On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

We introduce the class of $$\alpha $$ α -firmly nonexpansive and quasi $$\alpha $$ α -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $$\alpha $$ α -firmly nonexpansive operators coincide with so-called $$\alpha $$ α -averaged operators. For our more general setting, we show that $$\alpha $$ α -averaged operators form a subset of $$\alpha $$ α -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) $$\alpha $$ α -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) $$\alpha $$ α -firmly nonexpansive. Moreover, we will see that quasi $$\alpha $$ α -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates $$x_{n+1}:=Tx_{n}$$ x n + 1 : = T x n belong to the fixed point set $${{\,\mathrm{Fix}\,}}T$$ Fix T whenever the operator T is nonexpansive and quasi $$\alpha $$ α -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in $${{\,\mathrm{Fix}\,}}T$$ Fix T . Further, the projections $$P_{{{\,\mathrm{Fix}\,}}T}x_n$$ P Fix T x n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in $$L_p$$ L p , $$p \in (1,\infty ) \backslash \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } spaces on probability measure spaces.

Strong Convergence of Mann’s Iteration Process in Banach Spaces

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.

Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).

Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

Strong Convergence to a Solution of a Variational Inequality Problem in Banach Spaces

We consider the variational inequality problem for a family of operators of a nonempty closed convex subset of a 2-uniformly convex Banach space with a uniformly Gâteaux differentiable norm, into its dual space. We assume some properties for the operators and get strong convergence to a common solution to the variational inequality problem by the hybrid method proposed by Haugazeau. Using these results, we obtain several results for the variational inequality problem and the proximal point algorithm.

The Strong Convergence of a New Iterative Algorithm for Asymptotically Nonexpansive Mappings

In a real Banach space E with a uniformly differentiable norm, we prove that a new iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. The results in this paper improve and extend some recent results of other authors.

FRÉCHET DIFFERENTIABILITY OF THE NORM IN A SOBOLEV SPACE WITH A VARIABLE EXPONENT

Let Ω be a domain in ℝN, let [Formula: see text] be such that p(x) > 1 for all [Formula: see text], let W1,p(⋅) (Ω) be the Sobolev space with variable exponent p(⋅), let Γ0 be a dΓ-measurable subset of Γ = ∂Ω that satisfies dΓ-meas Γ0 > 0, and let UΓ0 = {u ∈ W1,p(⋅)(Ω); tr u = 0 on Γ0}. It is shown that the map u ∈ UΓ0 ↦ ‖u‖0,p(⋅), ∇ = ‖|∇u|‖0,p(⋅) is a Fréchet-differentiable norm on UΓ0, and a formula expressing the Fréchet derivative of this norm at any nonzero u ∈ UΓ0 is given. We also show that, if p(x) ≥ 2 for all [Formula: see text], (UΓ0, ‖u‖0,p(⋅), ∇) is uniformly convex. Using properties of duality mappings defined on Banach spaces having a Fréchet-differentiable norm, we give the explicit form of continuous linear functionals on (UΓ0, ‖u‖0,p(⋅), ∇). It is also shown that the space UΓ0 and its dual have the same Krein–Krasnoselski–Milman dimension.

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

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.

Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach spaceXwhich has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive inX. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.