On the Henstock-Kurzweil Integral for Riesz-space-valued Functions on Time Scales

We introduce and investigate the Henstock-Kurzweil (HK) integral for Riesz-space-valued functions on time scales. Some basic properties of the HK delta integral for Riesz-space-valued functions are proved. Further, we prove uniform and monotone convergence theorems.Comment: This is a preprint of a paper whose final and definite form is with 'J. Nonlinear Sci. Appl.', ISSN 2008-1898 (Print) ISSN 2008-1901 (Online). Article Submitted 17-Jan-2017; Revised 17-Apr-2017; Accepted for publication 19-Apr-2017. See [http://www.tjnsa.com

Semilocal convergence of Newton-like methods under general conditions with applications in fractional calculus

We present a semilocal convergence study of Newton-like methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies such as [5], [6], [7], [14] require that the operator involved is Fréchet-differentiable. In the present study we assume that the operator is only continuous. This way we extend the applicability of Newton-like methods to include fractional calculus and problems from other areas. Some applications include fractional calculus involving the Riemann-Liouville fractional integral and the Caputo fractional derivative. Fractional calculus is very important for its applications in many applied sciences

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights

We investigate the properties of a class of weighted vector-valued $L_p$-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal $L_p$-regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded \calH^\infty-calculus, interpolation theory, and operator sums.Comment: This is a preprint version. Published in Journal of Functional Analysis 262 (2012) 1200-122

Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem

By means of Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness of solutions to a boundary value problem for a discrete fractional mixed type sum-difference equation with the nonlinear term dependent on a fractional difference of lower order. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded and two representative examples are presented to illustrate the effectiveness of the main results

New criteria for the H∞H^\infty-calculus and the Stokes operator on bounded Lipschitz domains

We show that the Stokes operator A on the Helmholtz space Lp (Ω) for a bounded Lipschitz domain Ω ⊂ Rd, d ≥ 3, has a bounded H ∞- calculus if |1p − 1/2| ≤ 1/2d . Our proof uses a new comparison theorem A and the Dirichlet Laplace −∆ on Lp(Ω)d, which is based on “off-diagonal” estimates of the Littlewood-Paley decompositions of A and −∆. This comparison theorem can be formulated for rather general sectorial operators and is well suited to extrapolate the H ∞-calculus from L2(U ) to the Lp(U )-scale or part of it. It also gives some information on coincidence of domains of fractional powers