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This paper presents a numerical approach to solve fuzzy partial differential equation restrict to fuzzy boundary and initial conditions. The numerical solution to this problem is obtained by the finite difference method considering a particular type of fuzzy arithmetic called J 0 -interactive arithmetic. A study of the computational cost of the method is presented, as well as a comparison with the numerical solution obtained by the standard fuzzy arithmetic, which is associated with the Zadeh's extension principle. The paper focuses on the heat equation in order to illustrate the methodology. [ABSTRACT FROM AUTHOR]

In this work, we study a nonlocal erosion equation, which simulates the process of nanorelief formation. For a periodic boundary-value problem for the functional differential equation with partial derivatives, we examine local bifurcations in the cases where homogeneous equilibrium states change their stability. We prove that in the problem considered, subcritical bifurcations occur. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the smoothness of generalized solutions of the first boundary-value problem for a strongly elliptic functional differential equation containing orthotropic contraction transformations of the arguments of the unknown function in the leading part. The problem is considered in a disc, the coefficients of the equation are constant. Orthotropic contraction is understood as different contraction in different variables. Conditions for the conservation of smoothness on the boundaries of neighboring subdomains formed by the action of the contraction transformation group onto the disc are found in explicit form for any right-hand side from the Lebesgue space. [ABSTRACT FROM AUTHOR]

The stability of systems of linear autonomous functional differential equations of neutral type is studied. The study is based on the well-known representation of the solution in the form of an integral operator, the kernel of which is the Cauchy function of the equation under study. The definitions of Lyapunov, asymptotic, and exponential stability are formulated in terms of the corresponding properties of the Cauchy function, which allows us to clarify a number of traditional concepts without loss of generality. Along with the concept of asymptotic stability, a new concept of strong asymptotic stability is introduced. The main results are related to the stability with respect to the initial function from the spaces of summable functions. In particular, it is established that strong asymptotic stability with initial data from the space L1 is equivalent to the exponential estimate of the Cauchy function and, moreover, exponential stability with respect to initial data from the spaces Lp for any p ≥ 1. [ABSTRACT FROM AUTHOR]

A boundary value problem for a nonlinear second-order functional-differential equation on the interval with integral boundary conditions is considered. Sufficient conditions for the existence of at least one positive solution of the problem under consideration are found by using special topological tools. A nontrivial example in which the sufficient conditions for the unique solvability of the problem are satisfied is given. [ABSTRACT FROM AUTHOR]

In this work, we proposed some generalized Darbo type fixed point theorem by employing the concept of generalized operators and measure of non-compactness within the framework of partial order Banach spaces. Some fixed point theorems and their consequences are presented as part of our exploration. These theorems encompass the presentation of various fixed point theorems and the subsequent implications they entail. Further, we demonstrate the practical application of our findings by establishing the existence of solutions for delay differential equations. To substantiate our conclusions, we provide numerical estimations based on a real-world example. This research contributes to the advancement of fixed point theory in the context of partial order Banach spaces and showcases its practical relevance through a concrete application and empirical support. [ABSTRACT FROM AUTHOR]

We consider an initial-boundary value problem for a system of two differential equations, one of which is ordinary and the other a partial differential equation, where the connection between the equations is carried out through an integral functional. In this case, the boundary conditions contain higher-order time derivatives of the unknown functions. The initial-boundary value problem models the rotation of a mechanical system consisting of two rigid bodies connected by an elastic rod. The rotation is considered around the center of mass of one of the rigid bodies and carried out by a moment of external forces (control torque) applied to the axis of rotation of the rigid body. For the initial-boundary value problem, we introduce the concept of a generalized solution and prove theorems on the existence and uniqueness of a generalized solution and on the well-posedness of the problem formulation. We solve problems of optimal control of the rotation of a mechanical system from the initial state to the final state at a given point in time, minimizing the value of the control torque and minimizing the energy functional of the control torque. In this formulation, we also solve the performance problems under restrictions on the value of the control torque and on the value of the energy integral of the control torque. [ABSTRACT FROM AUTHOR]

We study a problem of damping a control system described by functional-differential equations of natural order and neutral type with nonsmooth complex coefficients on an arbitrary tree with global delay. The latter means that the delay propagates through internal vertices of the tree. Minimization of the energy functional of the system leads to a variational problem. We establish its equivalence to a certain self-adjoint boundary value problem on the tree for equations of order with nonlocal quasi-derivatives and multidirectional shifts of the argument as well as Kirchhoff-type conditions emerging at the internal vertices. The unique solvability of both problems is proved. [ABSTRACT FROM AUTHOR]

In this work, we prove a nonlocal Halanay inequality with an exact decay rate. This enables us to analyze behavior of solutions to some classes of nonlocal ODEs and PDEs involving unbounded delays. The obtained results extend and improve the previous ones proved for fractional differential equations and other nonlocal subdiffusion equations. [ABSTRACT FROM AUTHOR]

The oscillatory behavior of solutions of an even-order differential equation with a superlinear neutral term is considered using Riccati and generalized Riccati transformations, the integral averaging technique, and the theory of comparison. New sufficient conditions are established in the noncanonical case. An example is given to support our results. [ABSTRACT FROM AUTHOR]

A periodic boundary-value problem for an equation with a deviating spatial argument is considered. Using the Poincaré–Dulac method of normal forms, the method of integral manifolds, and asymptotic formulas, we examine a number of bifurcation problems of codimension 1 and 2. For homogeneous equilibrium states, we analyze possibilities of implementing critical cases of various types. The problem on the stability of homogeneous equilibrium states is studied and asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained. [ABSTRACT FROM AUTHOR]

An operational approach developed earlier for the spectral method that uses Legendre polynomials is generalized here for arbitrary systems of basis functions (not necessarily orthogonal) that satisfy only two conditions: the result of multiplication by or of differentiation with respect to is expressed in the same functions. All systems of classical orthogonal polynomials satisfy these conditions. In particular, we construct a spectral method that uses Chebyshev polynomials, which is most effective for numerical computations. This method is applied for numerical solution of the linear functional equations that appear in problems of generalized summation of series as well as in the problems of analytical continuation of discrete maps. We also demonstrate how these methods are used for solution of nonstandard and nonlinear boundary value problems for which ordinary algorithms are not applicable. [ABSTRACT FROM AUTHOR]

This manuscript focuses on a class of stochastic functional differential equations driven by time-changed Brownian motion. By utilizing the Lyapunov method, we capture some sufficient conditions to ensure that the solution for the considered equation is η-stable in the pth moment sense. Subsequently, we present some new criteria of the η-stability in mean square by using time-changed Itô formula and proof by contradiction. Finally, we provide some examples to demonstrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In a Banach space, we consider the Cauchy problem and the Dirichlet and Neumann boundary value problems for a functional-differential equation generalizing the Euler–Poisson–Darboux equation. A sufficient condition for the solvability of the Cauchy problem is proved, and an explicit form of the resolving operator is indicated, which is written using the Bessel and Struve operator functions introduced by the author. For boundary value problems in the hyperbolic case, we establish conditions imposed on the operator coefficient of the equation and the boundary elements that are sufficient for the unique solvability of these problems. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of d x a (s) + g (s , x a (s - ω (s))) = I x a (s) + f (s , x a (s - ϱ (s))) d s + ς (s) d ϖ H (s) , 0 ≤ s ≤ T , x a (s) = ζ (s) , - ρ ≤ s ≤ 0. The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples. [ABSTRACT FROM AUTHOR]

A new numerical-and-analytical method is developed for solving geometrically and physically nonlinear problems of bending shallow shells of complex shapes made from materials with different resistance to tension and compression. To linearize the initial nonlinear problem, the method of continuous continuation in the parameter associated with the external load was used. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear bending of a hollow shell (displacements, deformations, stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta– Merson method with automatic step selection. The initial conditions are found in the solution to the problem of geometrically linear deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution in the form of a formula, which solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. The problems of nonlinear deformation of a square cylindrical shell and a shell of complex shape with combined fixation conditions are solved. The influence of the direction of external loading, geometric shape, and fixation conditions on the stress-strain state is investigated. It is shown that failure to consider the different behaviors of the material in tension and compression leads to significant errors in calculating the stress-strain state parameters. [ABSTRACT FROM AUTHOR]

The purpose of this study is to establish fixed-point results for new interpolative contraction mappings in the setting of Busemann space involving a convex hull. To illustrate our findings, we also offer helpful and compelling examples. Finally, the theoretical results are applied to study the existence of solutions to fractional Navier–Stokes and fractional-functional differential equations as applications. [ABSTRACT FROM AUTHOR]

We prove a series of theorems on the asymptotic behavior of regular homeomorphic solutions of the nonlinear Cauchy–Riemann–Beltrami-type equation. Sufficient conditions for the logarithmic and exponential growth of regular solutions are obtained. [ABSTRACT FROM AUTHOR]

We obtain new particular solutions of nonlinear functional-differential equations of neutral type with linear deviations of the argument encountered in the theory of self-similar potentials and coherent states of quantum mechanics, and study some of their properties. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of an elliptic-type functional differential equation containing the contraction transformation of the arguments of the unknown function in its leading part, and the contractions are different for different arguments. Some necessary and sufficient conditions for the validity of a Gå rding-type inequality, which is an analog of the strong ellipticity condition, are presented in an explicit form. The Fredholm solvability and the structure of the spectrum of the first boundary value problem in the Sobolev spaces are studied. Sufficient conditions for the solvability of the equation in the Kondrat'ev weighted spaces on the plane are given. In the course of the proof, sufficient conditions for the invertibility of a finite-difference operator with variable coefficients on a line are obtained. Some concrete examples illustrating the results thus obtained are presented. [ABSTRACT FROM AUTHOR]

A method of extending the classical ARIMA and GARCH models to the continuous and nonlinear cases is considered. Stochastic functional differential equations, which are a natural generalization of the sums of independent random variables, are considered extended models. Along with the new model, a relevant optimization problem of estimating model parameters is considered, and the non-parametric problem is reduced to a parametric one. The new model was tested on real data, and the forecast results were compared with classical models. [ABSTRACT FROM AUTHOR]

We discuss the well-known Myshkis result on the stability of nonautonomous first-order delay differential equations, providing an extension to the general differential equations with aftereffect, and make comparison with available results. [ABSTRACT FROM AUTHOR]

A functional differential equation with a discrete retarded argument and a constant concentrated delay is considered. The problem of the asymptotic stability of this equation is reduced to the problem of location of the spectrum for the shift operator. Coefficient sufficient conditions for the asymptotic stability of this equation are obtained. The domain in the parameter space such that these conditions are necessary is obtained. [ABSTRACT FROM AUTHOR]

This study introduces a novel functional link neural network and explores the associated theorems for a class of nonlinear differential equations with fractional variable-order and time-varying delays. Lagrange polynomials, known for their efficiency in solving fractional calculus problems, are utilized as the neuro-solutions within the neural network structure. For an efficient learning process in artificial neural networks, appropriate activation functions play a crucial role. To handle the fractional derivative of variable-order and time-varying delays, it is essential to select the right activation function. Hence, numerical simulations are conducted to determine the optimal activation function for these equations. The training of the neural network is done using a modified Newton–Raphson method instead of the traditional supervised learning methods. As a result, the proposed functional link neural network provides more accurate results compared to conventional neural networks like the multilayer perceptron. [ABSTRACT FROM AUTHOR]

This study proposes a new numerical-and-analytical method for solving geometrically nonlinear problems of bending of complex-shaped plates made of functionally graded materials developed. The problem was formulated within the framework of a refined higher-order theory considering the quadratic law of distribution of transverse tangential stresses along the plate thickness. To linearize the nonlinear problem, we used the method of continuous continuation in the parameter associated with the external load. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear plate bending (displacements, strains, and stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient criteria for ensuring the existence of solutions and uniqueness for a class of nonlinear neutral Caputo fractional differential equations supplemented with infinite delay and nonlocal boundary conditions involving fractional derivatives. The theory of infinite delay and standard fixed point theorems are employed to obtain the existence results for the given problem. Examples will be constructed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We establish efficient conditions guaranteeing the existence of a solution of the periodic-type boundary-value problem for a two-dimensional system of nonlinear functional-differential equations in the case where the right-hand side of the system is the sum of positively homogeneous terms of degrees λ and 1/λ and other terms with relatively slow growth at infinity. The general results are reformulated in a special case of differential equations with maxima. [ABSTRACT FROM AUTHOR]

We obtain new properties of the solutions of a nonlinear functional-differential equation of neutral type with linear deviation of the argument encountered in quantum mechanics in the investigation of self-similar potentials and coherent states. Namely, we establish the conditions under which the solution of the Cauchy problem is either given on the entire positive semiaxis or exists only on a finite interval and study its asymptotic behavior. In addition, under certain conditions, for a sufficiently smooth solution, we present asymptotic representations of both the solution itself and its derivatives with an accuracy that depends on the smoothness of the solution. [ABSTRACT FROM AUTHOR]

We consider the optimal control problem for systems with infinite memory whose models are described by functional-differential equations. We prove the theorem on existence, uniqueness, and continuity of solutions for a system of functional-differential equations in which the delay interval is infinitely large. Sufficient conditions for the existence of optimal controls in the optimal control problems for systems with infinite memory are obtained in terms of the right-hand sides of the equations of motion and the function of performance criterion. [ABSTRACT FROM AUTHOR]

We study the local and global existence and uniqueness of solution and the local well-possedness for a general class of abstract fractional differential equations with state-dependent delay. Our studies are developed using a general abstract framework which allows us to present comprehensive results and to correct some important errors in the associate literature. Our results on the existence and uniqueness of solution are proved without assuming that the forcing terms are locally Lipschitz. In the last section, we present some examples of fractional in time partial differential equations motivated from the theory of population dynamics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions and periodic solutions for a class of semilinear neutral functional differential equations with state-dependent delay, in which the linear part is non-autonomous and generates a linear evolution operator. We first establish the existence and regularity of bounded solutions for the considered equation, and then we show by using Banach fixed point theorem that these solutions have periodicity property or asymptotic periodicity property respectively under some conditions. Finally, an example to illustrate the obtained results is given. [ABSTRACT FROM AUTHOR]

A computational method is developed for a family of functional differential equations in one independent variable with a single deviating argument, assumed to be a delay. These equations are mapped into a finite system of ODEs, a subsystem of which involves only the function controlling the delay. Key features include efficiency during method of steps, freedom from Jacobians and root finding techniques, and computing a continuous approximation. The nonlinear differential equations may be retarded, neutral or advanced. The method is established for state dependent delays, stiff equations, discontinuous initial history, a specific loss of monotonicity in the delay and extended naturally to distributed delays. When restricted to ODEs, the method extends naturally to PDEs; the hope is the delayed version here can eventually be extended to delayed PDEs. Conditions for convergence of the approximation are established, and results of numerical experiments are reported to indicate robustness of the implementation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of (ω , Q) -periodic mild solutions for the following problem x ′ (t) = A x (t) + f (t , x (t)) , t ∈ R , on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable C 0 -semigroup and the nonlinearity f satisfies suitable properties. The approaches are based on the well-known Banach contraction principle. In addition, a sufficient criterion is established for the existence and uniqueness of (ω , Q) -periodic mild solutions to the Hopfield-type neural network model. [ABSTRACT FROM AUTHOR]

This paper first investigates the equivalence of the space and translation invariance of Stepanov-like doubly weighted pseudo almost automorphic stochastic processes for nonequivalent weight functions; secondly, based on semigroup theory, fractional calculations, and the Krasnoselskii fixed-point theorem, we obtain the existence and uniqueness of Stepanov-like doubly weighted pseudo almost automorphic mild solutions for a class of nonlinear fractional stochastic neutral functional differential equations under non-Lipschitz conditions. These results enrich the complex dynamics of Stepanov-like doubly weighted pseudo almost automorphic stochastic processes. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of a periodic solution for a third-order neutral delay differential equation (NDDE) by applying Mawhin's continuation theorem of coincidence degree and analysis techniques. An illustrative example is given as an application to support our results. To confirm the accuracy of our results, we also present a plot of the behavior of the periodic solution. [ABSTRACT FROM AUTHOR]

The authors investigate the asymptotic stability in quadratic mean of the trivial solution of the Gikhman–Ito stochastic diffusion functional differential equations in terms of the eigenvalues of the matrix constructed from the coefficients of these equations. [ABSTRACT FROM AUTHOR]

In this paper, a neutral functional differential equation with multiple delays is considered. In a first step, we assumed some sufficient hypotheses to guarantee the existence of the Bogdanov–Takens and the triple-zero bifurcations. In a second step, the normal form of the two bifurcations is obtained by using the reduction on the center manifold and the theory of the normal form. Finally, we applied our study to a class of three-neuron bidirectional associative memory networks, its dynamic behaviors are studied and proved by an example and its numerical simulations. [ABSTRACT FROM AUTHOR]

The Auckland Numerical Ordinary Differential Equations (ANODE) series of conferences and workshops are organized every few years by a group of Auckland numerical analysts. ANODE 2023 was held in honor of John Butcher's 90th birthday and attracted colleagues, friends, and former students from around the world. The scientific program of ANODE 2023 reflected John Butcher's significant influence in the field of Numerical Analysis. This special issue includes 15 articles that cover a wide range of topics, showcasing John's contributions to the field and various applications. The editors express their gratitude to the authors, the editorial team, and the organizing committee for their support in completing this special issue. [Extracted from the article]

In this manuscript, we study the existence and uniqueness of solutions for a new neutral hybrid nonlinear differential equation in the context of a fractional generalized operator in the sense of ψ-Caputo. To emphasize the novelty of the manuscript, a pure technique of the noncompactness measures is applied to a hybrid system based on the notion of the modulus of continuity in Darbo's criterion that covers the existing results of other works published before. The Ulam–Hyers and generalized Ulam–Hyers stabilities are explored for the given neutral nonhybrid nonlinear problem. An application is prepared in the framework of an example to ensure the validity of theorems for different cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of pth-moment stability of stochastic functional differential equations with Markovian switching and impulsive control via comparison principle. Employing stochastic analysis theory and an impulsive delay differential inequality, we establish a new comparison principle for stochastic functional differential equations with Markovian switching and impulsive control. Using the comparison principle, we derive sufficient conditions for stochastic functional differential equations with Markovian switching and impulsive control by the stability of impulsive delay differential equations. An example is provided to show the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, a generalized Motsch-Tadmor model with piecewise interaction functions and fixed processing delays is investigated. According to functional differential equation theory and correlation properties of the stochastic matrix, we obtained sufficient conditions for the system achieving flocking, including an upper bound of the time delay parameter. When the parameter is less than the upper bound, the system achieves asymptotic flocking under appropriate assumptions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of periodic solutions for asymptotically linear second-order delay differential equations. We will establish an index theory for the linear system directly in the sense that we do not need to change the problem of the original linear system into the problem of an associated Hamiltonian system. By using the critical point theory and the index theory, some new existence results are obtained. [ABSTRACT FROM AUTHOR]

In a Banach space, we study the solvability of the pursuit problem in the sense of L.S. Pontryagin for a differential game whose dynamics is described by a functional-differential equation of neutral type. [ABSTRACT FROM AUTHOR]

In this paper, we study the parameter sensitivity analysis a class of fractional impulsive stochastic differential equation with S p -weighted pseudo almost periodic coefficients and infinite delay in Hilbert spaces. Firstly, a more appropriate concept of piecewise S p -weighted pseudo almost periodic in distribution for stochastic processes of class ∞ is introduced. The existence of piecewise weighted pseudo almost periodic mild solutions in distribution is presented by using α -order fractional resolvent operator theory, the stochastic analysis techniques and a fixed-point theorem. Secondly, the sensitivity properties of these infinite delay systems under non-Lipschitz conditions is also obtained. Finally, two examples are given for demonstration. [ABSTRACT FROM AUTHOR]

The Green's function method is a powerful technique to investigate the existence and uniqueness of the solution for two-point boundary value problems. The main task of this paper is to establish the maximal order of convergence of the Green's function method applied in the approximation of the solution for third- and fourth-order two-point boundary value problems with deviating argument. The developed iterative method uses a complete cubic spline interpolation procedure being able to provide both discrete and continuous approximation of the solution. Some numerical examples confirm the obtained theoretical results and illustrate the accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider a functional differential equation of parabolic type on a strip with a transformation of a spatial variable and boundary conditions with an oblique derivative. Using the Laplace and Fourier transforms, we obtain a representation of the problem under consideration in the form of a nonlinear integral equation. A special case of this representation is considered. The proved statements make it possible to implement iterative methods for obtaining approximate solutions of nonlinear partial differential equations taking into account the given conditions. The results show that the presented method is promising for solving similar problems. [ABSTRACT FROM AUTHOR]

We consider the problem of a zero-sum differential tracking game with a quadratic performance functional in which the plant subjected to uncontrolled disturbances is described by a nonlinear ordinary differential equation. The synthesis of optimal controls is known to necessitate online solving a scalar Bellman–Isaacs partial differential equation that contains information about the trajectory of the process to be monitored. The lack of information about this process over the entire control interval makes the synthesized controls unimplementable. An algebraic method is proposed for solving the Bellman-Isaacs equation, which contains the current value of the monitored process. As an illustration of the results obtained, we give the simulation of the behavior of a nonlinear system with two players with an open control horizon. [ABSTRACT FROM AUTHOR]

In this manuscript, the approximate controllability is encountered for a system involving instantaneous and non-instantaneous impulses. The theory of semigroup and Sadovskii's fixed point theorem are used to prove our main results. In the direction of controllability for a functional differential equations, a problem consists of either instant impulses or non-instant impulses separately. This is the reason that makes the study of the considered problem more interesting. We also make an observation related to high blood sugar level that is quite natural to relate with our problem. In the end, an example is given to validate the developed theory. [ABSTRACT FROM AUTHOR]

In this paper we study stability and convergence of stage-based interpolation Runge–Kutta (SBIRK) methods in which the functional term is approximated by using the stage-values for nonlinear Volterra functional differential equations (VFDEs). This type of methods includes some popular Runge–Kutta methods, for example, some continuous Runge–Kutta methods, as special cases. The uniqueness of solution to discrete algebraic equation is first proved. Unconditional contractivity and asymptotic stability of this class of methods for nonlinear VFDEs are obtained under the most general framework. A global error bound on infinite integration interval for arbitrarily variable step-sizes is also derived for the first time. Some efficient SBIRK methods including Radau IA, Radau IIA and Lobatto IIIC Runge–Kutta methods with appropriate interpolation operator are provided. Numerical results illustrate the effectiveness of the proposed methods for VFDEs and verify our theoretical findings. [ABSTRACT FROM AUTHOR]