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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]

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 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 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, 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]

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: 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: 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]

Abstract: By using a fixed point theorem of strict-set-contraction, some criteria are established for the existence of positive periodic solutions for a periodic neutral functional differential equation with impulses of the form [Copyright &y& Elsevier]

Abstract: In this work, we deal with a new existence theory for positive periodic solutions for two kinds of neutral functional differential equations by employing the Krasnoselskii fixed-point theorem. Applying our results to various mathematical models we improve some previous results. [Copyright &y& Elsevier]

Abstract: By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays where () are ω-periodic functions and () are ω-periodic functions with respect to their first arguments, respectively. [Copyright &y& Elsevier]

Abstract: By using the coincidence degree theory of Mawhin, we study a kind of high-order neutral functional differential equation with distributed delay as follows: Some new results on the existence of periodic solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a class of abstract neutral functional differential equations in a separable Hilbert space and study the approximation of solutions. An example is also given to illustrate the applications of the abstract results. [Copyright &y& Elsevier]

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

By means of the abstract continuation theory for k-contractions, some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation: dN/dt=N(t)a(t)−β(t)N(t)−∑lower limit j=1, upper limit n bj(t)Nt−σj(t)−∑lower limit i=1, upper limit m ci(t)N′t−τi(t). [Copyright &y& Elsevier]

The problem of periodic solutions for a kind of kth-order linear neutral functional differential equation is studied. By using the theory of Fourier expansions, a sufficient and necessary condition to guarantee the existence and uniqueness of periodic solution is obtained. Further, by applying this result and Schauder's fixed point principle, a kind of kth-order nonlinear neutral functional differential equation is investigated, and some new results on existence of the periodic solutions are given as well. These results improve and extend some known results in recent literature. [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. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of implicit neutral functional differential equations. Based on the continuous Runge­Kutta method (for ordinary differential equations) and the collocation method (for functional equations), two general one-step methods are formulated and their uniform order of approximation is discussed. Numerical stability of a class of Runge­Kutta-collocation methods is analyzed. In the case where a neutral equation can be expressed in an explicit as well as an implicit form, the advantage of constructing a numerical scheme directly from the implicit form is explained theoretically and demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

In this study, we investigate the dynamic behavior on a system of neutral functional differential equations in critical case. Combining monotonicity techniques and functional methods, we develop the delicate monotonicity with the addressed system and show the convergence for every pre-compact solution with some initial condition, which improves and complements some existing ones. [ABSTRACT FROM AUTHOR]

In this paper, the dissipativity of a class of nonlinear neutral functional differential equations is studied. A new generalized discrete Halanay-type inequality is derived and the dissipativity of implicit Euler method for the above-mentioned equations is given by applying this inequality. [ABSTRACT FROM AUTHOR]