Convex Banach Space Research Articles

Mordukhovich Derivatives of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces

In this paper, we investigate the properties of the Mordukhovich derivatives of the metric projection operator onto closed balls, closed and convex cylinders and positive cones in uniformly convex and uniformly smooth Banach spaces. We find the exact expressions for Mordukhovich derivatives of the metric projection operator in these spaces.

Best approximation theorems via relatively u-continuous and Jaggi nonexpansive mappings

In this paper, we introduced a new type of cyclic (noncyclic) mappings, called Jaggi relatively nonexpansive mappings and use to investigate the existence of best proximity points (pairs) in the framework of reflexive (strictly convex) Banach spaces by using a geometric concept of proximal normal structure. We also present a best proximity version of Schauder’s fixed point problem for non-self mappings. Finally, we prove the existence of a common best proximity point for an arbitrary family of cyclic relatively u-continuous mappings which are compact and then we obtain a generalization of Markov-Kakutani’s theorem.

Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces

We prove that if Y Y is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in ℓ 1 \ell _1 , or a normalized sequence with upper p-estimates for some p > 1 p>1 , then Y Y does not satisfy Lindenstrauss’ property B.

Conditional plasticity of the unit ball of the ℓ∞-sum of finitely many strictly convex Banach spaces

We prove that for any ℓ∞-sum Z=⨁i∈[n]Xi of finitely many strictly convex Banach spaces (Xi)i∈[n], an extremeness preserving 1-Lipschitz bijection f:BZ→BZ is an isometry, by constraining the componentwise behavior of the inverse g=f−1 with a theorem admitting a graph-theoretic interpretation. We also show that if X,Y are Banach spaces, then a bijective 1-Lipschitz non-isometry of type BX→BY can be used to construct a bijective 1-Lipschitz non-isometry of type BX′→BX′ for some Banach space X′, and that a homeomorphic 1-Lipschitz non-isometry of type BX→BX restricts to a homeomorphic 1-Lipschitz non-isometry of type BS→BS for some separable subspace S≤X.

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

Modified Inertial Krasnosel'skii-Mann Type Method for Solving Fixed PointProblems in Real Uniformly Convex Banach Spaces

We present an altered version of the inertial Krasnosel'skii-Mann algorithm and demonstrate convergence outcomes for mappings that are asymptotically nonexpansive within real, uniformly convex Banach spaces. To achieve our results, we skillfully construct the inequality in equation \eqref{6} and apply it accordingly. Our findings support and broadly generalize a number of significant findings from the literature. We demonstrate, as an application, the generation of maximal monotone operators' zeros via fixed point methods in Hilbert spaces. Additionally, we solve convex minimization issues using our fixed-point techniques.

Existence of sunny nonexpansive retractions and approximation of fixed points of a representation of nonexpansive mappings

This paper presents an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth uniformly convex Banach space with respect to a left-regular sequence of means defined on a subset of l∞(S). The main results are to establish an existence theorem of a sunny nonexpansive retraction and to create an algorithm for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces.

Convergence analysis of Picard–SP iteration process for generalized $$\alpha $$–nonexpansive mappings

AbstractIn this manuscript, we introduce a novel hybrid iteration process called the Picard–SP iteration process. We apply this new iteration process to approximate fixed points of generalized $$\alpha $$ α –nonexpansive mappings. Convergence analysis of our newly proposed iteration process is discussed in the setting of uniformly convex Banach spaces and results are correlated with some other existing iteration processes. The dominance of the newly proposed iteration process is exhibited with the help of a new numerical example. In the end, the comparison of polynomiographs generated by other well-known iteration processes with our proposed iteration process has been presented to make a strong impression of our proposed iteration process.

Asymptotic Behavior of Generalized CR−iteration Algorithm and Application to Common Zeros of Accretive Operators

The purpose of this study is to provide a generalized CR−iteration algorithm for finding common fixed points (CFPs) for nonself quasi-nonexpansive mappings (QNEMs) in a uniformly convex Banach space. The suggested algorithm’s convergence analysis is analyzed in uniformly convex Banach spaces. Keywords: Fixed point, CR−iterative algorithm, nonself QNEMs.

Convergence of Fibonacci–Ishikawa iteration procedure for monotone asymptotically nonexpansive mappings

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.

Some convergence results for G− mean nonexpansive mappings in uniformly convex Banach space endowed with graph

Some convergence results for G− mean nonexpansive mappings in uniformly convex Banach space endowed with graph

An Algorithm for Nonlinear Problems Based on Fixed Point Methodologies With Applications

This research presents a highly efficient fixed point algorithm for the computation of fixed points for a very general class of nonexpansive mappings called generalized (α, β)-nonexpansive mappings within the context of uniformly convex Banach space. Our research establishes both weak and strong convergence theorems of the scheme. Furthermore, we demonstrate that the class of generalized (α, β)-nonexpansive mappings contain many classes of nonlinear mappings of the classical literature. Then, we perform various numerical computations to prove the efficiency of the proposed approach. We also study the convergence analysis of the scheme for two dimensional space with taxicab norm. Moreover, we show that our new result gives an alternative approach for solving Caputo fractional differential equation in a novel mappings setting.

Convergence Analysis of Picard Thakur Hybrid Iterative Scheme for α-Nonexpansive Mappings in Uniformly Convex Banach Spaces

In this study, we investigate the convergence behavior of fixed points for generalized α-nonexpansive mappings using the Picard-Thakur hybrid iterative scheme. We obtain weak and strong convergence results for generalized α-nonexpansive mappings in a uniformly convex Banach space. Numerically, we demonstrate that the Picard-Thakur hybrid iterative scheme converges more rapidly than other well-known schemes. Additionally, we present findings on data dependence and provide a numerical example to illustrate the concept. The obtained results are expanded and generalized to be consistent with relevant findings in the existing literature.

A novel Noor iterative method of operators with property (E) as concerns convex programming applicable in signal recovery and polynomiography

This work proposes a novel Noor iterative scheme, called CT‐iteration, to approximate the fixed points in the new context of generalized nonexpansive mappings with property . We establish the strong and weak convergence results in a uniformly convex Banach space. Additionally, numerical examples of the iterative technique are demonstrated using a signal recovery application in a compressed sensing situation. Furthermore, we show the use of the proposed method to generate polynomiographs. The proposed algorithm has been implemented and tested via numerical simulation in MATLAB. The simulation results show the effectiveness of the proposed algorithm.

Hybrid Inertial Iterative Method for Fixed point, Variational Inequality and Generalized Mixed Equilibrium Problems in Banach Space

In this paper, we introduced a hybrid inertial iterative method which converges strongly to a common element of solution of generalized mixed equilibrium, variational inequality and fixed point problems in a two uniformly smooth and uniformly convex Banach space. Our hybrid inertial iterative method, techniques of proof and corollaries improves, extends and generalizes many results in the literature.

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

AbstractIn this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

Monotone Orbitally Nonexpansive and Cyclic Mappings in Partially Ordered Uniformly Convex Banach Spaces

Monotone Orbitally Nonexpansive and Cyclic Mappings in Partially Ordered Uniformly Convex Banach Spaces

Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application

This paper is aimed at proving the efficiency of a faster iterative scheme called $PC^*$-iterative scheme to approximate the fixed points for the class of Suzuki's Generalized non-expansive mapping in a uniformly convex Banach space. We will prove some weak and strong convergence results. It is justified numerically that the $PC^*$-iterative scheme converges faster than many other remarkable iterative schemes. We will also provide numerical illustrations with graphical representations to prove the efficiency of $PC^*$ iterative scheme. As an application of the solution of a fractional differential equation is obtained by using $PC^*$ iterative scheme.

A New faster Iteration Process to fixed points of mean-non expansive Mappings in Banach Space

Let X be a Banach space and F a subset of X. In this paper, we introduce a new iterative scheme to approximate fixed point of mean non-expansive mappings. We first prove that proposed iteration process is faster than all of Mann, Ishikawa, Picard, Agarwal, Noor and Thakur processes for contractive mappings. We also show that some weak and strong convergence theorems for mean non-expansive mappings. Using example presented in mean non-expansive mappings in Banach space. We compare the convergence behavior of the new iterative process with other iterative processes. Keywords. Mean non-expansive mappings, Iterative Process, Uniformly Convex Banach Space, Convergence Theorem.

Umbel convexity and the geometry of trees

For every p∈(0,∞), a new metric invariant called umbel p-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a “Poincaré-type” metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property (β). We explain how a relaxation of umbel p-convexity, called infrasup-umbel p-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants - fork p-convexity and infrasup-fork p-convexity - are introduced, and their relationship to Markov p-convexity and relaxations of the p-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram p-convexity inequality is proved for Heisenberg groups over p-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.