Resolvent Positive Operators and Positive Fractional Resolvent Families

This paper is concerned with positive α -times resolvent families on an ordered Banach space E (with normal and generating cone), where 0 < α ≤ 2 . We show that a closed and densely defined operator A on E generates a positive exponentially bounded α -times resolvent family for some 0 < α < 1 if and only if, for some ω ∈ ℝ , when λ > ω , λ ∈ ρ A , R λ , A ≥ 0 and sup λ R λ , A : λ ≥ ω < ∞ . Moreover, we obtain that when 0 < α < 1 , a positive exponentially bounded α -times resolvent family is always analytic. While A generates a positive α -times resolvent family for some 1 < α ≤ 2 if and only if the operator λ α − 1 λ α − A − 1 is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.

On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.

On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg–Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Dilatations of Linear Operators

The article is devoted to building various dilatations of linear operators. The explicit construction of a unitary dilation of a compression operator is considered. Then the J -unitary dilatation of a bounded operator is constructed by means of the operator knot concept of a bounded linear operator. Using the Pavlov method, we construct the self-adjoint dilatation of a bounded dissipative operator. We consider spectral and translational representations of the self-adjoint dilatation of a densely defined dissipative operator with nonempty set of regular points. Using the concept of an operator knot for a bounded operator and the Cayley transform, we introduce an operator knot for a linear operator. By means of this concept, we construct the J -self-adjoint dilatation of a densely defined operator with a regular point. We obtain conditions of isomorphism of extraneous dilations and their minimality.

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations Dtαxt=Axt+Dtα-1Ft,xt,Bxt, t∈R, where 1<α<2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

A unified treatment for Tikhonov regularization using a general stabilizing operator

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating [Formula: see text], a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖2 + ‖Lx‖2 ≥ γ‖x‖2 for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation [Formula: see text] where yδ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.

Complex powers of operators

We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists ? ? [0,?) such that the resolvent of A is bounded by O((1 + |?|)?) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (-A)b is the c.i.g. of an analytic semigroup of growth order r > 0.