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In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system's state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed. [ABSTRACT FROM AUTHOR]

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article's main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as U D - Ԓ -convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down ( U D ) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for Ԓ , we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard ( H H ) and Pachpatte types of inequalities using the concepts of coordinated U D - Ԓ -convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples. [ABSTRACT FROM AUTHOR]

This work considers fractional operators (derivatives and integrals) as surfaces  f (x , α)  subject to the function constraints defined by integer operators, which is a mandatory requirement of any fractional operator definition. In this respect, the problem can be seen as the problem of generating a surface constrained at some positive integer values of  α  for fractional derivatives and at some negative integer values for fractional integrals. This paper shows that by using the Theory of Functional Connections, all (past, present, and future) fractional operators can be approximated at a high level of accuracy by smooth surfaces and with no continuity issues. This practical approach provides a simple and unified tool to simulate nonlocal fractional operators that are usually defined by infinite series and/or complicated integrals. [ABSTRACT FROM AUTHOR]

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green's function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that the second derivative of the functions is monotone, absolutely convex, and concave. A section relating the results of exploration to generalized means and trapezoid formulas is included in the applications. We anticipate that the method presented in this study will inspire further research in this field. [ABSTRACT FROM AUTHOR]

In this paper, we derive some classical and fractional properties of the r R s matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. [ABSTRACT FROM AUTHOR]

This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context. [ABSTRACT FROM AUTHOR]

In 2012, new classes of analytic functions on U × U ¯ with coefficient holomorphic functions in U ¯ were defined to give a new approach to the concepts of strong differential subordination and strong differential superordination. Using those new classes, the extended Dziok–Srivastava operator is introduced in this paper and, applying fractional integral to the extended Dziok–Srivastava operator, we obtain a new operator D z − γ H m l α 1 , β 1 that was not previously studied using the new approach on strong differential subordinations and superordinations. In the present article, the fractional integral applied to the extended Dziok–Srivastava operator is investigated by applying means of strong differential subordination and superordination theory using the same new classes of analytic functions on U × U ¯. Several strong differential subordinations and superordinations concerning the operator D z − γ H m l α 1 , β 1 are established, and the best dominant and best subordinant are given for each strong differential subordination and strong differential superordination, respectively. This operator may have symmetric or asymmetric properties. [ABSTRACT FROM AUTHOR]

Fuzzy set theory, introduced by Zadeh, gives an adaptable and logical solution to the provocation of introducing, evaluating, and opposing numerous sustainability scenarios. The results described in this article use the fuzzy set concept embedded into the theories of differential subordination and superordination from the geometric function theory. In 2011, fuzzy differential subordination was defined as an extension of the classical notion of differential subordination, and in 2017, the dual concept of fuzzy differential superordination appeared. These dual notions are applied in this paper regarding the fractional integral applied to Dziok–Srivastava operator. New fuzzy differential subordinations are proved using known lemmas, and the fuzzy best dominants are established for the obtained fuzzy differential subordinations. Dual results regarding fuzzy differential superordinations are proved for which the fuzzy best subordinates are shown. These are the first results that link the fractional integral applied to Dziok–Srivastava operator to fuzzy theory. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to present classical second-order differential subordination knowledge extended in this study to include new results regarding third-order differential subordinations. The focus of this study is on the main problems examined by differential subordination theory. Hence, the new results obtained here reveal techniques for identifying dominants and the best dominant of certain third-order differential subordinations. Another aspect of novelty is the new application of the Gaussian hypergeometric function. Novel third-order differential subordination results are obtained using the best dominant provided by the theorems and the operator previously defined as Gaussian hypergeometric function's fractional integral. The research investigation is concluded by giving an example of how the results can be implemented. [ABSTRACT FROM AUTHOR]

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology. [ABSTRACT FROM AUTHOR]

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented. [ABSTRACT FROM AUTHOR]

In this study, we apply fractional calculus on certain convex functions and derive many novel mean-type inequalities by employing fractional calculus and convexity theory. In order to investigate fractional mean inequalities, we first build an identity in this study. Then, with its help, we derive many mean-type inequalities and estimate the error of HH inequality using a generalized version of RL-fractional integrals and certain classes of convex functions. The results obtained are validated by taking specific functions. Many mean-type inequalities that exist in the literature are generalized by the main results of this study. [ABSTRACT FROM AUTHOR]

In this paper, we introduce controlled S-metric-type spaces and give some of their properties and examples. Moreover, we prove the Banach fixed point theorem and a more general fixed point theorem in this new space. Finally, using the new results, we give two applications on Riemann–Liouville fractional integrals and Atangana–Baleanu fractional integrals. [ABSTRACT FROM AUTHOR]

This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ ∈ (0 , 1) by using the Picard iteration technique (PIT) and the semimartingale local time (SLT). [ABSTRACT FROM AUTHOR]

Sanford S. Miller and Petru T. Mocanu's theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. [ABSTRACT FROM AUTHOR]

A generalization of probability theory is proposed by using the Riemann–Liouville fractional integrals and the Caputo and Riemann–Liouville fractional derivatives of arbitrary (non-integer and integer) orders. The definition of the fractional probability density function (fractional PDF) is proposed. The basic properties of the fractional PDF are proven. The definition of the fractional cumulative distribution function (fractional CDF) is also suggested, and the basic properties of these functions are also proven. It is proven that the proposed fractional cumulative distribution functions generate unique probability spaces that are interpreted as spaces of a fractional probability theory of arbitrary order. Various examples of the distributions of the fractional probability of arbitrary order, which are defined on finite intervals of the real line, are suggested. [ABSTRACT FROM AUTHOR]

This study deals with a novel class of mean-type inequalities by employing fractional calculus and convexity theory. The high correlation between symmetry and convexity increases its significance. In this paper, we first establish an identity that is crucial in investigating fractional mean inequalities. Then, we establish the main results involving the error estimation of the Hermite–Hadamard inequality for composite convex functions via a generalized Riemann-type fractional integral. Such results are verified by choosing certain composite functions. These results give well-known examples in special cases. The main consequences can generalize many known inequalities that exist in other studies. [ABSTRACT FROM AUTHOR]

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched. [ABSTRACT FROM AUTHOR]

New results concerning fuzzy differential subordination theory are obtained in this paper using the operator denoted by D z − λ L α n , previously introduced by applying the Riemann–Liouville fractional integral to the convex combination of well-known Ruscheweyh and Sălăgean differential operators. A new fuzzy subclass D L n F δ , α , λ is defined and studied involving the operator D z − λ L α n . Fuzzy differential subordinations are obtained considering functions from class D L n F δ , α , λ and the fuzzy best dominants are also given. Using particular functions interesting corollaries are obtained and an example shows how the obtained results can be applied. [ABSTRACT FROM AUTHOR]

Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional derivative conforms to all prerequisites demanded by differential topology. Hence, the various mathematical forms, including those derivatives, do not lack the mathematical accuracy or defects of the well-known fractional derivatives. A summary of the Λ-fractional analysis is presented with its influence on the sources of differential equations, such as fractional differential geometry, field theorems, and calculus of variations. Λ-fractional ordinary and partial differential equations will be discussed. [ABSTRACT FROM AUTHOR]

Fuzzy differential subordination theory represents a generalization of the classical concept of differential subordination which emerged in the recent years as a result of embedding the concept of fuzzy set into geometric function theory. The fractional integral of Gaussian hypergeometric function is defined in this paper and using properties of the subordination chains, new fuzzy differential subordinations are obtained. Dominants of the fuzzy differential subordinations are given and using particular functions as such dominants, interesting geometric properties interpreted as inclusion relations of certain subsets of the complex plane are presented in the corollaries of the original theorems stated. An example is constructed as an application of the newly proved results. [ABSTRACT FROM AUTHOR]

The study done for obtaining the original results of this paper involves the fractional integral of the confluent hypergeometric function and presents its new applications for introducing a certain subclass of analytic functions. Conditions for functions to belong to this class are determined and the class is investigated considering aspects regarding coefficient bounds as well as partial sums of these functions. Distortion properties of the functions belonging to the class are proved and radii estimates are established for starlikeness and convexity properties of the class. [ABSTRACT FROM AUTHOR]

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions A n = f ∈ H U : f z = z + a n + 1 z n + 1 + ... , z ∈ U written simply A when n = 1 , the newly introduced class involves functions f ∈ A considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties. [ABSTRACT FROM AUTHOR]

The purpose of this research paper is first to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev's functionals by utilizing the proposed modified weighted-type fractional integral incorporating another function in the kernel F (θ) . For the weighted and extended Chebyshev's functionals, we also propose weighted fractional integral inequalities. With specific choices of ϖ (θ) and F (θ) as stated in the literature, one may easily study certain new inequalities involving all other types of weighted fractional integrals related to Chebyshev's functionals. Furthermore, the inequalities for all other type of fractional integrals associated with Chebyshev's functionals with certain choices of ϖ (θ) and F (θ) are covered from the obtained generalized weighted-type fractional integral inequalities. [ABSTRACT FROM AUTHOR]

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α -fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α -fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space R n , we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces L p (Ω) , Sobolev spaces W m , p (Ω) , and Hölder spaces C m , σ (Ω) , which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α -fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α -fractal function is a fractal function of similar kind. [ABSTRACT FROM AUTHOR]

The authors give some singular versions of the Gronwall–Bihari–Henry inequalities. They also establish a multivalued version of the Leray–Schauder alternative in strictly star-shaped sets. Based on these new fractional inequalities and fixed point theorem, they study an initial value problem for fractional differential inclusions with delay. [ABSTRACT FROM AUTHOR]

The fractional integral of confluent hypergeometric function is used in this paper for obtaining new applications using concepts from the theory of fuzzy differential subordination and superordination. The aim of the paper is to present new fuzzy differential subordinations and superordinations for which the fuzzy best dominant and fuzzy best subordinant are given, respectively. The original theorems proved in the paper generate interesting corollaries for particular choices of functions acting as fuzzy best dominant and fuzzy best subordinant. Another contribution contained in this paper is the nice sandwich-type theorem combining the results given in two theorems proved here using the two theories of fuzzy differential subordination and fuzzy differential superordination. [ABSTRACT FROM AUTHOR]

In the present paper, a new operator denoted by D z − λ L α n is defined by using the fractional integral of Sălăgean and Ruscheweyh operators. By means of the newly obtained operator, the subclass S n δ , α , λ of analytic functions in the unit disc is introduced, and various properties and characteristics of this class are derived by applying techniques specific to the differential subordination concept. By studying the operator D z − λ L α n , some interesting differential subordinations are also given. [ABSTRACT FROM AUTHOR]

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved. [ABSTRACT FROM AUTHOR]

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of "general nonlocal mappings" is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described. [ABSTRACT FROM AUTHOR]

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve. [ABSTRACT FROM AUTHOR]

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived. [ABSTRACT FROM AUTHOR]

The fractional integral is a function known for the elegant results obtained when introducing new operators; it has proved to have interesting applications. In the present paper, differential subordinations and superodinations for the fractional integral of the confluent hypergeometric function introduced in a previously published paper are presented. A sandwich-type theorem at the end of the original part of the paper connects the outcomes of the studies done using the dual theories. [ABSTRACT FROM AUTHOR]

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme. [ABSTRACT FROM AUTHOR]

Let A (p , n) be the class of f (z) which are analytic p-valent functions in the closed unit disk U ¯ = z ∈ C : z ≤ 1 . The expression B − m − λ f (z) is defined by using fractional integrals of order λ for f (z) ∈ A (p , n). When m = 1 and λ = 0 , B − 1 f (z) becomes Bernardi integral operator. Using the fractional integral B − m − λ f (z) , the subclass T p , n α s , β , ρ ; m , λ of A (p , n) is introduced. In the present paper, we discuss some interesting properties for f (z) concerning with the class T p , n α s , β , ρ ; m , λ. Also, some interesting examples for our results will be considered. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for a new nonlocal boundary value problem of integro-differential equations involving mixed left and right Caputo and Riemann–Liouville fractional derivatives and Riemann–Liouville fractional integrals of different orders. Our results rely on the standard tools of functional analysis. Examples are constructed to demonstrate the application of the derived results. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the 2-variables Konhauser matrix polynomials; then, we investigate some properties of these matrix polynomials such as generating matrix relations, integral representations, and finite sum formulae. Finally, we obtain the fractional integrals of the 2-variables Konhauser matrix polynomials. [ABSTRACT FROM AUTHOR]

Fractional calculus, albeit a synonym of fractional integrals and derivatives which have two main characteristics—singularity and nonlocality—has attracted increasing interest due to its potential applications in the real world. This mathematical concept reveals underlying principles that govern the behavior of nature. The present paper focuses on numerical approximations to fractional integrals and derivatives. Almost all the results in this respect are included. Existing results, along with some remarks are summarized for the applied scientists and engineering community of fractional calculus. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii's fixed point theorem, while the contraction mapping principle is employed to obtain the uniqueness of solutions for the problem at hand. The paper concludes with illustrative examples. [ABSTRACT FROM AUTHOR]

We establish sufficient criteria for the existence of solutions for a nonlinear generalized Langevin-type nonlocal fractional-order integral multivalued problem. The convex and non-convex cases for the multivalued map involved in the given problem are considered. Our results rely on Leray–Schauder nonlinear alternative for multivalued maps and Covitz and Nadler's fixed point theorem. Illustrative examples for the main results are included. [ABSTRACT FROM AUTHOR]

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces. [ABSTRACT FROM AUTHOR]

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo's derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications. [ABSTRACT FROM AUTHOR]

We develop the existence criteria for solutions of Liouville–Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are also presented. It is imperative to mention that our results correspond to the ones for a symmetric second-order nonlocal multipoint integral boundary value problem under suitable conditions (see the last section). [ABSTRACT FROM AUTHOR]

In the article, the authors present several inequalities of the Čebyšev type for conformable k-fractional integral operators. [ABSTRACT FROM AUTHOR]

In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of power-law long-range dependence in time by using the analogy with the concept of the long-range alpha-interaction. We also suggest the criterion of existence of PLT memory for frequency domain by using the concept of non-integer dimensions. For an economic process, for which it is known that an endogenous variable depends on an exogenous variable, the proposed criteria make it possible to identify the presence of the PLT memory. The suggested criteria are illustrated in various examples. The use of the proposed criteria allows apply the fractional calculus to construct dynamic models of economic processes. These criteria can be also used to identify the linear integro-differential operators that can be considered as fractional derivatives and integrals of non-integer orders. [ABSTRACT FROM AUTHOR]