Search

In this paper, we established two results concerning non-surjective (ε, s)-isometries of uniformly convex Banach spaces, which extended some known results of Dolinar and Jung. [ABSTRACT FROM AUTHOR]

In this paper, we established two results concerning non-surjective $ (\varepsilon, s) $-isometries of uniformly convex Banach spaces, which extended some known results of Dolinar and Jung.

In this paper, we introduce a new class of nonlinear mappings and compare it to other classes of nonlinear mappings that have appeared in the literature. We establish various existence and convergence theorems for this class of mappings under different assumptions in Banach spaces, particularly Banach spaces with a normal structure. In addition, we provide examples to substantiate the findings presented in this study. We prove the existence of a common fixed point for a family of commuting α\alpha -partially nonexpansive self-mappings. Furthermore, we extend the results reported by Suzuki (Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088–1095), Llorens-Fuster (Partially nonexpansive mappings, Adv. Theory Nonlinear Anal. Appl. 6 (2022), no. 4, 565–573), and Dhompongsa et al. (Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 350 (2009), no. 1, 12–17). Finally, we present an open problem concerning the existence of fixed points for α\alpha -partially nonexpansive mappings in the context of uniformly nonsquare Banach spaces.

We study the split common zero point problem with multiple output sets (SCZPPMOS, for short) in Banach spaces. We introduce two new cyclic projection algorithms for finding a solution to SCZPPMOS and establish the strong convergence of the sequences generated by them. [ABSTRACT FROM AUTHOR]

Given a nonempty subset of a normed space, a point of the unit sphere is said to be -acting if some ball touches the set by an analogue of the point , i.e., . Balayage theorems of geometric approximation theory are obtained in terms of -acting points. The concepts of -strictly convex and -uniformly convex spaces are introduced, density results for points of uniqueness of closed sets are obtained. In particular, we generalize one result of S. B. Stechkin, who characterized the spaces , where the set of points of uniqueness for any subset of is dense in it. We also show that if is a sun and if any -acting luminosity point is an LUR-point of the unit sphere, then has connected intersections with closed balls. A kind of converse of this result is proved: if a given set has connected intersections with open balls and if any -acting luminosity point is an LUR-point, then is a unimodal set, i.e., each local minimum of the distance function to M is a global one. [ABSTRACT FROM AUTHOR]

Abstract Let Y be a uniformly convex space with power type p, and let ( G , + ) $(G,+)$ be an abelian group, δ , ε ≥ 0 $\delta ,\varepsilon \geq 0$ , 0 < r < 1 $0< r

Let Y be a uniformly convex space with power type p, and let (G , +) be an abelian group, δ , ε ≥ 0 , 0 < r < 1 . We first show a stability result for approximate isometries from an arbitrary Banach space into Y. This is a generalization of Dolinar' results for (δ , r) -isometries of Hilbert spaces and L p (1 < p < ∞ ) spaces. As a result, we prove that if a standard mapping F : G → Y satisfies d (u , F (G)) ≤ δ ∥ u ∥ r for every u ∈ Y and | ∥ F (x) − F (y) ∥ − ∥ F (x − y) ∥ | ≤ ε , x , y ∈ G , then there is an additive operator A : G → Y such that ∥ F (x) − A x ∥ = o (∥ F (x) ∥) as ∥ F (x) ∥ → ∞. [ABSTRACT FROM AUTHOR]

Some results extending the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings are presented. [ABSTRACT FROM AUTHOR]

The Mann iteration with errors and Ishikawa iteration with errors have always been a hot topic of concern for many scholars. In this article, we explored the theorem of Chidume-Zegeye-Ntatin. The specific work was as follows: Firstly, the proof of the boundedness of sequence was improved in the theorem; Secondly, a simpler method was used to proved the theorem; Finally, the results extended and improve the related works. [ABSTRACT FROM AUTHOR]

In this paper, we bound the errors of kernel regularized regressions associating with 2 -uniformly convex two-sided RKBSs and differentiable σ (t) = | t | p p (1 < p ≤ 2) uniformly smooth losses. In particular, we give learning rates for the learning algorithm with loss V p (t) = | t | p p (1 < p ≤ 2). Also, we show a probability inequality and with which provide the error bounds for kernel regularized regression with loss V q (t) = | t | q q (q > 2). The discussions are comprehensive applications of the uniformly smooth function theory, the uniformly convex function theory and uniformly convex space theory. [ABSTRACT FROM AUTHOR]

Introduction Let be a nonempty subset of a normed linear space . A self-mapping is said to be nonexpansive provided that for all . In 1965, Browder showed that every nonexpansive self-mapping defined on a nonempty, bounded, closed and convex subset of a uniformly convex Banach space , has a fixed point. In the same year, Kirk generalized this existence result by using a geometric notion of normal structure. We recall that a nonempty and convex subset of a Banach space is said to have normal structure if for any nonempty, bounded, closed and convex subset of with , there exists a point for which . The well-known Kirk’s fixed point theorem states that if is a nonempty, weakly compact and convex subset of a Banach space which has the normal structure and is a nonexpansive mapping, then has at least one fixed point. In view of the fact that every nonempty, bounded, closed and convex subset of a uniformly convex Banach space has the normal structure, the Browder’ fixed point result is an especial case of Kirk’s theorem. Material and methods Let be a nonempty pair of subsets of a normed linear space . is said to be a noncyclic mapping if . Also the noncyclic mapping is called relatively nonexpansive whenever for any . Clearly, if , then we get the class of nonexpppansive self-mappings. Moreover, we note the noncyclic relatively nonexpansive mapping may not be continuous, necessarily. For the noncyclic mapping , a point is called a best proximity pair provided that In the other words, the point is a best proximity pair for if and are two fixed points of which estimates the distance between the sets and . The first existence result about such points which is an interesting extension of Browder’s fixed point theorem states that if is a nonempty, bounded, closed and convex pair in a uniformly convex Banach space and if is a noncyclic relatively nonexpansive mapping, then has a best proximity pair. Furthermore, a real generalization of Kirk’s fixed point result for noncyclic relatively nonexpansive mappings was proved by using a geometric concept of proximal normal structure, defined on a nonempty and convex pair in a considered Banach space. Results and discussion Let be a nonempty and convex pair of subsets of a normed linear space and be a noncyclic mapping. The main purpose of this article is to study of the existence of best proximity pairs for another class of noncyclic mappings, called noncyclic strongly relatively C-nonexpansive. To this end, we use a new geometric notion entitled -uniformly semi-normal structure defined on in a Banach space which is not reflexive, necessarily. To illustrate this geometric property, we show that every nonempty, bounded, closed and convex pair in uniformly convex Banach spaces has -uniformly semi-normal structure under some sufficient conditions. Conclusion The following conclusions were drawn from this research. We introduce a geometric notion of -uniformly semi-normal structure and prove that: Let be a nonempty, bounded, closed and convex pair in a strictly convex Banach space such that is nonempty and . Let be a noncyclic strongly relatively C-nonexpansive mapping. If has the -uniformly semi-normal structure, then has a best proximity pair. In the setting of uniformly convex in every direction Banach space , we also prove that: Let be a nonempty, weakly compact and convex pair in and be a noncyclic mapping such that for all with . If where is a projection mapping defined on then has -semi-normal structure. We present some examples showing the useability of our main conclusions. ./files/site1/files/42/8Abstract.pdf

In this paper, we present some fixed point results for a class of nonexpansive type and α-Krasnosel’skiĭ mappings. Moreover, we present some convergence results for one parameter nonexpansive type semigroups. Some non-trivial examples have been presented to illustrate facts.

In 1965, Browder proved the existence of a common fixed point for commuting families of nonexpansive mappings acting on nonempty bounded closed convex subsets of uniformly convex Banach spaces. The purpose of this paper is to extend this result to left amenable semigroups of nonexpansive mappings. [ABSTRACT FROM AUTHOR]

We prove that for every Banach space Y, the Besov spaces of functions from the n-dimensional Euclidean space to Y agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type q are continuously embedded in the Besov spaces of the same type if and only if Y has martingale cotype q. We interpret this as an extension of earlier results of Xu (J Reine Angew Math 504:195-226, 1998), and Martínez et al. (Adv Math 203(2):430-475, 2006). These two results combined give the characterization that Y admits an equivalent norm with modulus of convexity of power type q if and only if weakly differentiable functions have good local approximations with polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we prove the analog to Browder and Gohde fixed point theorem for G-nonexpansive mappings in complete hyperbolic metric spaces uniformly convex. In the linear case, this result is refined. Indeed, we prove that if X is a Banach space uniformly convex in every direction endowed with a graph G, then every G-nonexpansive mapping T : A → A, where A is a nonempty weakly compact convex subset of X, has a fixed point provided that there exists u0 2 A such that T(u0) and u0 are G-connected. [ABSTRACT FROM AUTHOR]

The paper investigates various properties of Moreau envelopes and proximal mappings of s -lower regular functions. In particular, for such a function on uniformly convex space, the Moreau envelope is shown to be differentiable with its derivative locally Hölder continuous. [ABSTRACT FROM AUTHOR]

Abstract: This paper provides a quantitative version of the classical image recovery problem to find an -approximate solution of the problem. The rate of asymptotic regularity of the iteration schemas, connected with the problem of image recovery, coincides with the existing optimal and quadratic bound for Krasnoselskii–Mann iterations. We then provide explicit effective and uniform bounds on the approximate fixed points of the mappings under consideration to be an approximate solution of the image recovery problem up to a uniform change from to . When combined, these results provide algorithms with explicit rates of convergence for the recovery of an -perturbation of the original image in different settings. [Copyright &y& Elsevier]

In this paper, we first introduce the class of generalized nonexpansive mappings in Banach spaces. This class contains both the classes of nonexpansive and α-nonexpansive mappings. In addition, we obtain some fixed point and coincidence point theorems for generalized nonexpansive mappings in uniformly convex Banach spaces. Our results extend some wellknown results in literature. [ABSTRACT FROM AUTHOR]

It is shown that if X is a uniformly convex Banach space and S a bounded linear operator on X for which ‖I−S‖=1, then S is invertible if and only if ‖I−12S‖

Abstract: The purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings. The results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature. More precisely, we give a rate of asymptotic regularity of an iteration schema due to Kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces. The rate only depends on an upper bound on the distance between the starting point and some common fixed point, a lower bound , the error and a modulus of uniform convexity. [Copyright &y& Elsevier]

The notion of a metric bead space was introduced in the preceding paper (L.Pasicki: Bead spaces and fixed point theorems, Topology Appl., vol. 156 (2009), 1811-1816) and it was proved there that every bounded set in such a space (provided the space is complete) has a unique central point. The bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces. It appears that normed bead spaces are identical with uniformly convex spaces. On the other hand the 'metric' approach leads to new elementary conditions equivalent to the uniform convexity. The initial part of the paper contains the proof that discus spaces (they seem to have a richer structure) are identical with bead spaces. [ABSTRACT FROM AUTHOR]

This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent. [ABSTRACT FROM AUTHOR]

We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z, then the coefficients of the nonconstant terms in the power series expansion about z cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum. [ABSTRACT FROM AUTHOR]

Abstract: The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point – the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler''s Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26). [Copyright &y& Elsevier]

We obtain new embedding theorems for Lorentz spaces of vector-valued martingales, thus generalizing the classical martingale inequalities. In contrast to earlier methods, we use martingale transformations defined by sequences of operators and identify the operator S(f) for a martingale f ranging in a Banach space X with the maximal operator for some ℓ( X)-valued martingale transform. The obtained inequalities are closely related to geometric properties of the Banach space in question. [ABSTRACT FROM AUTHOR]

In this paper we consider a family of regularization methods for linear operator equations with B-symmetric and B-positive operators. We establish convergence theorems for regularized solutions and their discrete approximations. [ABSTRACT FROM AUTHOR]

Abstract: The notion of a bead metric space defined here (see Definition 6) is a nice generalization of that of the uniformly convex normed space. In turn, the idea of a central point for a mapping when combined with the “single central point” property of the bead spaces enables us to obtain strong and elegant extensions of the Browder–Göhde–Kirk fixed point theorem for nonexpansive mappings (see Theorems 14–17). Their proofs are based on a very simple reasoning. We also prove two theorems on continuous selections for metric and Hilbert spaces. They are followed by fixed point theorems of Schauder type. In the final part we obtain a result on nonempty intersection. [Copyright &y& Elsevier]

Abstract: Characterizations for noncreasiness and uniform noncreasiness of the -direct sum of Banach spaces are given. As a consequence, the geometry of the infinite -direct sum and the Lebesgue–Bochner space with respect to these notions is investigated where . [Copyright &y& Elsevier]