Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces. (English) Zbl 06979944

Chang, Yong-Kui; Ponce, Rodrigo

J. Integral Equations Appl. 30, No. 3, 347-369 (2018).

Summary: Let \(\mathbb{X}\) be a Banach space. Let \(A\) be the generator of an immediately norm continuous \(C_0\)-semigroup defined on \(\mathbb{X}\). We study the uniform exponential stability of solutions of the Volterra equation \[ u'(t) = Au(t)+\int _0^t a(t-s)Au(s)\,ds,\quad t\geq 0,\;u(0)=x, \] where \(a\) is a suitable kernel and \(x\in \mathbb {X}\). Using a matrix operator, we obtain some spectral conditions on \(A\) that ensure the existence of constants \(C,\omega >0\) such that \(\|u(t)\|\leq Ce^{-\omega t}\|x\|\), for each \(x\in D(A)\) and all \(t\geq 0\). With these results, we prove the existence of a uniformly exponential stable resolvent family to an integro-differential equation with infinite delay. Finally, sufficient conditions are established for the existence and uniqueness of bounded mild solutions to this equation.

MSC:

47D06	One-parameter semigroups and linear evolution equations
45D05	Volterra integral equations
45N05	Abstract integral equations, integral equations in abstract spaces
47J35	Nonlinear evolution equations
45M10	Stability theory for integral equations

Keywords:

exponential stability; \(C_0\)-semigroups; mild solutions; almost periodic; Volterra equations; heat equation with memory

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