scholarly journals The Brezis–Browder Theorem in a general Banach space

2012 ◽  
Vol 262 (12) ◽  
pp. 4948-4971 ◽  
Author(s):  
Heinz H. Bauschke ◽  
Jonathan M. Borwein ◽  
Xianfu Wang ◽  
Liangjin Yao
Keyword(s):  
Banach Space ◽  
General Banach Space

On Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators

2005 ◽  
Vol 135 (1) ◽  
pp. 39-60 ◽  
Author(s):  
Davide di Giorgio ◽  
Alessandra Lunardi
Keyword(s):  
Banach Space ◽  
Explicit Formula ◽  
Fredholm Operator ◽  
Finite Dimension ◽  
Sufficient Conditions ◽  
Spectral Flow ◽  
Exponential Dichotomies ◽  
Sectorial Operators ◽  
Common Domain ◽  
General Banach Space

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

A contribution to the solution of the compact correction problem for operators on a Banach space

1989 ◽  
Vol 31 (2) ◽  
pp. 219-229
Author(s):  
Mícheál Ó Searcóid
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Compact Operator ◽  
Complex Number ◽  
Space Problem ◽  
Riesz Operators ◽  
Correction Problem ◽  
Bounded Below ◽  
General Banach Space

We consider the hypothesis that an operator T on a given Banach space can always be perturbed by a compact operator K in such a way that, whenever a complex number A is in the semi-Fredholm region of T + K, then T + K – λ is either bounded below or surjective. The hypothesis has its origin in the work of West [11], who proved it for Riesz operators on Hilbert space. In this paper, we reduce the general Banach space problem to one of considering only operators of a special type, operators which are, in a spectral sense, natural generalizations of the Riesz operators studied by West.

scholarly journals A Note on the Parabolic Differential and Difference Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Yasar Sozen ◽  
Pavel E. Sobolevskii
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Positive Operator ◽  
Difference Equations ◽  
Well Posedness ◽  
Holder Space ◽  
Strongly Positive Operator ◽  
Ill Posed ◽  
General Banach Space

The differential equationu'(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the strongly positive operatorAis ill-posed in the Banach spaceC(E)=C(ℝ,E)with norm‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder spaceCα(E)=Cα(ℝ,E)with norm‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα),0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme inC(ℝτ,E)spaces is proved. The well-posedness of this difference scheme inCα(ℝτ,E)spaces is obtained.

Convergence of Subdifferentials of Convexly Composite Functions

1999 ◽  
Vol 51 (2) ◽  
pp. 250-265 ◽  
Author(s):  
C. Combari ◽  
R. Poliquin ◽  
L. Thibault
Keyword(s):  
Banach Space ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Composite Functions ◽  
Kuratowski Convergence ◽  
General Banach Space

AbstractIn this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

scholarly journals Nonuniform dichotomy of evolutionary processes in Banach spaces

1983 ◽  
Vol 27 (1) ◽  
pp. 31-52 ◽  
Author(s):  
Petre Preda ◽  
Mihail Megan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Evolutionary Processes ◽  
General Banach Space

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C0 and (B, D) = [(Lp, Lq).

scholarly journals Characterizations of metric projections in Banach spaces and applications

1998 ◽  
Vol 3 (1-2) ◽  
pp. 85-103 ◽  
Author(s):  
Jean-Paul Penot ◽  
Robert Ratsimahalo
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Convex Subset ◽  
Metric Projection ◽  
Nonempty Closed Convex Subset ◽  
Convex Cones ◽  
Duality Mappings ◽  
Metric Projections ◽  
General Banach Space

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

Nonlocal Riemann–Liouville fractional evolution inclusions in Banach space

2020 ◽  
Vol 13 (08) ◽  
pp. 2050162
Author(s):  
Shamas Bilal ◽  
Tzanko Donchev ◽  
Nikolay Kitanov ◽  
Nasir Javaid
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Evolution Inclusions ◽  
Liouville Derivative ◽  
The Fixed Point Theorem ◽  
General Banach Space

In this paper, we study the existence of solutions for nonlocal semilinear fractional evolution inclusions involving Riemann–Liouville derivative in a general Banach space. The fixed point theorem for contractive valued multifunction is used. Illustrative example is provided.

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