Nonlinear Volterra Integro-differential Equations Research Articles

Energy methods for nonlinear hyperbolic volterra integrodifferential equations

(1979). Energy methods for nonlinear hyperbolic volterra integrodifferential equations. Communications in Partial Differential Equations: Vol. 4, No. 3, pp. 219-278.

Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces

SynopsisThe following theorem is proved: Let S(t), t≧0 be a dynamical system in an infinite dimensional Banach space X such that S(t) = S1(t)+S2(t) for t≧0, where (1) uniformly in bounded sets of x in X, and (2) S2(t) is compact for t sufficiently large. Then, if the orbit {S(t)x: t ≧0} of x ∈ X is bounded in X, it is precompact in X. Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

A nonlinear semigroup for a functional differential equation

A representation theorem is obtained for solutions of the nonlinear functional differential equation ( 1 ) u ′ ( t ) = F ( u t ) , t ⩾ 0 , u ( t ) = ϕ ( t ) , t ⩽ 0 , \begin{equation}\tag {$1$} u’(t) = F({u_t}), t \geqslant 0,\quad u(t) = \phi (t), t \leqslant 0,\end{equation} as a semigroup of nonlinear operators on a space of initial data X of “fading memory type.” Equation (1) is studied in the abstract setting of a Banach space E. The nonlinear functional F is a uniformly Lipschitz continuous mapping from X to E. The semigroup is constructed by transforming (1) to an abstract Cauchy problem ( C P ) w ′ ( t ) + A w ( t ) = 0 , w ( 0 ) = ϕ , \begin{equation}\tag {$CP$} w’(t) + Aw(t) = 0,\quad w(0) = \phi ,\end{equation} in the space X and applying a generation theorem of M. Crandall and T. Liggett to the operator A in X. The case when (1) is a nonlinear Volterra integrodifferential equation of infinite delay is given special consideration. The semigroup representation is used to obtain finite difference approximations for solutions of (CP) and to study the continuity of solutions of (1) with respect to perturbations of F and ϕ \phi .

On a differential game of evasion described by a class of nonlinear Volterra integrodifferential equations

On a differential game of evasion described by a class of nonlinear Volterra integrodifferential equations

On the asymptotic spectra of the bounded solutions of a nonlinear Volterra equation

We consider the asymptotic behavior of the bounded solutions of a nonlinear Volterra integrodifferential equation with a positive definite convolution kernel. Our main result states that (under appropriate assumptions) the asymptotic spectra of the solutions are contained in the set where the real part of the Fourier transform of the kernel vanishes. We also give a new asymptotic stability theorem, and present a new proof of a known result on the asymptotic behavior of the bounded solutions of a nonlinear, nondifferentiated Volterra equation.

Positive Definite Measures with Applications to a Volterra Equation

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation \begin{equation}\tag {$\ast $} x’(t) + \int _0^t {g(x(t - \tau ))\;d\mu (\tau ) = f(t),}\end{equation} with a positive definite kernel $\mu$. In particular, we give new sufficient conditions on the kernel $\mu$, which together with standard assumptions on f and g yield results on boundedness and asymptotic behavior of the solutions of $(\ast )$. Our proofs are based on the theory of distribution Fourier transforms.

Positive definite measures with applications to a Volterra equation

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation ( ∗ ) x ′ ( t ) + ∫ 0 t g ( x ( t − τ ) ) d μ ( τ ) = f ( t ) , \begin{equation}\tag {$\ast $} x’(t) + \int _0^t {g(x(t - \tau ))\;d\mu (\tau ) = f(t),}\end{equation} with a positive definite kernel μ \mu . In particular, we give new sufficient conditions on the kernel μ \mu , which together with standard assumptions on f and g yield results on boundedness and asymptotic behavior of the solutions of ( ∗ ) (\ast ) . Our proofs are based on the theory of distribution Fourier transforms.

Nonlinear Volterra Integral Equations with Positive Definite Kernels

Conditions are given for the asymptotic stability of the solution of a nonlinear Volterra integro-differential equation with convolution kernel. These conditions are less restrictive than most of the known ones for this equation.

An optimum phase-locked loop with a non-ideal filter

In communications engineering the problem of synchronization is vital. Any loss of synchronism is required to be restored in the shortest possible time. The concept of optimum PLL which utilizes a non-linear, memoryless feedback device was introduced by Ahmed and Cheng (1974). It was shown that this optimum PLL achieves synchronization in the minimum time if the filter were ideal. This paper considers the optimization of the PLL with non-ideal filter. The input sinusoid and the VCO output are assumed to have the same frequency but to differ in phase. The object of this paper is to investigate whether the optimum PLL introduced earlier (Ahmed and Cheng, p. 254) still retains its virtues when the ideal filter is replaced by a non-ideal one. The nonlinear Volterra integro-differential equation governing the PLL performance is solved by two different approaches and the solution indicates that the PLL in the presence of a non-ideal filter is optimal only within a certain range of variation of the parameter def...

Nonlinear Volterra integral equations with positive definite kernels

Conditions are given for the asymptotic stability of the solution of a nonlinear Volterra integro-differential equation with convolution kernel. These conditions are less restrictive than most of the known ones for this equation.

Numerical Methods for Nonlinear Volterra Integro-Differential Equations

Let $u'(x) = f(x,u(x),Iu(x))$ and $u(a) = \mu $, the initial value, where I is the Volterra functional $Iu(x) = \int_a^x {k(x,s,u(s))ds} $. Call this the general nonlinear Volterra integro-differential equation (VIDE). Approximations are given for the functional I and its derivatives. These approximations are applied to create one-step algorithms for the constructive solution of the VIDE problem. Let $x \in [a,b]$, integer $N \geqq 1$, and ${{h = (b - a)} / N}$. Define $x_n = a - nh$ for $n = 0(1)N$. A class of algorithms (each depending upon an order parameter $p \geqq 1$ and upon the step size h) are defined. For each fixed p and h, each algorithm recursively generates a sequence $\{ u_n \} $ which approximates $\{ u(x_n) \} $. Two error propagation theorems are presented. THEOREM 1. Under suitable smoothness conditions, $u_n = u(x_n ) + O(h^p )$, where$p \geqq 1$is the order parameter. The convergence is uniform on closed intervals. THEOREM 2. Under further smoothness conditions, $u_n = u(x_n ) + h^p e(x_n ) + O(h^{p + 1} )$, where the principal error function$e(x)$is defined as the solution of another Volterra integro-differential equation. The results of various numerical experiments, including 4th order Runge–Kutta type algorithms, are presented. These experiments serve to confirm the validity of our results.

Note on a Volterra integro-differential equation system

This note concerns a type of nonlinear Volterra integro-differential equation system which has only asymptotically zero solutions. It is a simple generalization of a one-dimensional result of J. J. Levin. It is a correction to another such note and gives a counterexample to that note.

On the numerical solution of nonlinear Volterra integro-differential equations

Piecewise polynomialss(x) of degreem≧2 and of continuity classC1 are used to obtain approximating functions to the exact solution of a given (ordinary) integro-differential equation of Volterra type. The unknown coefficients ofs(x) are computed recursively, by requiring thats(x) satisfy the integro-differential equation on a finite set of suitably chosen points. Results on the order of convergence of this method are given, together with a numerical illustration.

On a method for solving a system of nonlinear volterra integrodifferential equations

On a method for solving a system of nonlinear volterra integrodifferential equations

A nonlinear variation of constants formula for volterra equations

Abstract : A variation of constants formula is developed for nonlinear Volterra integral and integro-differential equations. This formula is suited to the study of various perturbation problems for Volterra equations. (Author)

On some nonlinear Volterra integrodifferential equations

On some nonlinear Volterra integrodifferential equations

On the Asymptotic Behavior of the Solution of a Nonlinear Integrodifferential Equation

Previous article Next article On the Asymptotic Behavior of the Solution of a Nonlinear Integrodifferential EquationStig-Olof LondenStig-Olof Londenhttps://doi.org/10.1137/0502034PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA theorem concerning the asymptotic behavior of the solution of a nonlinear Volterra integrodifferential equation is proved.[1] Richard Bellman and , Kenneth L. Cooke, Differential-difference equations, Academic Press, New York, 1963xvi+462 MR0147745 0105.06402 Google Scholar[2] Constantin Corduneanu, Sur une équation intégrale de la théorie du rélage automatique, C. R. Acad. Sci. Paris, 256 (1963), 3564–3567 MR0156169 0122.34102 Google Scholar[3A] J. J. Levin, The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. Math. Soc., 14 (1963), 534–541 MR0152852 0115.32403 CrossrefGoogle Scholar[3B] V. Lakshmikantham and , S. Leela, Differential and integral inequalities: Theory and applications. Vol. I: Ordinary differential equations, Academic Press, New York, 1969, 327–333 MR0379933 0177.12403 Google Scholar[4] J. J. Levin, The qualitative behavior of a nonlinear Volterra equation, Proc. Amer. Math. Soc., 16 (1965), 711–718 MR0190669 0141.11003 CrossrefISIGoogle Scholar[5] J. J. Levin, Boundedness and oscillation of some Volterra and delay equations, J. Differential Equations, 5 (1969), 369–398 10.1016/0022-0396(69)90051-5 MR0236642 0169.14102 CrossrefISIGoogle Scholar[6] J. J. Levin and , J. A. Nohel, Note on a nonlinear Volterra equation, Proc. Amer. Math. Soc., 14 (1963), 924–929 MR0157201 0151.16904 CrossrefGoogle Scholar[7] Stig-Olof Londen, On some nonlinear Volterra equations, Ann. Acad. Sci. Fenn. Ser. A VI, (1969), 74– MR0320659 0204.42903 Google Scholar[8] Stig-Olof Londen, On a Volterra integrodifferential equation, Soc. Sci. Fenn. Comment. Phys.-Math., 38 (1969), 1–4 MR0313742 0197.37901 Google Scholar[9] Stig-Olof Londen, On a nonlinear Volterra integrodifferential equation, Soc. Sci. Fenn. Comment. Phys.-Math., 38 (1969), 5–11 MR0313744 0191.40601 Google Scholar[10] Stig-Olof Londen, On a nonlinear retarded differential-difference equation. I, Soc. Sci. Fenn. Comment. Phys.-Math., 40 (1970), 47–63 MR0312030 0215.15602 Google Scholar[11] Stig-Olof Londen, On a nonlinear retarded differential-difference equation. II, Soc. Sci. Fenn. Comment. Phys.-Math., 40 (1970), 97–100 MR0312030 0218.34063 Google Scholar[12] John A. Nohel, Remarks on nonlinear Volterra equations, Proc. U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967), Benjamin, New York, 1967, 249–266 MR0222586 0179.44401 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails The asymptotic behavior of bounded solutions of a nonconvolution volterra equationJournal of Differential Equations, Vol. 31, No. 1 Cross Ref References Cross Ref On the asymptotic behavior of the solutions of a nonlinear volterra equationAnnali di Matematica Pura ed Applicata, Vol. 93, No. 1 Cross Ref On some nonlinear Volterra integrodifferential equationsJournal of Differential Equations, Vol. 11, No. 1 Cross Ref Volume 2, Issue 3| 1971SIAM Journal on Mathematical Analysis History Submitted:26 May 1970Published online:17 February 2012 InformationCopyright © 1971 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0502034Article page range:pp. 356-367ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

On Volterra’s Population Equation

On Volterra’s Population Equation