HÖLDER CONTINUOUS SOLUTIONS FOR SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

doi:10.1016/S0252-9602(11)60274-9

Abstract

Abstract:

We study H\"older continuous solutions for the second order integro-differential equations with infinite delay (P₁): u''(t)+ cu'(t)+∫^t_-∞β (t-s)u'(s)ds+∫^t_-∞γ(t-s)u(s)ds =Au(t)-∫^t_-∞δ(t-s)Au(s)ds+f(t) on the line R, where 0 < α< 1, A is a closed operator in a complex Banach space X, c ∈C is a constant, f ∈C^α (R，X) and β, γ, δ ∈L¹(R₊). Under suitable assumptions on the kernels β, γ and δ, we completely characterize the C^α-well-posedness of (P₁) by using operator-valued C^α-Fourier multipliers.

Key words: Fourier multiplier, C^α-well-posedness, integro-differential equations

CLC Number:

45N05

BU Shang-Quan. HÖLDER CONTINUOUS SOLUTIONS FOR SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES[J].Acta mathematica scientia,Series B, 2011, 31(3): 765-777.

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