Weak Noncompactness Research Articles

Leray Schauder Type Fixed Point Theorems in RWC-Banach Algebras and Application to Chandrasekhar Integral Equations

In this paper, the existence of fixed point results of Leray Schauder type for the sum and the product of nonlinear operators acting on RWC-Banach algebras under weak topology is proved. Our results are formulated in terms of a sequential characterization of the RWC-Banach algebra and the De Blasi measure of weak noncompactness. Application to Chandrasekhar Integral equations is also given.

Existence results for a nonlinear neutron transport equation with elastic and inelastic collision operators in L p -spaces

In this paper, we discuss the existence of solutions to a stationary neutron transport equation involving elastic and inelastic collision operators in L p - espaces ( 1 ≤ p < ∞ ) . For 1 < p < ∞ , we use the Krasnosel'slii fixed point theorem and the compactness which involved by the averaging result for neutron transport equation. For p = 1, our approach is different, it uses the measure of weak noncompactness of De Blasi, the concepts of Dunford–Pettis operators together with a recent version of Krasnosel'skii's fixed point theorem involving ws-compact and ww-compact operators and the weak compactness.

Strong Solutions and Mild Solutions for Sturm-Liouville Differential Inclusions

Existence results for a Cauchy problem driven by a semilinear differential Sturm-Liouville inclusion are achived by proving, in a preliminary way, an existence theorem for a suitable integral inclusion. In order to obtain this proposition we use a recent fixed point theorem that allows us to work with the weak topology and the De Blasi measure of weak noncompactness. So we avoid requests of compactness on the multivalued terms. Then, by requiring different properties on the map p involved in the Sturm-Liouville inclusion, we are able to establish the existence of both mild solutions and strong ones for the problem examinated. Moreover we focus our attention on the study of controllability for a Cauchy problem governed by a suitable Sturm-Liouville equation. Finally we precise that our results are able to study problems involving a more general version of a semilinear differential Sturm-Liouville inclusion.

Numerical and analytical approach to the Chandrasekhar quadratic functional integral equation using Picard and Adomian decomposition methods

&lt;p&gt;This research aimed to find numerical solutions to a type of nonlinear initial value problem (IVP) for hybrid fractional differential equations. Using the Adomian decomposition method (ADM) and the Picard method (PM), we studied the Chandrasekhar quadratic integral equation (QIE). Furthermore, we investigated existence and uniqueness results using measures of weak noncompactness. Through a set of examples and numerical simulations, a comparison was made between the results of the AMD and PM.&lt;/p&gt;

Some noncompact types of fixed point results in the generalized Banach spaces with respect to the G–weak topology contexts and applications

This research deals with Krasnoselskii’s fixed point theorem where the entries operators do not need to be G-weakly compact and contraction. These results were obtained by using the so-called generalized measure of weak noncompactness and some user-friendly lemmas. Moreover, these gained fixed point results are applied to study the existence of solutions of a coupled system for integral equations in the generalized Banach space mathcal{{ C }} ( [0,1], E_{1} )times mathcal{{ C }} ( [0,1], E_{2} ) .

Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

Generalized Schechter essential spectrum of the sum of two bounded operators involving measure of non-almost weak noncompactness

Generalized Schechter essential spectrum of the sum of two bounded operators involving measure of non-almost weak noncompactness

FIXED-POINT THEOREMS FOR MEIR–KEELER MULTIVALUED MAPS AND APPLICATION

This work is intended for a generalization of Darbo’s fixed-point theorem by using multivalued condensing operators, and a measure of weak noncompactness which does not necessarily possess the maximum property. Moreover, we prove some fixed-point theorems in Banach algebras which satisfy an appointed weak sequential condition. As an application, we establish the existence of solutions for a functional integral inclusion.

Equivalence of some properties in the theory of Banach algebras and applications

In this work, we establish a sequential characterization of the notion of relatively weak compactness of Banach algebras introduced recently by J. Banaś and L. Olszowy. Moreover, we show that this structure is one of the most important properties which could be lifted from a Banach algebra X to C(K,X) and L1(μ,X). In addition, fixed point theorems for the product of two nonlinear operators acting on RWC-Banach algebra are proved by the help of the De Blasi measure of weak noncompactness.

Solvability in weighted L1-spaces for the m-product of integral equations and model of the dynamics of the capillary rise

This article focuses on presenting a modified Hölder inequality and a new measure of noncompactness in L1N(R+) and applying them in proving two existence theorems (the existence and the uniqueness) of solutions for m-product of integral equations in the weighted Lebesgue spaces L1N(R+). This permits us to examine a model of the dynamics of the capillary rise of a fluid inside a tubular column in weighted Lebesgue spaces, which generalize the former results in the available literature. We utilize the analysis of fixed point hypothesis concerning a proper measure of weak noncompactness (MWNC) to get our outcomes. An example to validate our analysis is included.

Solutions of Neutral Differential Inclusions

Motivated by the study of neutral differential inclusions, we establish a new fixed point theorem for multivalued countably Meir-Keeler condensing mappings via an arbitrary measure of weak noncompactness which in turn include the fixed point theorems of Krasnoselskii and Dhage as special cases in non separable spaces.

A Measure of Weak Noncompactness in $$L^1({\mathbb {R}}^N)$$ and Applications

A Measure of Weak Noncompactness in $$L^1({\mathbb {R}}^N)$$ and Applications

Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces

Abstract The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space ( X , { | ⋅ | p } p ∈ Λ ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 𝜏-Krein–Šmulian property, where 𝜏 is a weaker Hausdorff locally convex topology of 𝑋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 𝜏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ ( R + ) × C ⁢ ( R + ) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result.

Weak demicompactness involving measures of weak noncompactness and invariance of the essential spectrum

In this paper, we show that an unbounded weakly S0-demicompact linear operator T, introduced in [18], acting on a Banach space, can be characterized by some measures of weak noncompactness. Moreover, our results are illustrated to discuss the relationship with Fredholm and upper semi-Fredholm operators as well as the stability of the essential spectrum of T.

Generalized relative essential spectra

The aim of the present paper is to give some spectral results about generalized Fredholm operators and the so called S-generalized Fredholm operators, where S is a given bounded linear operator acting on a Banach space X. When X possesses some properties, we provide then some sufficient conditions for which a bounded linear operator will be a generalized Fredholm. The obtained results are applied to characterize the so called generalized S-essential spectrum, in particular the generalized Jeribi S-essential spectrum [17]. These results are formulated by means of measure of weak noncompactness.

Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

EQUIVALENCE OF SEMI-NORMS RELATED TO SUPER WEAKLY COMPACT OPERATORS

AbstractWe study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

Convexification of super weakly compact sets and measure of super weak noncompactness

Convexification of super weakly compact sets and measure of super weak noncompactness

Meir-Keeler condensing operators and applications

Motivated by the open question posed by H. K. XU in [39] (Question 2:8), Belhadj, Ben Amar and Boumaiza introduced in [5] the concept of Meir-Keeler condensing operator for self-mappings in a Banach space via an arbitrary measure of weak noncompactness. In this paper, we introduce the concept of Meir- Keeler condensing operator for nonself-mappings in a Banach space via a measure of weak noncompactness and we establish fixed point results under the condition of Leray-Schauder type. Some basic hybrid fixed point theorems involving the sum as well as the product of two operators are also presented. These results generalize the results on the lines of Krasnoselskii and Dhage. An application is given to nonlinear hybrid linearly perturbed integral equations and an example is also presented.

Measure of super weak noncompactness in some Banach sequence spaces

The article aims to study measures of super weak noncompactness in Banach sequence spaces c0(Xi) and ℓp(Xi) (1≤p≤∞), and to characterize super weakly compact sets in these spaces by the measure σ. Specifically, in Banach spaces c0(Xi), ℓ∞(Xi), and in ℓ1(Xi) where Xi is super-reflexive, the formula of measure of super weak noncompactness σ is shown. In ℓp(Xi) (1<p<∞), a characterization of super weakly sets is proved in terms of measure of super weak noncompactness, and a regular measure, which is not equivalent to the measure σ, is constructed with specially chosen Xi.