• Title/Summary/Keyword: analytic semigroup

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INTEGRATION STRUCTURES FOR THE OPERATOR-VALUED FEYNMAN INTEGRAL

  • Jefferies, Brian
    • Journal of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.349-363
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    • 2001
  • The analytic in mass operator-valued Feynman integral is related to integration with respect to unbounded set functions formed from the semigroup obtained by analytic continuation of the heat semigroup and the spectral measure of multiplication by characteristics functions.

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BOUNDARY CONTROLLABILITY OF SEMILINEAR SYSTEMS IN BANACH SPACES

  • BALACHANDRAN, K.;ANANDHI, E.R.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.5 no.2
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    • pp.149-156
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    • 2001
  • Sufficient conditions for boundary controllability of semilinear systems in Banach spaces are established. The results are obtained by using the analytic semigroup theory and the Banach contraction principle. An example is provided to illustrate the theory.

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EXISTENCE OF MILD SOLUTIONS IN THE α-NORM FOR SOME PARTIAL FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

  • Jang, Hyun Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.3
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    • pp.393-401
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    • 2014
  • In this work, we discuss the existence of mild solutions in the ${\alpha}$-norm for some partial functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space X and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT

  • CHADHA, ALKA;PANDEY, DWIJENDRA N.
    • Journal of applied mathematics & informatics
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    • v.33 no.5_6
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    • pp.699-721
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    • 2015
  • This paper investigates the existence of mild solution for a fractional integro-differential equations with a deviating argument and nonlocal initial condition in an arbitrary separable Hilbert space H via technique of approximations. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal fractional integral equation onto finite dimensional space. The existence and uniqueness of solutions to each approximate integral equation is obtained by virtue of the analytic semigroup theory via Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We consider the Faedo-Galerkin approximation of the solution and demonstrate some convergenceresults. An example is also given to illustrate the abstract theory.

DEGENERATE VOLTERRA EQUATIONS IN BANACH SPACES

  • Favini, Angelo;Tanabe, Hiroki
    • Journal of the Korean Mathematical Society
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    • v.37 no.6
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    • pp.915-927
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    • 2000
  • This paper is concerned with degenerate Volterra equations Mu(t) + ∫(sub)0(sup)t k(t-s) Lu(s)ds = f(t) in Banach spaces both in the hyperbolic case, and the parabolic one. The key assumption is played by the representation of the underlying space X as a direct sum X = N(T) + R(T), where T is the bounded linear operator T = ML(sup)-1. Hyperbolicity means that the part T of T in R(T) is an abstract potential operator, i.e., -T(sup)-1 generates a C(sub)0-semigroup, and parabolicity means that -T(sup)-1 generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel k($.$) is degenerated or singular at t=0 using the results of Pruss[8] on analytic resolvents. Finally, we consider the case where $\lambda$ is a pole for ($\lambda$L + M)(sup)-1.

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ASYMPTOTIC STABILITY OF STRONG SOLUTIONS FOR EVOLUTION EQUATIONS WITH NONLOCAL INITIAL CONDITIONS

  • Chen, Pengyu;Kong, Yibo;Li, Yongxiang
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.319-330
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    • 2018
  • This paper is concerned with the global asymptotic stability of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. The discussion is based on analytic semigroups theory and the gradually regularization method. The results obtained in this paper improve and extend some related conclusions on this topic.

ON THE GENERALIZED ORNSTEIN-UHLENBECK OPERATORS WITH REGULAR AND SINGULAR POTENTIALS IN WEIGHTED Lp-SPACES

  • Imen Metoui
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.149-160
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    • 2024
  • In this paper, we give sufficient conditions for the generalized Ornstein-Uhlenbeck operators perturbed by regular potentials and inverse square potentials AΦ,G,V,c=∆-∇Φ·∇+G·∇-V+c|x|-2 with a suitable domain generates a quasi-contractive, positive and analytic C0-semigroup in Lp(ℝN , e-Φ(x)dx), 1 < p < ∞. The proofs are based on an Lp-weighted Hardy inequality and perturbation techniques. The results extend and improve the generation theorems established by Metoui [7] and Metoui-Mourou [8].

EXISTENCE OF SOLUTIONS OF QUASILINEAR INTEGRODIFFERENTIAL EVOLUTION EQUATIONS IN BANACH SPACES

  • Balachandran, Krishnan;Park, Dong-Gun
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.4
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    • pp.691-700
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    • 2009
  • We prove the local existence of classical solutions of quasi-linear integrodifferential equations in Banach spaces. The results are obtained by using fractional powers of operators and the Schauder fixed-point theorem. An example is provided to illustrate the theory.