Hausdorff Measure Of Noncompactness Research Articles

Differential equations with maximal monotone operators

The paper deals with multivalued differential equations in abstract spaces. Nonlocal conditions are assumed. The model includes an m-dissipative multioperator which generates an equicontinuous, not necessarily compact, semigroup. The regularity of the nonlinear term also depends on the Hausdorff measure of noncompactness. The existence of integral solutions is discussed, with a topological index argument. A transversality condition is required. The results are applied to a partial differential inclusion in a bounded domain in Rn with nonlocal integral conditions. The model also includes an m-dissipative but not necessarily compact semigroup generated by a suitable subdifferential operator.

Some results on matrix transformation and compactness for fibonomial sequence spaces

In this paper, we introduce the Fibonomial sequence spaces $$b_{0}^{r,s,F}$$ and $$b_{c}^{r,s,F}$$ and show that these are linearly isomorphic to the spaces $$c_{0}$$ and c, respectively. In addition, we present $$\alpha -$$ dual, $$\beta -$$ dual and $$\gamma -$$ dual for those spaces and characterize certain matrix classes. In the final section, we obtain some criteria for the compactness of certain matrix operators via Hausdorff measure of noncompactness on the space $$b_{0}^{r,s,F}.$$

Existence, uniqueness and Hyers-Ulam stability of random impulsive stochastic integro-differential equations with nonlocal conditions

<abstract><p>In this article, we study the existence and stability results of mild solutions for random impulsive stochastic integro-differential equations (RISIDEs) with noncompact semigroups and resolvent operators in Hilbert spaces. Initially, we prove the existence of mild solutions using Hausdorff measures of noncompactness and M$ \ddot{o} $nch fixed point theorem. Then, we explore the stability results which includes continuous dependence of initial conditions, Hyers-Ulam stability and mean-square stability of the system by developing some new analysis techniques and establishing an improved inequality. Finally, we propose an example to validate the obtained results.</p></abstract>

On generalization of Petryshyn's fixed point theorem and its application to the product of $ n $-nonlinear integral equations

<abstract><p>Regarding the Hausdorff measure of noncompactness, we provide and demonstrate a generalization of Petryshyn's fixed point theorem in Banach algebras. Comparing this theorem to Schauder and Darbo's fixed point theorems, we can skip demonstrating closed, convex and compactness properties of the investigated operators. We employ our fixed point theorem to provide the existence findings for the product of $ n $-nonlinear integral equations in the Banach algebra of continuous functions $ C(I_a) $, which is a generalization of various types of integral equations in the literature. Lastly, a few specific instances and informative examples are provided. Our findings can successfully be extended to several Banach algebras, including $ AC, C^1 $ or $ BV $-spaces.</p></abstract>

On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness

On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness

SOME GEOMETRIC CHARACTERIZATIONS OF EXTREMAL SETS IN HILBERT SPACES

Based on our previous result and by using the technique on α-minimal and χ-minimal sets with respect to the Kuratowski and Hausdorff measures of noncompactness, we give some new geometric characterizations of extremal sets in Hilbert spaces.

Absolute Lucas Spaces with Matrix and Compact Operators

The main purpose of this study is to introduce the absolute Lucas series spaces and to investigate their some algebraic and topological structure such as some inclusion relations, $BK-$ to this space, duals and Schauder basis. Also, the characterizations of matrix operators related to these space with their norms are given. Finally, by using Hausdorff measure of noncompactness, the necessary and sufficient conditions for a matrix operator on them to be compact are obtained.

Integrodifferential equations of Volterra type with nonlocal and impulsive conditions

This work is devoted to the study of a class of nonlocal impulsive integrodifferential equations of Volterra type. We investigate the situation when the resolvent operator corresponding to the linear part of {dxdt(t)=A(t)x(t)+∫0tΓ(t,s)x(s)ds+f(t,x(t)),t∈I=[0,T],t≠ti,i=1,2,3,…⁡,m,x(ti+)=x(ti)+Ji(x(ti)),x(0)=x0+g(x), is norm continuous. Our results are obtained by using the Hausdorff measure of noncompactness and fixed point theorems. An example is provided to illustrate the basic theory of this work.

Existence of solution of infinite systems of singular integral equations of two variables in C(I × I, ℓp) with I = [0, T], T > 0 and 1 < p < ∞ using Hausdorff measure of noncompactness

In this article, we discuss the solvability of infinite systems of singular integral equations of two variables in the Banach sequence spaces C(I ? I, ?p) with I = [0, T], T > 0 and 1 < p < ? with the help of Meir-Keeler condensing operators and Hausdorff measure of noncompactness. With an example, we illustrate our findings.

Matrix mappings and compact operators for Schröder sequence spaces

In this paper, we discuss the domain of a recently defined conservative matrix, constructed by means of the Schröder numbers in the spaces of $p-$absolutely summable sequences and bounded sequences. We determine the $\beta-$duals of the Banach spaces, introduced here, and present characterization of some matrix operators. Moreover, we give the characterization of certain compact operators via the Hausdorff measure of noncompactness.

Compact matrix operators on Banach space of absolutely $k$-summable series

In a more recent paper [23], Banach spaces of absolutely $k$-summable series $\left\vert T_{\phi }\right\vert _{k}$ have been introduced and studied by Sarıgöl and Agarwal. In this paper, we characterize matrix and compact operators from the space $\left\vert T_{\phi }\right\vert _{k}$ to the classical spaces $c_{0},c$ and $l_{\infty },$ by determining identities or estimates for operator norms and the Hausdorff measure of noncompactness.

Solvability of infinite systems of fractional differential equations in the space of tempered sequence space $m^beta(phi)$

The purpose of this article, is to establish the existence of solution of infinite systems of fractional differential equations in space of tempered sequence $m^beta(phi)$ by using techniques associated with Hausdorff measures of noncompactness. Finally, we provide an example to highlight and establish the importance of our main result.

On composition operators of Fibonacci matrix and applications of Hausdorff measure of noncompactness

The aim of the paper is introduced the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right).$ Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of new spaces and also construct the basis for the space $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$Finally we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators by applying the Hausdorff measure of noncompactness, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\},$ and $1\leq p<\infty.$

Characterisations of bounded linear and compact operators on the generalised Hahn space

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space hd, where d is an unbounded monotone increasing sequence of positive real numbers, into the spaces w0, w and w? of sequences that are strongly summable to zero, strongly summable and strongly bounded by the Ces?ro method of order one. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from hd into w, and identities for the Hausdorff measure of noncompactness of bounded linear operators from hd to w0, and use these results to characterise the classes of compact operators from hd to w and w0. Finally, we provide an example for an application of our results.

Tripled Fixed Points and Existence Study to a Tripled Impulsive Fractional Differential System via Measures of Noncompactness

In this paper, a tripled fractional differential system is introduced as three associated impulsive equations. The existence investigation of the solution is based on contraction principle and measures of noncompactness in terms of tripled fixed point and modulus of continuity. Our results are valid for both Kuratowski and Hausdorff measures of noncompactness. As an application, we apply the obtained results to a control problem.

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

In the present paper, the concept of almost periodic waves is introduced to discontinuous impulsive fractional inclusions involving Caputo fractional derivative. New results on the existence and uniqueness are established by using the theory of operator semigroups, Hausdorff measure of noncompactness, fixed point theorems and fractional calculus techniques. Applications to a class of fractional-order impulsive gene regulatory network (GRN) models are proposed to illustrate the results.

Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

A new sequence space related to the space ell _{p}, 1leq p<infty (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.

Existence results for impulsive delayed neutral stochastic functional differential equations with noncompact semigroup

In this paper, we study the existence of mild solutions for impulsive neutral stochastic functional differential equations driven by a fractional Brownian motion with noncompact semigroup and varying-time delays in a Hilbert space. With the help of the Hausdorff measure of noncompactness, the Mönch fixed-point theorem and some inequality technique, some new criteria to guarantee the mild solution for impulsive neutral stochastic functional differential equations are obtained. Moreover, some previous results will be significantly improved in our paper.

Hausdorff measure of noncompactness of matrix mappings on Cesàro spaces

In this study we establish some identities or estimates for operator norms and the Hausdorff measure of noncompactness of certain operators on spaces |C_{α}|_{k}, which have more recently been introduced in [14]. Further, by applying the Hausdorff measure of noncompactness, we establish the necessary and sufficient conditions for such operators to be compact and so the some well known results are generalized.

Complete controllability for a class of fractional evolution equations with uncertainty

&lt;p style='text-indent:20px;'&gt;In this paper, we study the complete controllability for a class of fractional evolution equations with a common type of fuzzy uncertainty. By using Hausdorff measure of noncompactness and Krasnoselskii's fixed point theorem in complete semilinear metric space, we give some sufficient conditions of the controllability for the fuzzy fractional evolution equations without involving the compactness of strongly continuous semigroup and the perturbation function. In addition, the controllable problem is considered in a subspace of fuzzy numbers in which the gH-differences always exist, that guarantees the satisfaction of hypotheses of the problem. An application example related to electrical circuit is given to illustrate the effectiveness of theoretical results.&lt;/p&gt;