It is shown that all solutions of the addressed system are globally exponentially convergent by employing the differential inequality technique and a novel argument. The obtained results improve and supplement existing ones. We also use numerical simulations to demonstrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Our aim in this work is to study the existence of solutions of a neutral functional differential equation with state-dependent delay. We use the Schauder's fixed point theorem to show the existence of global solutions. [ABSTRACT FROM AUTHOR]

For neutral delay differential equations of the form with $$g$$ defined on an open subset of the space $$C([-h,0],\mathbb {R}^n)\times C^1([-h,0],\mathbb {R}^n)$$ , we extend an earlier principle of linearized stability. The present result applies to a wider class of neutral differential equations with state-dependent delays which includes models for population dynamics with maturation delay. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method. The collocation method can be applied in two versions: the finite element method and the spectral element method. We give convergence results for the collocation method deduced by the convergence theory developed in [S. Maset, Numer. Math., (2015), pp. 1-31] for a general discretization of an abstract reformulation of the problems. Such convergence results are then applied in Part II [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794-2821] of this paper to boundary values problems for a particular type of neutral functional differential equations, namely, differential equations with deviating arguments. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solutions for a impulsive semilinear neutral functional integrodifferential equations with infinite delay in Banach spaces. The results are obtained by using the Hausdorff measure of noncompactness. Examples are provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

Abstract: An algorithm for calculating the third-order normal form of a nonresonant Hopf–Hopf singularity in a neutral functional differential equation (NFDE) is established. The van der Pol equation with extended delay feedback is investigated as an NFDE of second order. The existence of Hopf–Hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm. Periodic solutions and quasi-periodic solutions are found with the aid of the bifurcation diagram, and corresponding numerical illustrations are presented. With the breaking down of the 3-torus, a chaotic attractor appears in this NFDE of second order, following the Ruelle–Takens–Newhouse scenario which usually arises for an ordinary differential equation of order at least 4. This transition is shown via both theoretical and numerical approaches. [Copyright &y& Elsevier]

In this article, we show that the existence of a Lyapunov–Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability (GES) of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map on the right-hand side of the equation is Lipschitz on bounded sets. A link between GES and input-to-state stability is also provided. [ABSTRACT FROM AUTHOR]

By means of Mawhin's continuation theorem and some analysis methods, the existence of 2 kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above periodic solutions set. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the authors investigate some new properties of nonlinear difference operator in the first, which is associated with neutral functional differential equations. Then, by using these new properties, the problem of existence of periodic solutions for neutral functional differential system with nonlinear operator is studied. The interesting thing is that the periodic solution to the system is allowed to be not differentiable. [Copyright &y& Elsevier]

Abstract: In this paper, the authors first investigate some new properties of the -operator which is associated with neutral functional differential equations. Then, by using these new properties, the problem of existence of periodic solutions of the -operator neutral functional differential system is studied. The interesting thing is that the -operator is allowed to be unstable, which is generalized by the corresponding studies in the past under the crucial assumption that the -operator is stable. [Copyright &y& Elsevier]

Abstract: In this paper, we study weighted pseudo almost periodic solutions of neutral functional differential equations. By applying the properties of weighted pseudo almost periodic functions and the exponential dichotomy of linear systems as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of weighted pseudo-almost periodic solution of the equations. [Copyright &y& Elsevier]

Abstract: Consider the system of neutral functional differential equations where , and , . It is shown that if is nondecreasing on , and some additional assumptions hold, then the limit set of every bounded solution of such a system with some initial conditions is composed of -periodic solutions. Our results are new and complement some corresponding ones already known. [Copyright &y& Elsevier]

This paper addresses the computation of equivariant normal forms for some Neutral Functional Differential Equations (NFDEs) near equilibria in the presence of symmetry. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations (FDEs) and on the existence of center (or other invariant) manifolds. We illustrate our general results by some applications to a detailed case study of additive neurons with delayed feedback. [ABSTRACT FROM AUTHOR]

Abstract: Consider the system of neutral functional differential equations where , , and , . It is shown that if is strictly increasing on , and some additional assumptions hold, then every bounded solution of such a system tends to an equilibrium. Our results correct and extend some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a nonlinear neutral functional differential equation which arises from a transmission line model. By means of Krasnoselskii’s fixed point theorem, we obtain sufficient conditions of the existence of asymptotic periodic solutions. This paper is motivated by the difficulty encountered in the study of integrated electronic circuits which are composed of lossy transmission lines and nonlinear resistive loads with exponential – characteristics. We also correct one of the results in [V.G. Angelov, Lossy transmission lines terminated by -loads with exponential – characteristics, Nonlinear Anal. RWA 8 (2007) 579–589]. As an application, we show there is a periodic oscillation along the line when the load is a p–n diode. [Copyright &y& Elsevier]

Abstract: We study a nonlinear neutral functional differential equation. Applying the properties of almost periodic function and exponential dichotomy of linear system as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of almost periodic solution of the equations. [Copyright &y& Elsevier]

Abstract: By means of Mawhin’s continuation theorem, a kind of p-Laplacian neutral functional differential equation with multiple variable parameters as follows: is studied. The interesting thing is that the coefficient is not a constant, by analyzing some properties of operator and then using Mawhin’s continuation theorem, some new results on the existence of periodic solutions are obtained. [Copyright &y& Elsevier]

Abstract: Consider the system of neutral functional differential equations where , , . It is shown that if is nondecreasing on , and some additional assumptions hold, then the limit set of every bounded solution of such a system with some initial conditions is composed of -periodic solutions. Our results are new and complement some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper, we consider two types of second-order neutral functional differential equations. By choosing available operators and applying Krasnoselskii’s fixed point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations. [Copyright &y& Elsevier]

Abstract: Consider the neutral functional differential equation where , , . It is shown that if is nondecreasing on , and some additional assumptions hold, then the limit set of every bounded solution of such a equation with some initial conditions is composed of -periodic solutions. Our results are new and complement some corresponding ones already known. [Copyright &y& Elsevier]

In this work, we shall consider standard optimal control problems for a class of neutral functional differential equations in Banach spaces. As the basis of a systematic theory of neutral models, the fundamental solution is constructed and a variation of constants formula of mild solutions is established. We introduce a class of neutral resolvents and show that the Laplace transform of the fundamental solution is its neutral resolvent operator. Necessary conditions in terms of the solutions of neutral adjoint systems are established to deal with the fixed time integral convex cost problem of optimality. Based on optimality conditions, the maximum principle for time varying control domain is presented. Finally, the time optimal control problem to a target set is investigated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence of periodic solutions for a -Laplacian neutral functional differential equation, as follows It is meaningful that is not a constant function, and that the functions and are allowed to change signs, which are different from the corresponding ones of known literature. [Copyright &y& Elsevier]