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On Uniform Exponential Stability of Self adjoint Evolution Family: By Weak Rolewicz Approach

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Author Affiliations

  • 1Department of Mathematics University of Peshawar, Peshawar, PAKISTAN

Res. J. Recent Sci., Volume 4, Issue (2), Pages 68-71, February,2 (2015)

Abstract

In this article we prove that if P = {P(s,t)}s≥t≥0 is a self adjoint and strongly q-periodic continuous evolution family of bounded linear operators acting on a complex or real Hilbert space H then P is uniformly exponentially stable if for each unit vector x ∈ H the integral ∫n∞ ϕ(<P(s,0)x,x>) ds is bounded, whereϕ :R := [0,∞) → R+ is a non-decreasing function such that ϕ(0) = 0 and ϕ(s) >0 for all s ∈ (0∞) .

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