Real Integrability Conditions for the Nonuniform Exponential Stability of Evolution Families on Banach Spaces

Abstract

Let J be either ℝ or ℝ₊ := [0,∞). We prove that an evolution family U = {U(t, s)}_t≥s∈J which satisfies some natural assumptions is non-uniformly exponentially stable if there exist a positive real number α and a nondecreasing function φ : ℝ₊ → ℝ₊ with φ(t) positive for all positive t and such that for each s ∈ J, the following inequality

$$ \mathop {\sup }\limits_{t > s} \int_0^{t - s} {\varphi (e^{\alpha u} \left\| {U(s + u,s)x} \right\|)du = M_\varphi(s) < \infty } $$

holds true for all x ∈ X with ‖x‖ ≤ 1. We arrive at the same conclusion under the assumption that there exist three positive real numbers α, β and K such that for each t ∈ J the inequality

$$ \left( {\int_J {\chi ( - \infty ,t](\tau )e^{ - q\alpha \tau } \parallel U(t,\tau )^* x^* \parallel } )d\tau } \right)^{\tfrac{1} {q}}\leqslant Ke^{ - \beta t} $$

holds true, for all x* ∈ X* with ‖x*‖ ≤ 1 and for some q ≥ 1.

Part of this article was done while the author visited the School of Mathematical Sciences, Governmental College University, Lahore, Pakistan.