Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators

In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.

On the uniform ergodic for α−times integrated semigroups

Let $A$ be a generator of an $\alpha-$times integrated semigroup$(S(t))_{t\geq 0}$. We study the uniform ergodicity of $(S(t))_{t\geq 0}$ and we show that the range of $A$ is closed if and only if $\lambda R(\lambda,A)$ is uniformly ergodic.Moreover, we obtain that $(S(t))_{t\geq 0}$ is uniformly ergodic if and only if $\alpha=0$. Finally, we get that $\frac{1}{t^{\alpha+1}}\int_{0}^{t}S(s)ds$ converge uniformly for all $\alpha\geq 0$.

Cesàro and Abel ergodic theorems for integrated semigroups

Let {S(t)}t ≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ 2 R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t 2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.

Integrated Semigroups on Fréchet Spaces II: F_sBSV ([a,b],F)-space, FsLip_0,w([0,+∞),F)-space, Isometric Theorem and Application

In this article we will study the Riemann Stieltjes Laplace integral of vectorial functions in Fréchet spaces. Particularly we will prove a isometric theorem and a generation theorem for integrated semigroups on Fréchet spaces.

Spectral inclusions between $(\alpha+1)$-times integrated semigroups and its generators

We interest to $\alpha$-times integrated semigroups that are the solution of the $(\alpha+1)$-times integrated Cauchy problems. Particulary, we characterize partially the different spectrums of $\alpha$-times integrated semigroups using the spectrums of its generators.

Integrated Semigroups on Fréchet Spaces I: Bochner Integral, Laplace Integral and Real Representation Theorems

In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.

Integrated Semigroups on Fréchet Spaces I: Bochner Integral, Laplace Integral and Real Representation Theorems

In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.