New Measure Of Noncompactness Research Articles

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.

Existence of solution of functional Volterra-Fredholm integral equations in space L∞(ℝ+) and sinc interpolation to find solution

A new measure of noncompactness (Mnc) and Darbo fixed point theorem are utilized on the space L∞(ℝ+) to prove the existence of solution for functional Volterra–Fredholm integral equations. An example is given to confirm the validity of results. Furthermore, we propound an iterative algorithm by sinc interpolation to find the solution with an acceptable accuracy. In this algorithm, it is not necessary for the problem to be discretized to an algebraic system with unknown coefficients. We also use an iterative process to approximate a solution with exponential convergence.

Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations

First, we introduce a new measure of noncompactness in weighted Sobolev spaces $$W_{w}^{m,p}(\Omega )$$ , and then as an application, we study the existence of solutions for a class of the nonlinear convolution type integral equations using Darbo’s fixed point theorem associated with this new measure of noncompactness. Further, an example is presented to verify the effectiveness and applicability of our main results.

Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval

In this paper, we formulate necessary and sufficient conditions for relative compactness in the space BG({mathbb {R}}_+,E) of regulated and bounded functions defined on {mathbb R}_+ with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space BG({mathbb {R}}_+,E). We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps BG({mathbb {R}}_+,E) into BG({mathbb {R}}_+,E) and, additionally, be compact.

Weakly Asymptotic Stability for Fractional Delay Differential Mixed Variational Inequalities

In this paper, we introduce and study a new class of fractional delay differential mixed variational inequalities (FDDMVI, for short) formulated by a fractional delay evolution inclusion and a mixed variational inequality in infinite Banach spaces. By applying a new measure of noncompactness and fixed point theorem for a condensing set-valued map, we obtain the global solvability of a FDDMVI on the half-line. Furthermore, we apply the obtained results to establish a weakly asymptotic stability result of the zero mild solution to a FDDMVI. Finally, an example is given to demonstrate the main results.

Solvability for an infinite system of fractional order boundary value problems

We investigate the solvability for an infinite system of fractional order boundary value problems of differential equations in Banach sequence spaces l∞ and c. Our approach depends on Darbo’s fixed point theorem in conjunction with new measures of noncompactness in spaces C1(J,l∞) and C1(J,c).

On a measure of noncompactness in the Holder space Ck,γ(Ω) and its application

In this article, based on the concept of relative compactness in the Holder space Ck,γ(Ω) where k≥0, γ∈(0,1] and Ω is a compact subset of RN, we present a new measure of noncompactness on Ck,γ(Ω). Moreover, as an application we study the existence of solution of a nonlinear integral equation in the Holder space by applying Darbo fixed point theorem.

Existence of solutions for some classes of integro-differential equations in the Sobolev space $$W^{n,p}(\mathbb {R}_+)$$ W n , p ( R + )

In this paper, first, we introduce a new measure of noncompactness in the Sobolev space \(W^{n,p}(\mathbb {R}_+)\) and then, as an application, we study the existence of solutions for a class of the functional integral–differential equations using Darbo’s fixed point theorem associated with this new measure of noncompactness.

Construction of a measure of noncompactness in Sobolev spaces with an application to functional integral-differential equations

In this paper, first we introduce a measure of noncompactness in the Sobolev space W^{k,1}(Omega ) and then, as an application, we study the existence of solutions for a class of the functional integral-differential equations using Darbo’s fixed point theorem associated with this new measure of noncompactness. Further, two examples are presented to verify the effectiveness and applicability of our main results.

The behaviour of measures of noncompactness in $$L^\infty ({\mathbb {R}}^n)$$ L ∞ ( R n ) with application to the solvability of functional integral equations

In this work, we define a new measure of noncompactness on the space \(L^\infty ({\mathbb {R}}^n)\). In addition, we study the existence of solutions for a class of nonlinear functional integral equations of Fredholm type by using an extension of Darbo’s fixed point theorem associated with this new measure of noncompactness. Also, we will include an example which shows the efficiency of our results. The obtained results improve several ones obtained earlier.

MEASURES OF NONCOMPACTNESS IN A SOBOLEV SPACE AND INTEGRO-DIFFERENTIAL EQUATIONS

The aim of this paper is to introduce a new measure of noncompactness on the Sobolev space$W^{n,p}[0,T]$. As an application, we investigate the existence of solutions for some classes of functional integro-differential equations in this space using Darbo’s fixed point theorem.

Construction of measures of noncompactness of DC n [ J , E ] $\mathit{DC}^{n}[J,E]$ and C 0 n [ J , E ] $C^{n}_{0}[J,E]$ with application to the solvability of nth-order integro-differential equations in Banach spaces

In the present paper, we first investigate the construction of compact sets of $\mathit{DC}^{n}[J,E]$ and $C^{n}_{0}[J,E]$ , and then we introduce new measures of noncompactness on these spaces. In addition, as an application, we discuss the existence of solutions of initial value problems for nth-order nonlinear integro-differential equations of mixed type on an infinite interval in Banach spaces. We will also state an interesting example which shows that our results can apply for solving infinite systems of integro-differential equations.

Existence of solutions of functional integral equations of convolution type using a new construction of a measure of noncompactness on [formula omitted

Let Lp(R+) denote the space of Lebesgue integrable functions on R+ with the standard norm∥x∥p=(∫0∞|x(t)|pdt)1p.First, we define a new measure of noncompactness on the spaces Lp(R+) (1 ≤ p < ∞). In addition, we study the existence of entire solutions for a class of nonlinear functional integral equations of convolution type using Darbo’s fixed point theorem, which is associated with the new measure of noncompactness. We provide some examples to demonstrate that our results are applicable whereas the previous results are not.

Construction of a Measure of Noncompactness on $${BC(\Omega)}$$ B C ( Ω ) and its Application to Volterra Integral Equations

In the present paper, we investigate the construction of compact sets of bounded continuous functions on unbounded set \({\Omega}\) of \({\mathbb{R}^n}\), and then introduce a measure of noncompactness on this space. As an application, we study the existence of solutions for a class of nonlinear functional integral equations of Volterra type by using some fixed point theorems associated with this new measure of noncompactness. Obtained results generalize numerous other ones. We will also include some examples which show that our results are applicable where the previous ones are not.

Existence of solutions for some classes of integro-differential equations via measure of noncompactness

In this present paper, we introduce a new measure of noncompactness on the space consisting of all real functions which are n times bounded and continuously differentiable on R+. As an application, we investigate the problem of the existence of solutions for some classes of the functional integral-differential equations which enables us to study the existence of solutions of nonlinear integro-differential equations. In our considerations we apply the technique of measures of noncompactness in conjunction with Darbo's fixed point theorem. Finally, we give some illustrative examples to verify the effectiveness and applicability of our results.

Generalized quasi-Banach sequence spaces and measures of noncompactness

Given 0 &lt; s ≤ 1 and ψ an s-convex function, s – ψ -sequence spaces are introduced. Several quasi-Banach sequence spaces are thus characterized as a particular case of s – ψ -spaces. For these spaces, new measures of noncompactness are also defined, related to the Hausdorff measure of noncompactness. As an application, compact sets in s – ψ -interpolation spaces of a quasi-Banach couple are studied.

Generalized measures of noncompactness of sets and operators in Banach spaces

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.

Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness

Using the techniques of some new measures of noncompactness we prove in this paper some existence theorems concerning the global attractivity and ultimate positivity of the solu- tions for a nonlinear functional integral equation. Our investigations are placed in the Banach space of real-valued functions defined, continuous and bounded on unbounded intervals together with the applications of a recent measure theoretic fixed point theorem of Dhage (7). On one hand, our results generalize the attractivity results of Dhage (9) with a different method and the results of Banas and Rzepka (4) and Banas and Dhage (5) with similar method but under weaker conditions and on the other hand they are new to the literature as regards ultimate positivity of the solutions for nonlinear functional integral equations. A few realizations of the obtained results are also indicated.