scholarly journals Duality map characterisations for Opial conditions

1996 ◽  
Vol 53 (3) ◽  
pp. 413-417 ◽  
Author(s):  
Tim Dalby ◽  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Duality Mapping ◽  
Opial Condition ◽  
Opial’S Condition ◽  
Duality Map

We characterise Opial's condition, the non-strict Opial condition, and the uniform Opial condition for a Banach space X in terms of properties of the duality mapping from X into X*.

scholarly journals Nonlinear potentials in function spaces

2002 ◽  
Vol 165 ◽  
pp. 91-116 ◽  
Author(s):  
Murali Rao ◽  
Zoran Vondraćek
Keyword(s):  
Banach Space ◽  
Potential Theory ◽  
Finite Energy ◽  
Duality Mapping ◽  
Quadratic Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Nonlinear Potential Theory ◽  
Nonlinear Potential ◽  
Nonlinear Potentials

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

scholarly journals Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

2020 ◽  
pp. 56-71
Author(s):  
Jenica Cringanu
Keyword(s):  
Banach Space ◽  
Variational Method ◽  
Critical Points ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Existence Results ◽  
Duality Mapping ◽  
Topological Methods ◽  
Carathéodory Function ◽  
Direct Variational Method

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

scholarly journals Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

scholarly journals Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

2013 ◽  
Vol 21 (1) ◽  
pp. 183-200
Author(s):  
Prasit Cholamjiak ◽  
Yeol Je Cho ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
Gauge Functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

scholarly journals Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hiroko Manaka
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Duality Mapping ◽  
Nonlinear Mapping ◽  
Normalized Duality Mapping ◽  
Strongly Nonexpansive Mapping ◽  
Weak Convergence Theorem

LetEbe a smooth Banach space with a norm·. LetV(x,y)=x2+y2-2 x,Jyfor anyx,y∈E, where·,·stands for the duality pair andJis the normalized duality mapping. We define aV-strongly nonexpansive mapping byV(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists aV-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

scholarly journals The Γ-opial property

2006 ◽  
Vol 73 (3) ◽  
pp. 473-476 ◽  
Author(s):  
Monika Budzyńska ◽  
Tadeusz Kuczumow ◽  
Małgorzata Michalska
Keyword(s):  
Banach Space ◽  
Compact Subset ◽  
Short Paper ◽  
Opial Condition ◽  
Opial Property

In this short paper we show that if (X, ∥ · ∥) is a Banach space, Γ a norming set for X and C is a nonempty, bounded and Γ sequentially compact subset of X, then in C the Γ-Opial condition for nets is equivalent to the Γ-Opial condition.

scholarly journals Smoothness and the asymptotic-norming properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Asplund Space ◽  
Duality Mapping

We study some smoothness properties of a Banach space X that are related to the weak* asymptotic-norming properties of the dual space X*. These properties imply that X is an Asplund space and are related to the duality mapping of X.

scholarly journals Geometrical implications of upper semi-continuity of the duality mapping on a Banach space

1978 ◽  
Vol 79 (1) ◽  
pp. 99-109 ◽  
Author(s):  
John Giles ◽  
David Gregory ◽  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Duality Mapping

scholarly journals A-properness and fixed point theorems for dissipative type maps

1999 ◽  
Vol 4 (2) ◽  
pp. 83-100 ◽  
Author(s):  
K. Q. Lan ◽  
J. R. L. Webb
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Index ◽  
Fixed Point Theorems ◽  
Duality Mapping ◽  
Uniformly Convex ◽  
Point Index ◽  
Weakly Inward ◽  
Convex Subsets

We obtain newA-properness results for demicontinuous, dissipative type mappings defined only on closed convex subsets of a Banach spaceXwith uniformly convex dual and which satisfy a property called weakly inward. The method relies on a new property of the duality mapping in such spaces. New fixed point results are obtained by utilising a theory of fixed point index.

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