Measure Of Noncompactness Research Articles

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

In this paper, utilizing the technique of generalized Darbo's fixed‐point theorem associated with measure of noncompactness in Banach space, we analyze the existence of solution for a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator. The existing result was obtained to strengthen the ones mentioned previously in the literature. An example for a class of nonlinear functional integral equations is also presented to validate our main result. Finally, we propose the numerical method formed by the modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

On nullity and fullness of measures of noncompactness

On nullity and fullness of measures of noncompactness

An analysis of approximate controllability for Hilfer fractional delay differential equations of Sobolev type without uniqueness

This study focused on the approximate controllability results for the Hilfer fractional delay evolution equations of the Sobolev type without uniqueness. Initially, the Lipschitz condition is derived from the hypothesis, which is represented by a measure of noncompactness, in particular, nonlinearity. We also examined the continuity of the solution map of the Sobolev type of Hilfer fractional delay evolution equation and the topological structure of the solution set. Furthermore, we prove the approximate controllability of the fractional evolution equation of the Sobolev type with delay. Finally, we provided an example to illustrate the theoretical results.

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 $$b_{0}^{r,s,F}.$$

On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings

The purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation T(u,C(u))=u using Kannan-type equicontractive mappings, T:A×C(A)¯→Y, where C is a compact mapping, A is a bounded, closed and convex subset of a Banach space Y. To achieve this objective, the authors have presented Sadovskii’s theorem, which utilizes the measure of noncompactness. The relevance of the obtained results has been illustrated through the consideration of various initial value problems.

Measures of noncompactness in the study of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations

Measures of noncompactness in the study of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations

Existence results for anti-periodic fractional coupled systems with $$p-$$Laplacian operator via measure of noncompactness in Banach spaces

The aim of this paper is to investigate the existence of anti-periodic solutions for a nonlinear coupled system of $$\Psi -$$ Caputo fractional differential equations with $$p-$$ Laplacian operator in a Banach space. The proofs of our main results are based on Mönch’s fixed point theorem and some basic tools of $$\Psi -$$ Caputo fractional calculus paired with measures of noncompactness. As an application, our theoretical results have been illustrated by providing an interesting example.

SOLVABILITY OF QUADRATIC HADAMARD-TYPE FRACTIONAL INTEGRAL EQUATIONS IN ORLICZ SPACES

We demonstrate some properties of Hadamard fractional operators such as boundedness, monotonicity, continuity, and acting conditions in Orlicz spaces Lφ. We apply these properties with a proper measure of noncompactness (MNC) to inspect the existence of monotonic solutions of some general, but abstract, form of quadratic Hadamard-type fractional integral equations in Lφ. We also discuss the uniqueness of the solution.

New discussion on the existence and controllability of fractional evolution inclusion of order 1 < r < 2 without compactness

In this article, we investigate the existence and controllability results for fractional evolution inclusions in Banach spaces. By eliminating compactness hypotheses on the semigroup produced by the linear component and any requirements on the multivalued nonlinearity defined in terms of measures of noncompactness, we apply a new method to arrive at the existence of mild solutions and controllability results. In fact, to satisfy the existence and controllability requirements, we propose a fixed‐point strategy for weakly sequentially closed maps. Finally, a concrete example is provided to highlight our theoretical findings.

An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space

In the current contribution, an appropriate quantity connected to the space of all convergent sequences is provided and shown to be a measure of noncompactness in a Banach space. Through the application of the fixed point theorems of Darbo and Meir–Keeler, this amount is used to discuss whether a solution to an infinite system of fractional Sturm–Liouville operators exists. We offer a numerical example as an application of the key finding in the study.

Remarks on incorrect measure of noncompactness in $\operatorname{BC}(\mathbb{R}_+\times\mathbb{R}_+)$

In this paper, we indicate an error occurring in a few articles. This error concerns a certain measure of noncompactness in the space \operatorname{BC}(\mathbb{R}_+\times\mathbb{R}_+) . We show how this error could be corrected. Moreover, we propose a more convenient approach using the more general Fréchet space C(\mathbb{R}_+\times\mathbb{R}_+) in which we indicate a simpler measure of noncompactness.

Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of the controllability of mild solutions, it is usual practice to apply Krasnoselskii’s theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness. A fractional Caputo approach of order between 1 and 2 was used to construct our model. The families of linear operators cosine and sine, which are strongly continuous and uniformly bounded, are used to achieve the mild solution. To make our results seem to be applicable, a numerical example is provided.

Qualitative Behaviour of Stochastic Integro-differential Equations with Random Impulses

In this paper, we study the existence and some stability results of mild solutions for random impulsive stochastic integro-differential equations (RISIDEs) with noncompact semigroups in Hilbert spaces via resolvent operators. Initially, we prove the existence of mild solution for the system is established by using Mönch fixed point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results includes continuous dependence of solutions on initial conditions, exponential stability and Hyers–Ulam stability for the aforementioned system are investigated. Finally, an example is proposed to validate the obtained results.

On solutions of a hybrid generalized Caputo-type problem via the noncompactness measure in the generalized version of Darbo’s criterion

In this manuscript, we study the existence and uniqueness of solutions for a new neutral hybrid nonlinear differential equation in the context of a fractional generalized operator in the sense of ψ-Caputo. To emphasize the novelty of the manuscript, a pure technique of the noncompactness measures is applied to a hybrid system based on the notion of the modulus of continuity in Darbo’s criterion that covers the existing results of other works published before. The Ulam–Hyers and generalized Ulam–Hyers stabilities are explored for the given neutral nonhybrid nonlinear problem. An application is prepared in the framework of an example to ensure the validity of theorems for different cases.

Analytical Contribution to a Cubic Functional Integral Equation with Feedback Control on the Real Half Axis

Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects. In this work, we present a study of a cubic functional integral equation with a general and new constraint that may help in investigating some real problems. We discuss the existence of solutions for an equation that involves a control variable in the class of bounded continuous functions BC(R+), by applying the technique of measure of noncompactness on R+. Furthermore, we establish sufficient conditions for the continuous dependence of the state function on the control variable. Finally, some remarks and discussion are presented to demonstrate our results.

Mild solutions of semilinear evolution equation and their applications in second‐order hyperbolic PDE

The principal goal of this work is to solvability of the mild solution for the second‐order hyperbolic PDE with initial/boundary‐value problem with nonlocal condition of the form where is open. Our analysis relies on the technique of measure of noncompactness.

Asymptotic stability of nonlinear neutral delay integro-differential equations

Abstract In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.

Approximate controllability of stochastic delay differential systems driven by Poisson jumps with instantaneous and noninstantaneous impulses

AbstractControl systems in which instantaneous and noninstantaneous impulses occur simultaneously are difficult to handle. In this article, we investigate the solvability and approximate controllability for a new category of stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses. Utilizing the theory of fundamental solution, stochastic analysis, the measure of noncompactness, and the fixed‐point approach, we establish the presence of a mild solution for the proposed system. We have also constructed a new set of sufficient constraints that assures approximate controllability of the considered system. Next, we discuss the existence of a solution and approximate controllability for an impulsive deterministic control system in which the nonlinear term contains spatial derivatives. Lastly, two examples are presented to encapsulate the abstract results.

Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness

Abstract In this article, some new generalizations of Darbo’s fixed-point theorem are given and the solvability of an infinite system of weighted fractional integral equations of a function with respect to another function is studied. Also, with the help of a proper example, we illustrate our findings.

Weak Solution for a Fractional Langevin Inclusion with the Katugampola–Caputo Fractional Derivative

In this work, we examine the existence of weak solution for a class of boundary value problems involving fractional Langevin inclusion with the Katugampola–Caputo fractional derivative under specified conditions contain the Pettis integrability assumption. The Mönch fixed point theorem is used with the weak noncompactness measure approach to investigate the existence results. In order to illustrate our results, we present an example.