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This paper aims to investigate the existence of solution of a generalized Hadamard proportional fractional integral equations involving non-singular kernels. First, we introduce Hadamard fractional integral equations. Then, we demonstrate the existence of the solution of Hadamard proportional fractional integral equations based on the concept of a measure of noncompactness and Petryshyn's fixed point theorem. After that, we provide suitable examples which evidence the existing result. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the existence and stability of a solution to the time fractional parameters Sc,Ec,Ic,Ac,Tc,R$$ \left({S}_c,{E}_c,{I}_c,{A}_c,{T}_c,R\right) $$ of Atangana–Baleanu fractional ordered 2019‐nCoV model. For the existence of the solution, we derive Darbo‐type fixed‐point theorems and its corollaries for (α,β)$$ \left(\alpha, \beta \right) $$‐admissible and ψ$$ \psi $$‐condensing operator in the setting of modular space. Moreover, the stability analysis and behavior of solution are analyzed by the numerical technique governed by Toufik and Atangana [1]. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p -proportional ω -weighted κ -Hilfer derivative includes an exponential function, D a , λ σ , ρ , p , κ , ω , and then we consider a non-instantaneous impulse differential inclusion containing D a , λ σ , ρ , p , κ , ω with order σ ∈ (1 , 2) and of kind ρ ∈ [ 0 , 1 ] in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the (p , ω , κ) -generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the (p , ω , κ) -generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator D a , λ σ , ρ , p , κ , ω , our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigated Caputo fractional integro-differential equations with non-instantaneous impulses and nonlocal conditions. By employing the solution operator, the Mönch fixed point theorem, and the stepwise estimation method, we eliminated the Lipschitz condition of the nonlinear term, while also dispensing with the requirement for the compressibility coefficient condition $ 0 < k < 1 $. The main results presented represented a generalization and enhancement of previous findings. Furthermore, an example was provided to verify the application of our main results.

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, Φ -Hilfer, fractional derivative of order μ ∈ (1 , 2) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted Φ -Laplace transform, w-weighted ψ -convolution and the measure of non-compactness. Since the operator, the w-weighted Φ -Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider the fractional delayed evolution equation of order γ ∈ (1 , 2) in ordered Banach space. In the absence of assumptions about the compactness of cosine families or related sine families, the existence results of positive solutions are studied by using some fixed point theorems and monotone iterative method under the conditions that nonlinear function satisfies the non-compactness measure conditions and some appropriate growth conditions or order conditions. [ABSTRACT FROM AUTHOR]

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ -Hilfer fractional derivative, D 0 , t σ , v , ψ , w , of order μ ∈ (1 , 2) , in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D 0 , t σ , v , ψ , w of order μ   ∈ (1 , 2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ -Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D 0 , t σ , v , ψ , w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one function with respect to another variable order function. It is essential to keep in mind that the symmetry of a transformation for differential equations is connected to local solvability, which is synonymous with the existence of solutions. As a consequence, existence requirements for weighted fractional derivative of a function with respect to another function of constant order are necessary. Moreover, the stability with in Ulam–Hyers–Rassias sense is reviewed. The outcomes are derived using the Kuratowski measure of non-compactness. A model illustrates the trustworthiness of the observed results.

In this work, we show the existence of solutions for a higher order implicit fractional differential equations with multi-point boundary conditions at resonance. Our results are obtained by utilizing the (Kuratovski) measure of non-compactness and abstract continuation theorem for k-set contractions. Finally, an example is given to illustrate the applicability of theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy type problem in Banach spaces for various classes of functional inclusions with causal multioperators. Based on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. As an application, we obtain generalizations of existence theorems for a Cauchy type problem for semilinear second order differential inclusions of this type and semilinear differential inclusions of the fractional order 1 < q < 2. [ABSTRACT FROM AUTHOR]

We study the long time behavior of solutions for time-fractional pseudo-parabolic equations involving time-varying delays and nonlinear pertubations, where the nonlinear term is allowed to have superlinear growth. Concerning the associated linear problem, we establish a variation-of-parameters formula of mild solutions and prove some regularity estimates of resolvent operators. In addition, thanks to local estimates on Hilbert scales, fixed point arguments and a new Halanay type inequality, we obtain some results on the global solvability, stability, dissipativity and the existence of decay solutions to our problem. [ABSTRACT FROM AUTHOR]

A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new $ \psi $-Caputo operator. The novelty of this work, in addition to the $ \psi $-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the $ \psi(\kappa) $ function.

We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of non-negative matrices that define operators on Banach sequences, we obtain additional results. Our results extend the results of several authors that appeared relatively recently. [ABSTRACT FROM AUTHOR]

In this work, we define a new notion of μ F -contraction in Banach spaces, by modeling with measure of non-compactness the well-known concept of F -contraction given by Wardowski in metric spaces. We prove a fixed-point theorem and a common fixed-point theorem for our contraction. Our results extend the well-known Darbo contraction principle and large contractions in the literature. Furthermore, the paper contains examples and applications that show the validity of our results. [ABSTRACT FROM AUTHOR]

This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one function with respect to another variable order function. It is essential to keep in mind that the symmetry of a transformation for differential equations is connected to local solvability, which is synonymous with the existence of solutions. As a consequence, existence requirements for weighted fractional derivative of a function with respect to another function of constant order are necessary. Moreover, the stability with in Ulam–Hyers–Rassias sense is reviewed. The outcomes are derived using the Kuratowski measure of non-compactness. A model illustrates the trustworthiness of the observed results. [ABSTRACT FROM AUTHOR]

In this article a novel concept of C*-algebra-valued measure of non-compactness is defined. Using this concept, the well known fixed point theorems of Darbo, Sadovskii and Krasnoslskii are generalized. We present non-trivial examples and applications to validate the real utilization of our results. [ABSTRACT FROM AUTHOR]

In this manuscript, we examine the existence and the stability of solutions to the boundary value problem of Riemann-Liouville fractional differential equations of variable order. The obtained new results are based on the fixed point theorem of Darbo and Kuratowski's metric of noncompactness (MNK) with the help of piece-wise constant functions. In addition, the derived fundamental results are proven suitable because they satisfy the Ulam-Hyers Rassias stability sufficient conditions. Several numerical examples were discussed too to demonstrate the reasonableness and effectiveness of the observed results. [ABSTRACT FROM AUTHOR]

A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new ψ-Caputo operator. The novelty of this work, in addition to the ψ-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the ψ(1) function. [ABSTRACT FROM AUTHOR]

In this research article, we deal with the global convergence of successive approximations (s.a) as well as the existence of solutions to a fractional $ {(p, q)} $-difference equation. Then, we discuss the existence result of the solutions of Caputo-type $ {(p, q)} $-difference fractional vector-order equations in a Banach space. Also, we prove a theorem on the global convergence of successive approximations to the unique solution of our problem. Finally, the application of the main results is demonstrated by presenting numerical examples.

In this paper, we study the existence and stability of solutions for impulsive pantograph fractional integro-differential equation via ψ-Hilfer fractional derivative in a appropriate Banach space. Our approach is based on fixed point theorems of Darbo’s and Mönch via Kuratowski measure of non-compactness. An example is given to illustrate our approach. [ABSTRACT FROM AUTHOR]

In this research article, we deal with the global convergence of successive approximations (s.a) as well as the existence of solutions to a fractional (p , q) -difference equation. Then, we discuss the existence result of the solutions of Caputo-type (p , q) -difference fractional vector-order equations in a Banach space. Also, we prove a theorem on the global convergence of successive approximations to the unique solution of our problem. Finally, the application of the main results is demonstrated by presenting numerical examples. [ABSTRACT FROM AUTHOR]

In the existing article, the existence of solutions to nonlinear fractional differential inclusions in the sense of the Atangana-Baleanu-Caputo (ABC) fractional derivatives in Banach space is studied. The investigation of the main results relies on the set-valued issue of Mönch fixed point theorem incorporated with the Kuratowski measure of non-compactness. A simulated example is proposed to explain the obtained results.

This manuscript is related to deriving some necessary and appropriate conditions for qualitative results about a class of Sturm-Liouville (S-L) boundary value problems (BVPs) with the p -Laplacian operator under a fractional ϑ -Caputo type derivative. For the required results, we use Mönch's fixed point theorem with a measuring of non-compactness. Here, it is important to mention that the aforesaid equations belong to a highly significant class of problems that have many of the same properties and applications to solving various problems of dynamics and wave equations theory. For the demonstration of our theoretical results, we provide an example.

The goal of this manuscript is to study the existence theory of solution for a nonlinear boundary value problem of tripled system of fractional order hybrid sequential integro-differential equations. The analysis depends on some results from fractional calculus and fixed point theory. As a result, we generalized Darbo's fixed point theorem to form an updated version of tripled fixed point theorem to investigate the proposed system. Also, Hyres-Ulam and generalized Hyres-Ulam stabilities results are established for the considered system. For the illustration of our main results, we provide an example.

This work is dedicated to the study of a second order evolutionary partial differential variational-like inequality in Banach space. We obtain the mild solution of our problem by applying the concept of strongly continuous cosine family of bounded linear operators, fixed point theorem for condensing multi-valued operators and measure of non-compactness. It is proved that the solution set of mixed variational-like inequalities is non-empty, bounded, closed and convex.

In this paper we make an attempt to prove common fixed point theorem for two and three mappings in Banach algebra by using measure of non-compactness. In the second section of the paper we prove the common fixed point theorem for two continuous linear operators on Banach algebra and also an effort is made to prove common fixed point theorem for three continuous commuting operators in Banach algebra. [ABSTRACT FROM AUTHOR]

In this work, we concentrate on the existence of the solutions set of the following problem c D q α σ (t) ∈ F (t , σ (t) , c D q α σ (t)) , t ∈ I = [ 0 , T ] σ 0 = σ 0 ∈ E , as well as its topological structure in Banach space E. By transforming the problem posed into a fixed point problem, we provide the necessary conditions for the existence and compactness of solutions set. Finally, we present an example as an illustration of main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional ( κ , ϕ )-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space C ([ 1 , T ]) arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo's fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations. [ABSTRACT FROM AUTHOR]

We study the existence of decay solutions for a class of abstract fractional mobile-immobile equations involving impulses and nonlinear perturbations in Hilbert spaces, where the nonlinear terms are supposed to be of superlinear growth. Concerning the associated linear problem, we make use of the theory of completely positive functions to establish a variation of parameters formula in terms of the relaxation functions and to prove the differentiability of resolvent families. In addition, based on local estimates on Hilbert scales and fixed point arguments, we obtain some results on the global existence and the existence of a compact set of decay solutions to our problem. [ABSTRACT FROM AUTHOR]

In the existing article, the existence of solutions to nonlinear fractional differential inclusions in the sense of the Atangana-Baleanu-Caputo (ABC) fractional derivatives in Banach space is studied. The investigation of the main results relies on the set-valued issue of Monch fixed point theorem incorporated with the Kuratowski measure of non-compactness. A simulated example is proposed to explain the obtained results. [ABSTRACT FROM AUTHOR]

The goal of this manuscript is to study the existence theory of solution for a nonlinear boundary value problem of tripled system of fractional order hybrid sequential integro-differential equations. The analysis depends on some results from fractional calculus and fixed point theory. As a result, we generalized Darbo's fixed point theorem to form an updated version of tripled fixed point theorem to investigate the proposed system. Also, Hyres-Ulam and generalized Hyres-Ulam stabilities results are established for the considered system. For the illustration of our main results, we provide an example. [ABSTRACT FROM AUTHOR]

This manuscript is related to deriving some necessary and appropriate conditions for qualitative results about a class of Sturm-Liouville (S-L) boundary value problems (BVPs) with the p-Laplacian operator under a fractional ϑ-Caputo type derivative. For the required results, we use Mönch's fixed point theorem with a measuring of non-compactness. Here, it is important to mention that the aforesaid equations belong to a highly significant class of problems that have many of the same properties and applications to solving various problems of dynamics and wave equations theory. For the demonstration of our theoretical results, we provide an example. [ABSTRACT FROM AUTHOR]

We propose a new concept of condensing operators by using a notion of measure of non-compactness in the setting of Banach spaces and establish a new generalization of Darbo's fixed point theorem. We also show the applicability of our results to integral equations. A concrete example will be presented to support the application part. [ABSTRACT FROM AUTHOR]

This work is dedicated to the study of a second order evolutionary partial differential variational-like inequality in Banach space. We obtain the mild solution of our problem by applying the concept of strongly continuous cosine family of bounded linear operators, fixed point theorem for condensing multi-valued operators and measure of non-compactness. It is proved that the solution set of mixed variational-like inequalities is non-empty, bounded, closed and convex. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions are investigated in this study. The fractional integro-differential equations contain a finite number of Riemann–Liouville fractional integral and non-integral type nonlinearities, as well as Caputo differential operators of various orders subject to fractional boundary conditions on an infinite interval. At the boundary conditions, we use sub-strip and multi-point contribution. There are various techniques to solve such type of differential equations and one of the most common is known as symmetry analysis. The symmetry analysis has widely been used in problems involving differential equations, although determining the symmetries can be computationally intensive compared to other methods. Therefore, we employ the degree theory due to the Mawhin involving measure of a non-compactness technique to arrive at our desired findings. An interesting pertinent problem has also been provided to demonstrate the applicability of our results. [ABSTRACT FROM AUTHOR]

Let X be a rearrangement-invariant Banach function space on the unit circle \mathbb {T} and let H[X] be the abstract Hardy space built upon X. We prove that if the Cauchy singular integral operator (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau)}{\tau -t}\,d\tau is bounded on the space X, then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator aI+bH with a,b\in \mathbb {C}, acting on the space X, coincide. We also show that similar equalities hold for the backward shift operator (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H[X]. Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss fixed point theorems for a new χ-set contraction condition in partially ordered Banach spaces, whose positive cone K $\mathbb{K}$ is normal, and then proceed to prove some coupled fixed point theorems in partially ordered Banach spaces. We relax the conditions of a proper domain of an underlying operator for partially ordered Banach spaces. Furthermore, we discuss an application to the existence of a local fractional integral equation.

In this paper, at first, we introduce $alpha_{mu}$-admissible, $Z_mu$-contraction and $N_{mu}$-contraction via simulation functions. We prove some new fixed point theorems for defined class of contractions via $alpha$-admissible simulation mappings, as well. Our results can be viewed as extension of the corresponding results in this area. Moreover, some examples and an application to functional integral equations are given to support the obtained results.

In this paper, we consider a class of abstract fractional differential inclusion with finite delay in which the multi-valued nonlinearity is possibly superlinear. We analyze some sufficient conditions that ensure the global solvability of problem. Our main result is the existence of a compact set of decay solutions to our problem by estimating the measure of noncompactness and using the fixed point theory for a condensing map. The obtained results will be applied to a concrete polytope fractional differential system. [ABSTRACT FROM AUTHOR]