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This paper intends to demonstrate the boundedness of the fractional bilinear Hardy operator and its adjoint on the $ \lambda $-central Morrey space with variable exponents. Analogous outcomes for their commutators are derived when the symbol functions are elements of the $ \lambda $-central bounded mean oscillation ($ \lambda $-central BMO) space.

In this paper the authors present three different Lyapunov-type inequalities for a higher-order Caputo fractional differential equation with identical boundary conditions marking the inaugural instance of such an approach in the existing literature. Their findings extend and complement certain prior results in the literature.

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order $ \lambda $ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular.

Abstract In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ${M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot)}(w)}$ . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.

In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.

We prove boundedness and compactness for the iterated commutators of the $ \theta $-type Calderón-Zygmund singular integral and its fractional variant on the weighed Morrey spaces.

In this article, we present new fractional integral inequalities via Euler’s beta function in terms of ss-convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable ss-convexity. The results obtained in this study extended other related results reported in the literature.

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples.

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.

The contribution of fractional calculus in the development of different areas of research is well known. This article presents investigations involving fractional calculus in the study of analytic functions. Riemann-Liouville fractional integral is known for its extensive applications in geometric function theory. New contributions were previously obtained by applying the Riemann-Liouville fractional integral to the convolution product of multiplier transformation and Ruscheweyh derivative. For the study presented in this article, the resulting operator is used following the line of research that concerns the study of certain new subclasses of analytic functions using fractional operators. Riemann-Liouville fractional integral of the convolution product of multiplier transformation and Ruscheweyh derivative is applied here for introducing a new class of analytic functions. Investigations regarding this newly introduced class concern the usual aspects considered by researchers in geometric function theory targeting the conditions that a function must meet to be part of this class and the properties that characterize the functions that fulfil these conditions. Theorems and corollaries regarding neighborhoods and their inclusion relation involving the newly defined class are stated, closure and distortion theorems are proved, and coefficient estimates are obtained involving the functions belonging to this class. Geometrical properties such as radii of convexity, starlikeness, and close-to-convexity are also obtained for this new class of functions.

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.

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.

In this article, we introduce the notions of generalized fractional integrals for the interval-valued functions (IVFs) of two variables. We establish Hermite-Hadamard (H-H) type inequalities and some related inequalities for co-ordinated convex IVFs by using the newly defined integrals. The fundamental benefit of these inequalities is that these can be turned into classical H-H inequalities and Riemann-Liouville fractional H-H inequalities, and new kk-Riemann-Liouville fractional H-H inequalities can be obtained for co-ordinated convex IVFs without having to prove each one separately.

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 UD-Ԓ-convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down (UD) 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 (HH) and Pachpatte types of inequalities using the concepts of coordinated UD-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.

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system $ \mathcal{D}^{\alpha}_{t}X(t) = AX(t) $ described by fractional differential equations, where $ \mathcal{D} $ is the fractional derivative in the Caputo-Fabrizio sense and $ A = (a_{ij})_{2\times2} $ is nonsingular coefficient matrix with $ a_{ij}\in\mathbb{R} $. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system.

In this paper, we derive a new fractional integral identity and using this identity as an auxiliary result, some new trapezium like inequalities essentially using the class of generalized γ-convex functions are established. In order to show the efficiency of the obtained results, we discuss and present some special cases and applications.

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.

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.

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 The present article discusses the boundedness criteria for the fractional Hardy operators on weighted variable exponent Morrey–Herz spaces M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ${M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(w)}$ .

Abstract This paper deals with Al-Salam fractional q-integral operator and its application to certain q-analogues of Bessel functions and power series. Al-Salam fractional q-integral operator has been applied to various types of q-Bessel functions and some power series of special type. It has been obtained for basic q-generating series, q-exponential and q-trigonometric functions as well. Various results and corollaries are provided as an application to this theory.

In this paper, we establish several differential subordinations regarding the operator 𝐷^(−𝜆)_𝑧 𝑆𝑅^(𝑚,𝑛) defined using the fractional integral of the differential operator 𝑆𝑅^(𝑚,𝑛), obtained as a convolution product of Salagean operator 𝑆^𝑚 and Ruscheweyh derivative 𝑅^𝑛. By means of the newly obtained operator, a new subclass of analytic functions denoted by 𝒮𝓡_(m,n,𝜆)(𝛿) is introduced and various properties and characteristics of this class are derived, makinguse of the concept of differential subordination.

In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.

In this paper, we derive some classical and fractional properties of the rRs 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.

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 Dz−γHmlα1,β1 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 Dz−γHmlα1,β1 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.

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.

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.

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.

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.

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.

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 The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function G $\mathcal{G}$ in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of ω ( θ ) $\omega (\theta )$ and G ( θ ) $\mathcal{G}(\theta )$ as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of ω ( θ ) $\omega (\theta )$ and G ( θ ) $\mathcal{G}(\theta )$ .

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. We have included some recent work on the topic dealing with sequential, Caputo-Fabrizio, Hadamard, Hilfer, Katugampola, hyprid, or nested fractional derivatives. We have also focused on Lyapunov-type inequalities for problems containing fractional derivatives with respect to a certain function, systems of fractional differential equations, and conformable derivatives. Furthermore, Lyapunov-type inequalities for fractional differential equations based on non Green function approach are discussed.

Let Tn be the class of functions f which are defined by a power series f(z)=z+an+1zn+1+an2zn+2+…f\left( z \right) = z + {a_{n + 1}}{z^{n + 1}} + {a_n}2{z^{n + 2}} + \ldots for every z in the closed unit disc 𝕌¯\bar {\mathbb{U}}. With m different boundary points zs, (s = 1,2,...,m), we consider αm ∈ eiβ𝒜−j−λf(𝕌), here 𝒜−j−λ is the generalized Alexander integral operator and 𝕌 is the open unit disc. Applying 𝒜−j−λ, a subclass Bn(αm,β,ρ; j, λ) of Tn is defined with fractional integral for functions f. The object of present paper is to consider some interesting properties of f to be in Bn(αm,β,ρ; j, λ).

Abstract In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.

In the article, we establish a Simpson-type generalized identity containing multi-parameters and derive some new estimates for the generalized Simpson’s quadrature rule via the Raina fractional integrals. As applications, we provide several inequalities for the f -divergence measures and probability density functions.

In this article we show that, when the delay approaches zero, the solution of multidimensional delay differential equations driven by a Holder continuous function of order 1/3

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.

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.

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.

Infant hydrocephalus is a clinically abnormal clinical state with an accumulation of fluid in cavities (ventricles) deep within the brain. Hence, pressure is increased inside the skull. The ventricles widen due to the excess fluid applying pressure upon the (parenchyma) brain tissues. The infant brain tissue is described by a biomechanics model as a hyperelastic, Λ-fractional viscoelastic material, trying to describe the various conditions developing hydrocephalus. Λ-fractional continuum mechanics is applied with time variables due to viscosity and space fractional variables due to porosity. The simultaneous influence of both the viscosity and porosity of the membrane material (parenchyma) increases the cerebrospinal fluid’s pressure, causing the fluid’s accumulation in the brain.

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 The present paper aims to establish certain new classes of integral inequalities for a class of n ( n∈N $n\in \mathbb{N}$) positive continuous and decreasing functions by utilizing the generalized fractional conformable integral operators (FCIO) recently defined by Khan and Khan. From these results, we also derive several particular cases.

Abstract Our aim in this present paper is to establish several Chebyshev type inequalities involving generalized fractional conformable integral operator recently introduced by T.U. Khan and M.A. Khan (J. Comput. Appl. Math. 346:378–389, 2019). Also, we present Chebyshev type inequalities involving Riemann–Liouville type fractional conformable integral operators as a particular result of our main result.

This paper is devoted to studying the existence of solutions of a nonlocal initial value problem involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. Illustrative examples are also given.

In this paper, we discuss the existence and uniqueness of solutions for a new kind of Langevin equation involving Riemann-Liouville as well as Caputo fractional derivatives, and variable coefficient, supplemented with nonlocal-terminal fractional integro-differential conditions. The proposed study is based on modern tools of functional analysis. We also extend our discussion to the associated inclusions problem. For the applicability of the obtained results, several examples are constructed. Some interesting observations are also presented.

Abstract In this paper, we present sufficient criteria ensuring the existence and uniqueness of solutions for nonlinear impulsive multi-order Caputo-type generalized fractional differential equations supplemented with nonlocal integro-initial value conditions involving generalized fractional integrals. Extremal solutions for the given problem are also discussed. The main tools of our study include Krasnoselskii’s fixed point theorem, Banach contraction mapping principle and monotone iterative technique. Examples are constructed for illustrating the obtained results.

Abstract In the present research, we develop some integral inequalities of Hermite–Hadamard type for differentiable η-convex functions. Moreover, our results include several new and known results as particular cases.

In this article we use the method of lower and upper solutions combined with the fixed point theorem by Bohnnenblust-Karlin to show the existence of solutions for initial-value problems of impulsive Caputo-Hadamard fractional differential inclusions of order in (0,1).

In this paper, we introduce a new kind of generalized fractional integral boundary conditions and develop the existence theory for a fractional di erential equation involving generalized Caputotype fractional derivative equipped with these conditions. We also study the inclusion case of the given problem. Examples are constructed to demonstrate the application of the obtained results.