where D is the Caputo fractional derivative of order α with  < α < , A :D(A)⊂ X → X is the infinitesimal generator of a resolvent Sα(t), t ≥ , B : U → X is a bounded linear operator, u ∈ L([,b],U), X and U are two real Hilbert spaces, J–α t h denotes the  – α order fractional integral of h ∈ L([,b],X). The controllability problem has attracted a lot of mathematicians and engineers’ att...

We exhibit an example of a bounded linear operator on a Banach space which admits an everywhere defined local resolvent with continuous derivatives of all orders. Let T be a bounded linear operator acting on a complex Banach space X. It is well known that the resolvent z 7→ (T − z)−1 defined on the complement of the spectrum σ(T ) is unbounded. More precisely, ‖(T − z)−1‖ → ∞ whenever z approac...

In this paper, we consider a system of generalized resolvent equations involving generalized pseudocontractive mapping with corresponding system of variational inclusions in real Banach spaces. We establish an equivalence between the system of generalized resolvent equations and the system of variational inclusions using the concept of H(.,.)-co-accretive mapping. Furthermore, we prove the exis...

Spectrum of the second-order differential operator with periodic point interactions in L2 R is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spe...

For any univariate polynomial P whose coefficients lie in an ordinary differential field F of characteristic zero, and for any constant indeterminate α, there exists a nonunique nonzero linear ordinary differential operator R of finite order such that the αth power of each root of P is a solution of Rzα = 0, and the coefficient functions of R all lie in the differential ring generated by the co...

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addit...

We discuss the problem of deriving estimates for the resolvent of the linear operator associated with three dimensional perturbations of plane Couette flow, and determining its dependence on the Reynolds number R. Depending on the values of the parameters involved, we derive estimates analytically. For the remaining values of the parameters, we prove that deriving estimates for the resolvent ca...

Abstract. We analyze the behavior of the trace of the resolvent of an elliptic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal extension of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first component and expand ...

In this paper, we consider a more general form of variational inclusions, called generalized variational inclusion (for short, GVI). In connection with GVI, we also consider a generalized resolvent equation with H-resolvent operator, called H-resolvent equation (for short, H-RE). We suggest iterative algorithms to compute the approximate solutions of GVI and H-RE. The existence of a unique solu...