872 research outputs found
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
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
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
-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces.
These spaces arise naturally in the context of maximal -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 -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
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