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Register a new userFractional powers approach of operators for higher order abstract Cauchy problems
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In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $ngeqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.
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We give stability and consistency results for higher order Grunwald-type formulae used in the approximation of solutions to fractional-in-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier multipliers to gain regularity and stability results. We then generalise the theory to the case where the first derivative operator is replaced by the generator of a bounded group on an arbitrary Banach space.
This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein inequalities for higher order Laplacians. We show also that the heat semi-group associated to powers larger than one of the laplacian does not satisfy the strict maximum principle in general. Finally, in a suitable range we provide several positive results.
In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.
Given two arbitrary sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ of real numbers satisfying $$|lambda_1|>|mu_1|>|lambda_2|>|mu_2|>...>| lambda_j| >| mu_j| to 0 ,$$ we prove that there exists a unique sequence $c=(c_n)_{ninZ_+}$, real valued, such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ of symbols $c=(c_{n})_{nge 0}$ and $tilde c=(c_{n+1})_{nge 0}$ respectively, are selfadjoint compact operators on $ell^2(Z_+)$ and have the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Gamma_c$ and of $Gamma_{tilde c}$ in terms of the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$. More generally, given two arbitrary sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ of positive numbers satisfying $$rho_1>sigma_1>rho_2>sigma_2>...> rho_j> sigma_j to 0 ,$$ we describe the set of sequences $c=(c_n)_{ninZ_+}$ of complex numbers such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ are compact on $ell ^2(Z_+)$ and have sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ respectively as non zero singular values.
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