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Continuous data dependence for an abstract Volterra integro-differential equation in Hilbert space with applications to viscoelasticity. (English) Zbl 0342.45014

MSC:
45K05 Integro-partial differential equations
47Gxx Integral, integro-differential, and pseudodifferential operators
45N05 Abstract integral equations, integral equations in abstract spaces
74Hxx Dynamical problems in solid mechanics
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References:
[1] C.M. Dafermos , An Abstract Volterra Equation with Applications to Linear Viscoelasticity , J. Diff. Eqs. , 7 ( 1970 ), pp. 554 - 569 . MR 259670 | Zbl 0212.45302 · Zbl 0212.45302 · doi:10.1016/0022-0396(70)90101-4
[2] R.J. Knops - L.E. Payne , Growth Estimates for Solutions of Evolutionary equations in Hilbert Space with Applications in Elastodynamics , Arch. Rat. Mech. Anal. , 41 ( 1971 ), pp. 369 - 398 . MR 330731 | Zbl 0227.35017 · Zbl 0227.35017 · doi:10.1007/BF00281873
[3] R.J. Knops - L.E. Payne , Uniqueness in Classical Elastodynamics , Arch. Rat. Mech. Anal. , 32 ( 1968 ), pp. 349 - 355 . MR 219261 | Zbl 0159.56201 · Zbl 0159.56201
[4] R.J. Knops - L.E. Payne , On Uniqueness and Continuous Dependence in Dynamical Problems of Linear Thermoelasticity , Int. J. Solids Structures , 6 ( 1969 ), pp. 1173 - 1184 . Zbl 0209.56605 · Zbl 0209.56605 · doi:10.1016/0020-7683(70)90054-5
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[7] C.E. Beevers , Uniqueness and Stability in Linear Viscoelasticity , ZAMP , 26 ( 1975 ), pp. 177 - 186 . MR 366154 | Zbl 0314.73035 · Zbl 0314.73035 · doi:10.1007/BF01591506
[8] C.E. Beevers , Some Continuous Dependence Results in the Linear Dynamic Theory of Anisotropic Viscoelasticity , Journal de Mécanique , 14 ( 1975 ), pp. 1 - 13 . MR 386404 | Zbl 0324.73031 · Zbl 0324.73031
[9] A.C. Murray - M.H. Protter , The Asymptotic Behavior of Solutions of Second Order Systems of Partial Differential Equations , J. Diff . Eqs. , 13 ( 1973 ), pp. 57 - 80 . MR 328291 | Zbl 0257.35017 · Zbl 0257.35017 · doi:10.1016/0022-0396(73)90032-6
[10] R.J. Knops - L.E. Payne , Continuous Data Dependence for the Equations of Classical Elastodynamics , Proc. Camb. Phil. Soc. , 66 ( 1969 ), pp. 481 - 491 . MR 270604 | Zbl 0184.51004 · Zbl 0184.51004
[11] F. Bloom , Continuous Dependence on Initial Geometry for a Class of Abstract Equations in Hilbert Space , to appear in the J. Math. Anal. Appl. MR 442401 | Zbl 0347.35024 · Zbl 0347.35024 · doi:10.1016/0022-247X(77)90207-4
[12] W.A. Day , On the Monotonicity of the Relaxation Functions of Viscoelastic Materials , Proc. Camb. Phil. Soc. , 67 ( 1970 ), pp. 503 - 508 . MR 250545 | Zbl 0202.25302 · Zbl 0202.25302
[13] F. Bloom , On Stability in Linear Viscoelasticity , Mechanics Research Communications , 3 ( 1976 ), pp. 143 - 150 . Zbl 0367.73051 · Zbl 0367.73051 · doi:10.1016/0093-6413(76)90001-X
[14] F. Bloom , Growth Estimates for Solutions to Initial-Boundary Value Problems in Viscoelasticity , to appear in the J. Math. Anal. Appl. Zbl 0361.45010 · Zbl 0361.45010 · doi:10.1016/0022-247X(77)90074-9
[15] F. Bloom , Stability and Growth Estimates for Volterra Integrodifferential Equations in Hilbert Space , Bull. A.M.S. , 82 ( 1976 ), # 5. Zbl 0329.45016 · Zbl 0329.45016 · doi:10.1090/S0002-9904-1976-14127-4
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