Volterra Functional Equations Research Articles

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

The stability of nonlinear delay integro-differential equations in the sense of Hyers-Ulam

Abstract In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.

On Stackelberg equilibrium in the sense of program strategies in Volterra functional operator games

For a nonlinear Volterra functional operator equation controlled by two players with the help of finite dimensional program controls with integral objective functionals we prove existence of Stackelberg equilibrium (in the style of M.S.Nikol'skiy). On this way we use our formerly proved results on continuous dependence of the state and functionals on finite dimensional controls and also classical Weierstrass theorem. The property of being singleton for the minimizer set of the first player is proved by the scheme of M.S. Nikol'skiy applied earlier for a linear ordinary differential equation.

Volterra Functional Equations in the Theory of Optimization of Distributed Systems. On the Problem of Singularity of Controlled Initial–Boundary Value Problems

Volterra Functional Equations in the Theory of Optimization of Distributed Systems. On the Problem of Singularity of Controlled Initial–Boundary Value Problems

Volterra functional equations and optimization of distributed systems. Special optimal controls

Volterra functional equations and optimization of distributed systems. Special optimal controls

Darbo type best proximity point results via simulation function with application

Primarily this work intends to prove the best proximity point (pair) results using the concept of measure of noncompactness and simulation functions. The obtained results generalize and extend some present state of the art on Darbo type fixed point theorems. The main results are applied to actualize the optimum solutions of a system of nonlinear mixed Fredholm–Volterra functional integro-differential equations with local initial conditions.

Floquet Theory and Stability of Periodic Solutions of Renewal Equations

We prove the validity of a Floquet theory and the existence of Poincare maps for periodic solutions of renewal equations, also known as Volterra functional equations. Our approach is based on sun–star perturbation theory of dual semigroups and relies on a spectral isolation property and on the regularity of the semiflow. This contributes a new chapter to the stability analysis, in analogy with ordinary and retarded functional differential equations as well as the case of equilibria.

Вольтерровы функциональные уравнения в проблеме устойчивости существования глобальных решений распределенных управляемых систем

Earlier the author proposed a rather general form of describing controlled initial–boundary value problems (CIBVPs) by means of Volterra functional equations (VFE) z(t)=f(t,A[z](t),v(t) ), t≡{t^1,⋯,t^n }∈Π⊂R^n, z∈L_p^m≡(L_p (Π) )^m, where f(.,.,.):Π×R^l×R^s→R^m; v(.)∈D⊂L_k^s – control function; A:L_p^m (Π)→L_q^l (Π)- linear operator; the operator A is a Volterra operator for some system T of subsets of the set Π in the following sense: for any H∈T, the restriction A├ [z]┤|_H does not depend on the values of ├ z┤|_(Π\H); (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets H∈T). This continuation is realized with the help of the chain {H_1⊂H_2⊂⋯⊂H_(k-1)⊂H_k≡Π}, where H_i∈T, i=¯(1,k.) A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case p=q=k=∞ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator F_v [z(.) ](t)≡f(t,A[z](t),v(t)) satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space L_∞^m (H) such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case 1≤p,q,k ≤∞, (this case covers a much wider class of CIBVP), the operator F_v; as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space L_p^m (H), for which the operator F_v is a contraction operator. The corresponding basic theorem (equivalent norm theorem) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.

Qualitative theory for Volterra difference equations

We investigate lp boundedness, the topological structure of solutions set and the asymptotic periodicity of Volterra functional difference equations. The theoretical results are complemented with a set of applications.

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.

Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer-Resource, Cannibalistic and Trophic Models.

With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer-resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class, we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur. The methods combine curve continuation, ODE solvers and test functions. Finally, we apply the methods to the above models using existing data for Daphnia magna consuming Algae and for Perca fluviatilis feeding on Daphnia magna. In particular, we validate the methods by deriving expressions for equilibria and bifurcations with respect to which we compute errors, and by comparing the obtained curves with curves that were computed earlier with other methods. We also present new curves to show how the methods can easily be applied to derive new biological insight. Schemes of algorithms are included.

On Renewal Equations in Population Biology

On Renewal Equations in Population Biology Odo Diekmann (Mathematical Institute, Utrecht University) The first aim of this talk is to show, by way of examples, how ubiquitous Renewal Equations are in the mathematical formulation of models in Population Dynamics and Infectious Disease Epidemiology. The second aim is to advocate viewing Renewal Equations as Delay Equations, on par with Delay Differential Equations. A third aim is to draw attention to a class of numerical methods that allow to use standard ODE tools for a numerical bifurcation analysis. [1] A. Lotka On an integral equation in population analysis Annals of Math. Stat. 10 (1939) 144-161 [2] W.O. Kermack, A.G. McKendrick A contribution to the mathematical theory of epidemics Proc. Roy. Soc. A 115 (1927) 700-721 doi : 10.1016/S0092-8240(05)80040-0 [3] W. Feller On the Integral Equation of Renewal Theory Annals of Math. Stat. 12 (1941) 243-267 [4] W. Feller An introduction to probability theory and its applications, Vol. II (Wiley, second edition, 1971) [5] J.A.J. Metz, O. Diekmann (eds.) The Dynamics of Physiologically Structured Populations Springer Lecture Notes in Biomathematics 68, 1986. downloadable from : http://webarchive.iiasa.ac.at/Research/EEP/Metz2Book.html [6] O.Diekmann, M.Gyllenberg, J.A.J.Metz, H.R.Thieme On the formulation and analysis of general deterministic structured population models. I. Linear theory J.Math.Biol. 36 (1998) 349 - 388 [7] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz, H.R. Thieme On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory J. Math. Biol. 43 (2001) 157 -189 [8] O. Diekmann, Ph. Getto, M. Gyllenberg Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars SIAM J. Math. Anal. 39 (2007) 1023-1069 [9] O. Diekmann, M. Gyllenberg, J.A.J. Metz, S. Nakaoka, A.M. de Roos Daphnia revisited : local stability and bifurcation theory for physiologically structured population models explained by way of an example J. Math. Biol. 61 (2010) 277-318 [10] O. Diekmann, M. Gyllenberg Equations with infinite delay : Blending the abstract and the concrete J. Diff. Equa. 252 (2012) 819-851 [11] D. Breda, O. Diekmann, W.F. de Graaf, A. Pugliese, R. Vermiglio On the formulation of epidemic models (an appraisal of Kermack and McKendrick) Journal of Biological Dynamics 6:sup2 (2012) 103-117 DOI:10.1080/17513758.2012.716454 [12] O. Diekmann & K. Korvasova Linearization of solution operators for state-dependent delay equations : a simple example Discrete and Continuous Dynamical Systems A 36 (1) 2016 137-149 doi:10.3934/dcds.2016.36.137 [13] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, R. Vermiglio Pseudospectral discretization of nonlinear delay equations : new prospects for numerical bifurcation analysis SIAM J. Applied Dynamical Systems (2016)15(1): 1-23 DOI. 10.1137/15M1040931 [14] O. Diekmann, Ph. Getto, Y. Nakata On the characteristic equation λ = α1 + (α2 + α3λ) e−λ and its use in the context of a cell population model J. Math. Biol. (2016) 72:877–908 DOI 10.1007/s00285-015-0918-8 [15] A. Calsina, O. Diekmann, J.Z. Farkas Structured populations with distributed recruitment: from PDE to delay formulation Mathematical Methods in the Applied Sciences DOI: 10.1002/mma.3898

An hp-spectral collocation method for nonlinear Volterra functional integro-differential equations with delays

In this paper, we propose a multistep Legendre–Gauss spectral collocation method for the nonlinear Volterra functional integro-differential equations (VFIDEs) with vanishing delays. This method is easy to implement and possesses the high-order accuracy. We also provide a rigorous convergence analysis of the hp-version of the multistep spectral collocation method under H1-norm. Numerical results confirm the theoretical analysis.

Stability analysis of Volterra functional equations by realization theory

The method of realization of input–output operators in the form of abstract discrete-time control systems and the frequency-domain method are used for the stability/instability analysis of a class of nonlinear Volterra functional equations. To this end, we construct an associated time-invariant abstract discrete-time control system in some weighted function spaces.

On a nonlocal boundary value problem of a coupled system of Volterra functional integro-dierential equations.

In this paper, we study the existence of a unique solution for a nonlocal boundary- value problem of coupled system of Volterra functional integro-dierential equations.

Computing the Eigenvalues of Realistic Daphnia Models by Pseudospectral Methods

This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.

On the asymptotic stability of Volterra functional equations with vanishing delays

We analyze the asymptotic stability of solutions of linear Volterra integral equations with general continuous convolution kernels and vanishing delays. The analysis is based on an extension of the variation-of-parameter formula for non-delay Volterra integral equations and on energy function techniques. The delay integral equations studied in this paper will be of interest in the (still open) stability analysis of numerical methods (e.g. collocation and Runge-Kutta-type methods) for Volterra integral equations with vanishing delays.

Variational Iteration Method for Volterra Functional Integrodifferential Equations with Vanishing Linear Delays

Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay functionθ(t)vanishes inside the integral limits such thatθ(t)=qtfor0<q<1,t≥0. Either the approximate solutions that are converging to the exact solutions or the exact solutions of three test problems are obtained by using this presented process. The numerical solutions and the absolute errors are shown in figures and tables.

On Spectral Method for Volterra Functional Integro-Differential Equations of Neutral Type

The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence.

Numerical solution of volterra functional integral equation by using cubic B‐spline scaling functions

In this article, we consider a class of nonlinear functional integral equations which has rather general form and contains a lot of particular cases such as functional equations and nonlinear integral equations of Volterra type. We use a combination of a fixed point method and cubic semiorthogonal B‐spline scaling functions to solve the integral equation numerically. We provide an error analysis for the method which shows that the approximate solution converges to the exact solution. Some numerical results for several test problems are given to confirm the accuracy and the ease of implementation of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 699–722, 2014