Memory approximate controllability properties for higher order Hilfer time fractional evolution equations

In this paper we study the approximate controllability of fractional partial differential equations associated with the so-called Hilfer type time fractional derivative and a non-negative selfadjoint operator $A$ with a compact resolvent on $L^2(\Omega)$, where \Omega\subset\RR^N ($N\geq 1$) is an open set. More precisely, we show that if $0\le\nu\le 1$, $1<\mu\le 2$ and \Omega\subset\RR^N is an open set, then the system \begin{equation*} \begin{cases} \D^{\mu,\nu}_tu+Au=f\chi_{\omega}\;\;&\mbox{ in }\;\Omega\times(0,T),\\ (I_t^{(1-\nu)(2-\mu)}u)(\cdot,0)=u_0 &\mbox{ in }\;\Omega,\\ (\partial_tI_t^{(1-\nu)(2-\mu)}u)(\cdot,0)=u_1 &\mbox{ in }\;\Omega, \end{cases} \end{equation*} is memory approximately controllable for any $T>0$, $u_0\in D(A^{1/\mu})$, $u_1\in L^2(\Omega)$ and any non-empty open set $\omega\subset\Omega$. The same result holds for every $u_0\in D(A^{1/2})$ and $u_1\in L^2(\Omega)$.Comment: arXiv admin note: text overlap with arXiv:2003.0818

Periodic solutions of integro-differential equations in vector-valued function spaces

AbstractOperator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue–Bochner spaces Lp, the Besov spaces Bp,qs (and related spaces such as the Hölder–Zygmund spaces Cs) and the Triebel–Lizorkin spaces Fp,qs. We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations

On the Boundary Value Theorem for Holomorphic Semigroups

this paper, the abstract Gevrey space D(

On analytic semigroups and cosine functions in Banach spaces

If $A$ generates a bounded cosine function on a Banach space $X$ then the negative square root $B$ of $A$ generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by $B$. This characterization relies on new results on the inversion of the vector-valued conjugate potential transform

Well-posedness of degenerate integro-differential equations in function spaces

We use operator-valued Fourier multipliers to obtain characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We treat periodic vector-valued Lebesgue, Besov and Trieblel-Lizorkin spaces. We observe that in the Besov space context, the results are applicable to the more familiar scale of periodic vector-valued H\"older spaces. The equation under consideration are important in several applied problems in physics and material science, in particular for phenomena where memory effects are important. Several examples are presented to illustrate the results

On analytic semigroups and cosine functions in Banach spaces

If $A$ generates a bounded cosine function on a Banach space $X$ then the negative square root $B$ of $A$ generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by $B$. This characterization relies on new results on the inversion of the vector-valued conjugate potential transform