Measures of noncompactness and some applications

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space [Formula: see text] using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space [Formula: see text] are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to [Formula: see text]. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with [Formula: see text] dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

In this paper, we prove some common fixed point theorems for various types of compatible mappings in the setting of complete multiplicative metric spaces.

This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii's fixed point theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions, and mixed-type conditions, respectively.

The aim of this work is to study the existence of solutions to a class of fractional differential equations with a mixed of fractional integral boundary conditions involving the Hilfer fractional derivative. The proof is based on Monch's fixed point theorem and the technique of measures of noncompactness. Two examples illustrating the main results are also constructed.

In the present paper, we prove a fixed point theorem in fuzzy metric spaces through weak Compatibility.

Under the idea of a measure of noncompactness, some fixed point results are proposed and a generalization of Darbo’s fixed point theorem is given in this manuscript. Furthermore, some novel quadruple fixed points results via a measure of noncompactness for a general class of functions are presented. Ultimately, the solutions to a system of non-linear functional integral equations by the fixed point results obtained are discussed, and non-trivial examples to illustrate the validity of our study are derived.

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

Banach’s contraction mapping principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis with special applications to the theory of differential and integral equations. Because the underlined space of this theorem is a metric space, the theory that developed following its publication is known as the metric fixed point theory. Over the last one hundred years, many people have tried to generalize the definition of a metric space. In this paper, we survey the most popular generalizations and we discuss the recent uptick in some generalizations and their impact in metric fixed point theory.

Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called “equivalent” if there exist positive constants b and c such that bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets \({S\subset X}\). If such constants do not exist, the measures of noncompactness are “inequivalent.” We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (Ω, Σ, μ) is a general measure space, 1 ≤ p ≤ ∞, and X = Lp(Ω, Σ, μ); or if K is a compact Hausdorff space and X = C(K); or if K is a compact metric space, 0 < λ ≤ 1, and X = C0,λ(K), the Banach space of Holder continuous functions with Holder exponent λ. We also prove the existence of such inequivalent measures of noncompactness if Ω is an open subset of \({\mathbb{R}^n}\) and X is the Sobolev space Wm,p(Ω). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f : C→C and from the closely related issue of giving the proper definition of the “cone essential spectral radius” of such maps. These questions are considered in the companion paper [28]; see, also, [27].

‎We propose to investigate the solutions of system of functional-integral‎ ‎equations in the setting of measure of noncompactness on real-valued bounded and continuous Banach space‎. To achieve this‎, ‎we first establish some new Darbo type fixed and coupled fixed point results for‎ $mu$-set $(omega,vartheta)$-contraction operator‎ ‎using arbitrary measure of noncompactness in Banach spaces‎. An example is given in support for the solutions of a pair of system of functional-integral‎ equations‎.

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra and geometry as well as discrete and computational mathematics. This interdisciplinary theory par excellence provides insight and powerful tools for the solvability aspects of central problems in many areas of current interest in mathematics where topological considerations play a crucial role. Indeed, existence for linear and nonlinear problems is commonly translated into fixed point problems; for example, the existence of solutions to elliptic partial differential equations, the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory.Themetric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. However, the Polish mathematician Stefan Banach is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications in industrial fields such as image processing engineering, physics, computer science, economics, and telecommunications. A particular interest in fixed points for set-valued operators developed towards the mid-20th century with the celebrated extensions of the Brouwer and Lefschetz theorems by Kakutani and Eilenberg-Montgomery, respectively. The Banach contraction principle was later on extended to multivalued contractions by Nadler. The fixed point theory for multivalued maps found numerous applications in control theory, convex and nonsmooth optimization, differential inclusions, and economics. The theory is also used prominently in denotational semantics (e.g., to give meaning to recursive programs). In fact, it is still too early to truly estimate the importance and impact of set-valued fixed point theorems in mathematics in general as the theory is still growing and finding renewed outlets. This special issue adds to the development of fixed point theory by focusing on most recent contributions. It includes works on nonexpansive mappings in Banach and metric spaces, multivalued mappings in Banach and metric spaces, monotone mappings in ordered spaces, multivalued mappings in ordered spaces, and applications to such nonmetric spaces as modular spaces, as well as applications to logic programming and directed graphs.

Monotonic solutions of Urysohn integralequation on unbounded interval

In this paper, we prove the local attractivity of solutions for a certain nonlinear Volterra type functional integral equations. We rely on a measure theoretic fixed point theorem of Dhage (2008) for nonlinear D-set-contraction in Banach spaces. Finally, we furnish an example to validate all the hypotheses of our main result and to ensure the existence and ultimate attractivity of solutions for a numerical nonlinear functional integral equation.

In this paper, we prove some common fixed point theorems in 0-complete partial metric spaces. Our results extend and generalize many existing results in the literature. Some examples are included which show that the generalization is proper.