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U uvodnom poglavlju dan je pregled nekih općenitih rezultata iz područja topologije i funkcionalne analize potrebnih u ostatku rada. Poseban naglasak je na kompaktnosti u različitim prostorima od koje su navedene četiri vrste za koje se ispostavlja da su ekvivalentne u metričkim prostorima. U širokoj klasi beskonačnodimenzionalnih normiranih prostora zatvorena jedinična kugla nije kompaktna, što nas motivira na promatranje nekih drugih topologija na tom prostoru. U drugom poglavlju dokazani su Arzela-Ascolijev i Frechet-Kolmogorovljev teorem kojima su dane karakterizacije relativne kompaktnosti u prostorima C(K), odnosno \(L^p(\mathbb{R}^n)\). Zatim je promatran suodnos konveksnosti i kompaktnosti, posebno u vidu ekstremnih točaka takvih skupova. Tu smo dokazali poznati Krein-Milmanov teorem o egzistenciji ekstremnih točaka. U trećem poglavlju su promatrane slabe topologije inducirane familijom preslikavanja i dokazana su brojna svojstva tako dobivenih topologija. Zatim smo proučili posebne slučajeve kada je ta familija jednaka dualu normiranog prostora, odnosno kada je jednaka slici kanonskog ulaganja prostora u njegov bidual, čime redom dobivamo slabu, odnosno slabu* topologiju. Posljednje poglavlje posvećeno je najvažnijim rezultatima o kompaktnosti u slabim i slabim* topologijama. Prvo smo dokazali Banach-Alaogluov teorem o slaboj* kompaktnosti zatvorene jedinične kugle duala bilo kojeg normiranog prostora. Pokazane su brojne posljedice tog teorema poput činjenice da slaba* topologija duala Banachovog prostora ima Heine-Borelovo svojstvo, kao i Kakutanijev teorem koji daje karakterizaciju refleksivnosti preko slabe kompaktnosti zatvorene jedinične kugle. Nakon toga smo se dotaknuli Eberlein-Šmulianovog teorema kojim smo pokazali ekvivalenciju četiri tipa kompaktnosti u slabim topologijama (koje općenito nisu metrizabilne) te Krein-Šmulianovog teorema kojim smo pokazali da se u Banachovim prostorima svojstvo slabe kompaktnosti ne gubi uzimanjem zatvorene konveksne ljuske. Rad smo završili Jamesovim teoremom kojim je dana zanimljiva karakterizacija refleksivnosti i slabe kompaktnosti pomoću svojstva dostizanja supremuma funkcionala dualnog prostora. In the opening chapter we gave an overview of some general results in topology and functional analysis required in the rest of the thesis. The emphasis was on compactness in different settings: we considered four types which turn out to be equivalent in metric spaces. We showed that in the wide class of infinite-dimensional normed spaces closed unit ball is not compact, which motivated us to consider some other topologies on these spaces. The second chapter is devoted to compactness criteria in some commonly used normed spaces. We began by proving the Arzela-Ascoli and Frechet-Kolmogorov theorems that give characterizations of relative compactness in C(K) and \(L^p(\mathbb{R}^n)\) spaces, respectively. Then we took a look at the relationship between convexity and compactness, especially with respect to extreme points of such sets. Here we proved the famous Krein-Milman theorem on the existence of extreme points. In the third chapter we started by defining weak topologies induced by a family of mappings and proved many properties of these topologies. Then we considered special cases when this family is either the dual of a normed space or the image of the space under the canonical embedding into its bidual which gave us the weak and weak* topology. In the final chapter we covered the main results regarding compactness in weak and weak* topologies. First we proved the Banach-Alaoglu theorem on weak* compactness of the closed unit ball in the dual of any normed space. We showed many consequences of this theorem, such as the fact that the weak* topology has the Heine-Borel property, as well as Kakutani’s theorem which gives us a characterization of reflexivity in terms of weak compactness of the closed unit ball. We continued by proving the Eberlein- Šmulian theorem which states that the four types of compactness are equivalent in the weak topology (which isn’t generally metrizable) and the Krein-Šmulian theorem which shows that taking closed convex hulls preserves weak compactness in Banach spaces. We concluded the thesis with James’ theorem which gives us an interesting criterion for reflexivity and weak compactness by considering the supremum attaining functionals of the dual space.

Let (X, · ) be a (real) Banach space. We refer to [38] or [28] as some introduction to the general theory of Banach spaces. Note that, as usual in the case, all the results we discuss here remain valid for complex scalars with possibly different constants. Let I be a countable set with possibly some ordering we refer to whenever considering convergence with respect to elements of I (wich will be denoted by limi→∞). Definition 1 We say that countable system of vectors is biorthogonal if for i, j ∈ I we have

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

In Chapter 16 we give applications to various topics of Banach space theory, the most spectacular of which is a sharpened version of a celebrated result of Jean Bourgain on the Λ p problem.

We study the Bishop-Phelps-Bollob\'as property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a~uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop-Phelps-Bollob\'as theorem holds for operators from $\ell_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute., Comment: 26 pages

We prove the following result: Let M be a subset of a Banach space X with the following property: there is e ≥ 0 such that f is bounded on M whenever f is a real-valued and σ ( X ″ , X ′ ) -continuous function on X + e B X ″ . Then M is 2e-weakly relatively compact, that is, the σ ( X ″ , X ′ ) -closure of M is contained in X + 2 e B X ″ . This theorem is a quantitative extension of a theorem on weak compactness due to M. Valdivia that corresponds to the case e = 0 .

In this work, we give sufficient conditions for the existence of a mild solution for some impulsive integro-differential equations in Banach spaces. We study the existence without assuming the Lipschitz condition on the nonlinear term f. The compactness on the C0-semigroup(T(t))t≥0 in a Banach space is not needed. We use Hausdorff’s measure of noncompactness, resolvent operators and Darbo’s fixed point Theorem to obtain the main result of this work. Keywords: Impulsive integro-differential equations, Mild solutions, Sadovskii’s fixed point Theorem, Resolvent operators, Noncompactness measures

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is “the generalized Neumann lemma” which is quite different from the method in [12] where “the generalized Banach lemma” was used. By the method of the perturbation analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

Let X be a Banach space with an 1-unconditional Schauder basis and without isomorphic copies of l 1 ⊕ c 0 . We obtain an equivalent condition to weak compactness by means of a fixed-point theorem. Namely: a closed convex bounded subset C of X is weakly compact if and only if every cascading nonexpansive mapping T : C → C has a fixed point. We particularize our results when C is the closed unit ball of the Banach space X, obtaining a new characterization of reflexivity. Note that weak compactness is independent of the underlying equivalent norm and that every Banach space with an unconditional Schauder basis can be renormed to be 1-unconditional.

The main aim of this article is to establish analogues of Landau’s theorem for solutions to the$\overline{\unicode[STIX]{x2202}}$-equation in Dirichlet-type spaces.

In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

The objective in this study is to use generalized iterative procedures in order to approximate solutions of an equation on a Banach space setting. In particular, we present a semi-local convergence analysis for these methods. Some applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

The aim of this work is to present new abstract fixed point theorems in ordered Banach spaces and on Banach algebras. The main existence results generalize Krasnosel’skii’s fixed point theorem of compression and expansion to a class of $$\alpha $$ -homogeneous operators and monotone operators. Two applications to nonlinear Hammerstein integral equations, one of which is of fractional order, are provided together with a numerical example.

In this paper we study the Hardy–Lorentz spaces for Banach space valued martingales. The dual spaces are characterized and several martingale inequalities are established. The proofs mainly depend on the classical tool of atomic decompositions. As usual, these conclusions are closely related to the geometrical properties of the underlying Banach spaces.

We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E . The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.

In this work, we prove the weak and strong convergence of a sequence generated by a modified S-iteration process for finding a common fixed point of two G-nonexpansive mappings in a uniformly convex Banach space with a directed graph. We also give some numerical examples for supporting our main theorem and compare convergence rate between the studied iteration and the Ishikawa iteration.

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].

Let${\mathcal C}[{\mathcal X}]$be any class of operators on a Banach space${\mathcal X}$, and let${Holo}^{-1}({\mathcal C})$denote the class of operatorsAfor which there exists a holomorphic functionfon a neighbourhood${\mathcal N}$of the spectrum σ(A) ofAsuch thatfis non-constant on connected components of${\mathcal N}$andf(A) lies in${\mathcal C}$. Let${{\mathcal R}[{\mathcal X}]}$denote the class of Riesz operators in${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators$A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$(the class of all upper or lower [semi] Fredholm operators) by commuting operators in$B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi= 1(B− μi) is Riesz, then there exist decompositions${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$and$B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$such that: (i) If λ ≠ 0, then$\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0)$A\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-B_0\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), whereB0= ⊕mi= 1(Bi− μi); (iii) if$\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then$A-\lambda B\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$).

We prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear in...

In this article, fixed points of generalized asymptotically quasi-ϕ-nonexpansive mappings and equilibrium problems are investigated based on a monotone projection algorithm. Strong convergence theorems are established without the aid of compactness in the framework of reflexive Banach spaces.

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball BX* of X* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.

In this paper, we introduce the concept of a new nonlinear mapping called demigeneralized in a Banach space. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a common fixed point for a family of the new nonlinear mappings in a Banach space. We apply this result to obtain new strong convergence theorems in a Hilbert space and a Banach space, respectively.

In this paper a nonsmooth mean value theorem in asplund spaces, under convexity, using the properties of limiting subdifferentials is established. We research on a kind of mean value theorem and prove that this theorem for set-valued mappings under convexity of domein in banach spaces. This theorem is use full to establish new results in convex analysis.

The Livsic Theorem for Holder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism ${\sigma :X\to X}$ that satisfies the Anosov Closing Lemma and a Holder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. The map $a(x)$ is a coboundary with a Holder continuous transition function if and only if $a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)$ is the identity for each periodic point $p=\sigma^n p$.