Darbo Fixed Point Theorem Research Articles

Abstract random differential equations with state-dependent delay using measures of noncompactness

This paper is devoted to the existence of random mild solutions for a general class of second-order abstract random differential equations with state-dependent delay. The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. An application related to partial random differential equations with state-dependent delay is presented.

Analysis of nonlinear implicit fractional differential equations with the Atangana-Baleanu derivative via measure of non-compactness

<p>In this study, we proved existence results for nonlinear implicit fractional differential equations with the Caputo version of the Atangana-Baleanu derivative, subject to the boundary and nonlocal initial conditions. The Kuratowski's measure of non-compactness and its associated fixed point theorems–Darbo's fixed point theorem and Mönchh's fixed point theorem, are the foundation for the analysis in this paper. We support our results with examples of nonlinear implicit fractional differential equations involving the Caputo version of the Atangana-Baleanu derivative subject to both boundary and nonlocal initial conditions. In addition, we provide solutions to the problems we considered.</p>

Solvability of product of $ n $-quadratic Hadamard-type fractional integral equations in Orlicz spaces

<abstract><p>The current study demonstrated and studied the existence of monotonic solutions, as well as the uniqueness of the solutions for a general and abstract form of a product of $ n $-quadratic fractional integral equations of Hadamard-type in Orlicz spaces $ L_\varphi $. We utilized the analysis of the measure of non-compactness associated with Darbo's fixed-point theorem and fractional calculus to obtain the results.</p></abstract>

Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases

<abstract><p>This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.</p></abstract>

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

Random differential hyperbolic equations of fractional order in Fréchet spaces

Abstract In the present paper, we provide some existence results for the Darboux problem of partial fractional random differential equations in Fréchet spaces with an application of a generalization of the classical Darbo fixed point theorem and the concept of measure of noncompactness.

Existence and controllability of integrodifferential equations with non-instantaneous impulses in Fréchet spaces

In this paper, we investigate existence of mild solutions to a non-instantaneous integrodifferential equation via resolvent operators in the sense of Grimmer in Fréchet spaces. Utilizing the technique of measures of noncompactness in conjunction with the Darbo's fixed point theorem, we present sufficient criteria ensuring the controllability of the given problem. An illustrative example is also discussed.

Solvability of the product of n-integral equations in Orlicz spaces

We study the existence of a.e. monotonic Lφ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\varphi $$\\end{document}-solutions for the product of n-quadratic integral equations. As indicated by the different continuity properties of the considered operators in Orlicz spaces, we study three different cases in which the generating N-functions satisfy the conditions Δ′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta '$$\\end{document}, Δ2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _2,$$\\end{document} and Δ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _3$$\\end{document}. The method adopted in this paper consists of an appropriate application of some measure of noncompactness and the Darbo fixed point theorem for solving operators acting on the product of n-Orlicz spaces.

Existence of mild solutions of second-order impulsive differential equations in Banach spaces

We discuss the existence of solutions for second-order impulsive differential equation with nonlocal conditions in Banach spaces. Our approach is based on the generalization of Schauder fixed pointprinciple that is Darbo fixed point theorem. An example is also presented to for illustration.

A System of High-Order Fractional Differential Equations with Integral Boundary Conditions

The existence of a solution for a system of two nonlinear high-order fractional differential equations including the Atangana-Baleanu-Caputo derivative with integral boundary conditions, is proved. Simultaneously, we discuss the existence of a solution by applying the Schauder fixed point theorem and a generalized Darbo fixed point theorem, which involves the concept of measure of noncompactness. The paper also contains some examples that illustrate the application of the main result.

Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space

<abstract><p>In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e.,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{fredholm} w(\zeta_1) = y(\zeta_1)+w(\zeta_1)\int_{\bf J}\kappa(\zeta_1, \varphi)(T_1w)(\varphi)d\varphi+z(\zeta_1)\int_{\bf J}h(\varphi, (T_2w)(\varphi))d\varphi, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>for $ \zeta_1\in{\bf J} = [0, 1], $ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.</p></abstract>

Discontinuous solutions of delay fractional integral equation via measures of noncompactness

<abstract><p>This article considers the existence and the uniqueness of monotonic solutions of a delay functional integral equation of fractional order in the weighted Lebesgue space $ L_1^N({\mathbb{R}}^+) $. Our analysis uses a suitable measure of noncompactness, a modified version of Darbo's fixed point theorem, and fractional calculus in the mentioned space. An illustrated example to show the applicability and significance of our outcomes is included.</p></abstract>

On the measure of noncompactness in $L_p(\mathbb{R}^+)$ and applications to a product of $n$-integral equations

In this article, we prove a new compactness criterion in the Lebesgue spaces $L_p({\mathbb{R}}^+), 1 \leq p < \infty$ and use such criteria to construct a measure of noncompactness in the mentioned spaces. The conjunction of that measure with the Hausdroff measure of noncompactness is proved on sets that are compact in finite measure. We apply such measure with a modified version of Darbo fixed point theorem in proving the existence of monotonic integrable solutions for a product of $n$-Hammerstein integral equations $n\geq 2$.

Existence of Atangana-Baleanu fractional neutral Volterra integro-differential equations with non-instantaneous impulses

The main focus of this manuscript is centered around Atangana-Baleanu semilinear fractional integrodifferential equations accompanied by non-instantaneous impulses. The outcomes are demonstrated using the Darbo fixed point theorem along with its results when the measure of non-compactness collaborates. Eventually, a demonstration example is proposed.

Existence of a bounded variation solution of a nonlinear integral equation in L1(R+)

In this paper we study the existence of a unique solution of a nonlinear integral equation in the space of bounded variation on an unbounded interval by using measure of noncompactness and Darbo fixed point theorem.

Existence of a solution in the Holder space for a nonlinear functional integral equation

This article is entirely devoted to the application of the measure of noncompactness defined in the Holder space. Here the emphasis is on the study of the nonlinear functional integral equation with changed arguments. Precisely, the existence of a solution is obtained by employing the Darbo fixed point theorem under certain hypotheses. Finally, we provide a tangible example which supports our results.

Existence of an infinite system of fractional hybrid differential equations in a tempered sequence space

In this article, the existence of the solution of infinite system of fractional hybrid differential equations with initial conditions by reducing them to integral equations in the space \(C(I, m(\theta )^{\beta })\) is discussed based on Darbo fixed point theorem. In the end, a suitable example to strengthen the article 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.

The measure of noncompactness in a generalized coupled fixed point theorem and its application to an integro-differential system

This paper discusses the existence of a solution for the system of integro-differential equations. By using a certain measure of noncompactness through the generalized Darbo fixed point theorem, we deduce conditions that guarantee the existence of a solution for the system. The advantage of the Darbo fixed point theorem is it helps us to impose weaker conditions in equations, because the contraction condition reduces to continuous and self-mapping conditions. To illustrate our results, we provide multiple examples and solve them by using the artificial small parameter method, with tables and figures that demonstrate the exactitude of the results.

An existence result involving both the generalized proportional Riemann-Liouville and Hadamard fractional integral equations through generalized Darbo's fixed point theorem

&lt;abstract&gt;&lt;p&gt;In this paper, we propose and prove an extension and generalization, which extends and generalizes the Darbo's fixed point theorem (DFPT) in the context of measure of noncompactness (MNC). Thereafter, we use DFPT to investigate the existence of solutions to mixed-type fractional integral equations (FIE), which include both the generalized proportional $ (\kappa, \tau) $-Riemann-Liouville and Hadamard fractional integral equations. We've included a suitable example to strengthen the article.&lt;/p&gt;&lt;/abstract&gt;