A note on boundary conditions for nonlinear operators

We investigate boundary conditions for strict-$\psi$-contractive and $\psi$-condensing operators. We derive results on the existence of eigenvectors with positive and negative eigenvalues and we obtain fixed point theorems for classes of noncompact opera\-tors.

Recensione: MR2817284 Dhompongsa, S.; Nanan, N. Fixed point theorems by ways of ultra-asymptotic centers. Abstr. Appl. Anal. 2011, Art. ID 826851, 21 pp. (Reviewer: Diana Caponetti

Paper review

Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1]$

In this paper for any $\varepsilon >0$ we construct a new proper $k$-ball-contractive retraction of the closed unit ball of the Banach space $C^m [0,1]$ onto its boundary with $k < 1+ \varepsilon$, so that the Wośko constant $W_\gamma (C^m [0,1])$ is equal to $1$.

Proper 1-ball contractive retractions in Banach spaces of measurable functions

In this paper we consider the Wosko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k > 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct. 1. Introduction Let X be an infinite-dimensional Banach space with unit closed ball B(X) and unit sphere S(X). It is well known that, in this setting, there is a retraction of B(X) onto S(X), that is, a continuous mapping R : B(X) ! S(X) with Rx = x for all x 2 S(X). In (4) Benyamini and Sternf…

On some parameters related to weak noncompactness in L1(μ,E)

A weak measure of noncompactness γU is defined in a Banach space in terms of convex compactness. We obtain relationships between the measure γU(A) of a bounded set A in the Bochner space L1(μ,E) and two parameters Π(A) and Λ1(A).

Recensione: MR2928500 Cascales, Bernardo; Kalenda, Ondřej F. K.; Spurný, Jiří A quantitative version of James's compactness theorem. Proc. Edinb. Math. Soc. (2) 55 (2012), no. 2, 369–386. (Reviewer: Diana Caponetti)

Paper review

Compactness in Groups of Group-Valued Mappings

We introduce the concepts of extended equimeasurability and extended uniform quasiboundedness in groups of group-valued mappings endowed with a topology that generalizes the topology of convergence in measure. Quantitative characteristics modeled on these concepts allow us to estimate the Hausdorff measure of noncompactness in such a contest. Our results extend and encompass some generalizations of Fr&eacute;chet&ndash;&Scaron;mulian and Ascoli&ndash;Arzel&agrave; compactness criteria found in the literature.

On some parameters related to weak noncompactness in L1(μ,E)

Abstract A weak measure of noncompactness γU is defined in a Banach space in terms of convex compactness. We obtain relationships between the measure γU (A) of a bounded set A in the Bochner space L1 (μ,E) and two parameters Π(A) and Δ1(A). Then the criterion for relative weak compactness due to Ulger [19] and Diestel-Ruess-Schachermayer [11] is recovered.

Rearrangement and convergence in spaces of measurable functions

We prove that the convergence of a sequence of functions in the space of measurable functions, with respect to the topology of convergence in measure, implies the convergence -almost everywhere ( denotes the Lebesgue measure) of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space , and also on Orlicz spaces with respect to a finitely additive extended real-valued set function. In the space and in the space , of finite elements of an Orlicz space of a -additive set function, we introduce some parameters which estimate the Hausdorff measure of noncompactness. We obtain some relations involving these parameters when passing from a bounded set of , or , to th…

On proper k-ball contractive retractions in the Banach space BC(R+)

Description of the limit set of Henstock–Kurzweil integral sums of vector-valued functions

Abstract Let f be a function defined on [ 0 , 1 ] and taking values in a Banach space X . We show that the limit set I HK ( f ) of Henstock–Kurzweil integral sums is non-empty and convex when the function f has an integrable majorant and X is separable. In the same setting we give a complete description of the limit set.

Recensione: MR2739903 Haddadi, Mohammad Reza; Mazaheri, Hamid; Labbaf Ghasemi, Mohammad Hussein Relation between fixed point and asymptotical center of nonexpansive maps. Fixed Point Theory Appl. 2011, Art. ID 175989, 6 pp. (Reviewer: Diana Caponetti)

Paper review

An extension of Guo's theorem via k--contractive retractions

Abstract Let X be a infinite-dimensional Banach space. We generalize Guo's Theorem [D.J. Guo, Eigenvalues and eigenvectors of nonlinear operators, Chinese Ann. Math. 2 (1981) 65–80 [English]] to k- ψ -contractions and condensing mappings, under a condition which depends on the infimum k ψ of all k ⩾ 1 for which there exists a k- ψ -contractive retraction of the closed unit ball of the space X onto its boundary.

Recensione: MR3198633 Reviewed Olszowy, Leszek A family of measures of noncompactness in the space L1loc(R+) and its application to some nonlinear Volterra integral equation. Mediterr. J. Math. 11 (2014), no. 2, 687–701. (Reviewer: Diana Caponetti)

On the measure of solvability of the identity operator

Eigenvectors of k-psi-contractive wedge operators

We present new boundary conditions under which the fixed point index of a strict-$\psi$-contractive wedge operator is zero. Then we investigate eigenvalues and corresponding eigenvectors of k-$\psi$-contractive wedge operators.

Boundary conditions for k-$psi$-contractive maps

On Boundary Conditions for Wedge Operators on Radial Sets

We present a theorem about calculation of fixed point index for k-$\psi$-contractive operators with 0 < k <1 defined on a radial set of a wedge of an infinite dimensional Banach space. Then results on the existence of eigenvectors and nonzero fixed points are obtained.

Recensione: MR2826706 Abbas, Mujahid; Hussain, Nawab; Rhoades, Billy E. Coincidence point theorems for multivalued f -weak contraction mappings and applications. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105 (2011), no. 2, 261–272. (Reviewer: Diana Caponetti)

MR2645846 (2011f:46031) Day, Jerry B.; Lennard, Chris A characterization of the minimal invariant sets of Alspach's mapping. Nonlinear Anal. 73 (2010), no. 1, 221–227. (Reviewer: Diana Caponetti)

Weakly compact, convex subsets in a Banach space need not have the fixed point property for nonexpansive mappings, as shown by D.E. Alspach in [Proc. Amer. Math. Soc. 82 (1981), no. 3, 423–424; MR0612733 (82j:47070)], where the example of a weakly compact, convex subset $C$ of $L_1[0,1]$ and of a nonexpansive self mapping $T$ on $C$ fixed point free is provided. Then, by Zorn's lemma, there exist weakly compact, convex, $T$-invariant fixed point free subsets of the set $C$ which are minimal with respect to these properties. But these minimal invariant sets have not been explicitly characterized. In the paper under review the authors give an explicit formula for the $n$th power $T^n$ of the …

A Note on the Measure of Solvability

A remark on weakly convex continuous mappings in topological linear spaces

Abstract Let C be a compact convex subset of a Hausdorff topological linear space and T : C → C a continuous mapping. We characterize those mappings T for which T ( C ) is convexly totally bounded.

Recensione: MR3038069 Reviewed Banaś, Józef; Ben Amar, Afif Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets. Comment. Math. Univ. Carolin. 54 (2013), no. 1, 21–40. (Reviewer: Diana Caponetti)

An extension of Guo's theorem via k-psi-contractive retraction

Let X be a infinite-dimensional Banach space. We generalize Guo''s Theorem [D.J. Guo, Eigenvalues and eigenvectors of nonlinear operators, Chinese Ann. Math. 2 (1981) 65–80 [English]] to k-ψ-contractions and condensing mappings, under a condition which depends on the infimum kψ of all k \ge1 for which there exists a k-ψ-contractive retraction of the closed unit ball of the space X onto its boundary.

Examples of proper k-ball contractive retractions in F-normed spaces

Abstract Assume X is an infinite dimensional F -normed space and let r be a positive number such that the closed ball B r ( X ) of radius r is properly contained in X . The main aim of this paper is to give examples of regular F -normed ideal spaces in which there is a 1-ball or a ( 1 + e ) -ball contractive retraction of B r ( X ) onto its boundary with positive lower Hausdorff measure of noncompactness. The examples are based on the abstract results of the paper, obtained under suitable hypotheses on X .

A note on the admissibility of modular function spaces

Abstract In this paper we prove the admissibility of modular function spaces E ρ considered and defined by Kozlowski in [17] . As an application we get that any compact and continuous mapping T : E ρ → E ρ has a fixed point. Moreover, we prove that the same holds true for any retract of E ρ .

On the integration of Riemann-measurable vector-valued functions

We confine our attention to convergence theorems and descriptive relationships within some subclasses of Riemann-measurable vector-valued functions that are based on the various generalizations of the Riemann definition of an integral.

Monotonicity and total boundednessin spaces of measurable functions

Abstract We define and study the moduli d(x, 𝓐, D) and i(x, 𝓐,D) related to monotonicity of a given function x of the space L 0(Ω) of real-valued “measurable” functions defined on a linearly ordered set Ω. We extend the definitions to subsets X of L 0(Ω), and we use the obtained quantities, d(X) and i(X), to estimate the Hausdorff measure of noncompactness γ(X) of X. Compactness criteria, in special cases, are obtained.

Eigenvectors of k–ψ-contractive wedge operators

Abstract We present new boundary conditions under which the fixed point index of a strict- ψ -contractive wedge operator is zero. Then we investigate eigenvalues and corresponding eigenvectors of k – ψ -contractive wedge operators.

On Regulated Solutions of Impulsive Differential Equations with Variable Times

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.

On the admissibility of the space L_{0}(A, X) of vector-valued measurable functions

We prove the admissibility of the space L_0(A,X) of vector-valued measurable functions determined by real-valued finitely additive set functions defined on algebras of sets.

On a step method and a propagation of discontinuity

In this paper we analyze how to compute discontinuous solutions for functional differential equations, looking at an approach which allows to study simultaneously continuous and discontinuous solutions. We focus our attention on the integral representation of solutions and we justify the applicability of such an approach. In particular, we improve the step method in such a way to solve a problem of vanishing discontinuity points. Our solutions are considered as regulated functions.

On some parameters related to weak noncompactness in L1(μ,E)

A measure of weak noncompactness γU is defined in a Banach space X in terms of convex compactness. We obtain relationships between the measure γU(A) of a bounded set A in the Bochner space L1(μ,E) and two parameters Π(A) and Λ1(A) related, respectively, to uniform integrability and weak-tightness. The criterion for relative weak compactness in L1(μ,E) is recovered.

MR2449047 (2009j:47108) Chermisi, Milena; Martellotti, Anna Fixed point theorems for middle point linear operators in $L^1$. Fixed Point Theory Appl. 2008, Art. ID 648591, 13 pp. (Reviewer: Diana Caponetti) 47H10 (47H09)

In the paper under review the notion of middle point operator is introduced. The authors prove that for a given nonempty, bounded, $\rho$-closed, convex subset K of L1(μ), where $\rho$ is the metric of the convergence locally in measure, if T from (K, $\rho$) to(K, $\rho$) is a continuous, $\rho$-nonexpansive, middle point linear operator, then T has at least one fixed point in K. To prove the theorem they use results of A. V. Bukhvalov [in Operator theory in function spaces and Banach lattices, 95–112, Birkh¨auser, Basel, 1995; MR1322501 (95m:46123)] and M. Furi and A. Vignoli [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 48 (1970), 195–198; MR0279792 (43 #5513)]. Then they …

MR2580162 (2011b:46030) Martinón, Antonio A note on measures of nonconvexity. Nonlinear Anal. 72 (2010), no. 6, 3108–3111. (Reviewer: Diana Caponetti), 46B20 (52A05 54B20)

Eisenfeld and Lakshmikantham [Yokohama Math. J. 24 (1976), no.1-2, 133-140; MR0425704 (54$\#$13657)] defined the measure of nonconvexity $\alpha(C)$ of a subset $C$ of a Banach space $X$ to be the Hausdorff distance $h(C, {\rm conv} C)$ between the set $C$ and its convex hull. In this note the author, for a nonempty bounded subset $C$ of $X$, defines a measure of nonconvexity $\beta(C)$ as the Hausdorff distance of $C$ to the family $bx(X)$ of all nonempty bounded convex subsets of $X$, i.e. $ \beta(C)= \inf_{K \in bx(X)}h(C,K ). $ The author studies the properties of $\beta$. He shows that $\alpha$ and $ \beta$ are equivalent, but not equal in the general case.

MR2543732 (2010g:46038) Colao, Vittorio; Trombetta, Alessandro; Trombetta, Giulio Hausdorff norms of retractions in Banach spaces of continuous functions. Taiwanese J. Math. 13 (2009), no. 4, 1139–1158. (Reviewer: Diana Caponetti)

A retraction $R$ from the closed unit ball of a Banach space $X$ onto its boundary is called $k$-ball contractive if there is $k \ge 0$ such that $ \gamma_X(RA) \le k \gamma_X(A) $ for each subset $ A$ of the closed unit ball, where $\gamma_X$ denote the Hausdorff (ball) measure of noncompactness. In the paper under review the authors consider the problem of evaluating the Wo\'{s}ko constant, which is the infimum of all numbers $k$'s for which there is a $k$-ball contractive retraction from the closed unit ball onto the sphere, in Banach spaces of real continuous functions defined on domains which are not necessarily bounded or finite dimensional. The paper extends some previous results val…

MR2370688 (2009e:46013) Navarro-Pascual, J. C.; Mena-Jurado, J. F.; Sánchez-Lirola, M. G. A two-dimensional inequality and uniformly continuous retractions. J. Math. Anal. Appl. 339 (2008), no. 1, 719--734. (Reviewer: Diana Caponetti) 46B20 (46E40)

Let X be an infinite-dimensional uniformly convex Banach space and let BX and SX be its closed unit ball and unit sphere, respectively. The main result of the paper is that the identity mapping on BX can be expressed as the mean of n uniformly continuous retractions from BX onto SX for every n >= 3. Then, the authors observe that the result holds under a property weaker than uniform convexity, satisfied by any complex Banach space, so that the result generalizes that of [A. Jim´enez-Vargas et al., Studia Math. 135 (1999), no. 1, 75–81; MR1686372 (2000b:46025)]. As an application the extremal structure of spaces of vector-valued uniformly continuous mappings is studied.

MR2595826 (2011c:46026) Domínguez Benavides, T. The Szlenk index and the fixed point property under renorming. Fixed Point Theory Appl. 2010, Art. ID 268270, 9 pp. (Reviewer: Diana Caponetti)

It is known that not every Banach space can be renormed so that the resultant space satisfies the weak Fixed Point Property (w-FPP). In the paper under review the author gives a further contribution to identify classes of Banach spaces which can be renormed to satisfy the w-FPP. Let $X$ be a Banach space and $X^*$ its dual. The dual norm is $UKK^*$ if for every $\varepsilon >0$ there is $\theta(\varepsilon)>0$ such that every $u$ in the closed unit ball $B_{X^*}$ of $X^*$ with $\|u\| > 1 - \theta(\varepsilon)$ has a weak$^*$ open neighborhood $\mathcal{U}$ with diam$(B_{X^*}\cap\mathcal{U})< \epsilon$. In [Bull. Lond. Math. Soc. 42 (2010), no. 2, 221--228; MR2601548] M. Raya showed that if …