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We develop the martingale theory in the framework of Lorentz amalgam spaces. Atomic decompositions for the martingale Hardy Lorentz amalgam spaces are established. As applications of atomic decompositions, the dual spaces of the martingale Hardy Lorentz amalgam spaces are characterized. Furthermore, when the stochastic basis is regular, the boundedness of the fractional integrals on martingale Hardy Lorentz amalgam spaces is proved. The results obtained here generalized the corresponding known results in martingale Hardy amalgam spaces and various classical martingale Hardy spaces. [ABSTRACT FROM AUTHOR]

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) . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient criteria for ensuring the existence of solutions and uniqueness for a class of nonlinear neutral Caputo fractional differential equations supplemented with infinite delay and nonlocal boundary conditions involving fractional derivatives. The theory of infinite delay and standard fixed point theorems are employed to obtain the existence results for the given problem. Examples will be constructed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch's fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. [ABSTRACT FROM AUTHOR]

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability. [ABSTRACT FROM AUTHOR]

Let p ∈ [ 1 , ∞ ] , q ∈ [ 1 , ∞) , s ∈ Z + : = N ∪ { 0 } , α ∈ R , and β ∈ (0 , 1) . In this article, the authors first find a reasonable version I ~ β of the (generalized) fractional integral I β on the special John–Nirenberg–Campanato space via congruent cubes, J N (p , q , s) α con (R n) , which coincides with the Campanato space C α , q , s (R n) when p = ∞ . To this end, the authors introduce the vanishing moments up to order s of I β . Then the authors prove that I ~ β is bounded from J N (p , q , s) α con (R n) to J N (p , q , s) α + β / n con (R n) if and only if I β has the vanishing moments up to order s. The obtained result is new even when p = ∞ and s ∈ N , namely, the Campanato space. Moreover, the authors show that I β can be extended to a unique continuous linear operator from the Hardy-kind space H K (p , q , s) α + β / n con (R n) , the predual of J N (p ′ , q ′ , s) α + β / n con (R n) with 1 p + 1 p ′ = 1 = 1 q + 1 q ′ , to H K (p , q , s) α con (R n) if and only if I β has the vanishing moments up to order s. The proof of the latter boundedness strongly depends on the dual relation (H K (p , q , s) α con (R n)) ∗ = J N (p ′ , q ′ , s) α con (R n) , the good properties of molecules of H K (p , q , s) α con (R n) , and the crucial criterion for the boundedness of linear operators on H K (p , q , s) α con (R n) . [ABSTRACT FROM AUTHOR]

In this paper, we develop the martingale theory in the framework of grand Lebesgue spaces for 0 < p ≤ 1 . Atomic decompositions for the grand martingale Hardy spaces are established. With the help of the atomic decompositions, the dual spaces of the grand martingale Hardy spaces are characterized and some martingale inequalities are deduced. Furthermore, when the stochastic basis is regular, the boundedness of the fractional integrals on grand martingale Hardy spaces is proved. The results obtained here unify and generalize the previous results of grand martingale Hardy spaces. [ABSTRACT FROM AUTHOR]

In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form. [ABSTRACT FROM AUTHOR]

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) . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces. And the corresponding commutators generated by BMO function are also considered. [ABSTRACT FROM AUTHOR]

We obtain the boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents K ˙ p (⋅) , q (⋅) α (⋅) , such as some sublinear operators, the fractional integral and its commutator. [ABSTRACT FROM AUTHOR]

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 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) and G (θ) 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) and G (θ) . [ABSTRACT FROM AUTHOR]

We study integral operators with kernels K (x , y) = k 1 (x - A 1 y) ⋯ k m (x - A m y) , k i (x) = Ω i (x) | x | n / q i where Ω i : R n → R are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and n q 1 + ⋯ + n q m = n - α , 0 ≤ α < n . We obtain the boundedness of this operator from L p (·) into L q (·) for 1 q (·) = 1 p (·) - α n , for certain exponent functions p satisfying weaker conditions than the classical log-Hölder conditions. [ABSTRACT FROM AUTHOR]

In this paper, based on the "fuzzy" calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C1 criterion according to the Ross classification. The new calculus is tightly related to the continuous values of the continuous spin S = 1 and can generalize the expression for the fractional values of the shifting discrete index. This calculus can be interpreted as the appearance of the hidden states corresponding to unobservable values of S = 1. Many well-known formulas can be generalized and receive a new extended interpretation. In particular, one can factorize any rectangle matrix and receive the "perfect" filtering formula that allows transforming any (deterministic or random) function to another arbitrary function and vice versa. This transformation can find unexpected applications in data transmission, cryptography and calibration of different gadgets and devices. One can also receive the hybrid ("centaur") formula for the Fourier (F-) transformation unifying both expressions for the direct and inverse F-transformations in one mathematical unit. The generalized Dirichlet formula, which is obtained in the frame of the new calculus to allow selecting the desired resonance frequencies, will be useful in discrete signals processing, too. The basic formulas are tested numerically on mimic data. [ABSTRACT FROM AUTHOR]

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of α-fractionally convex functions. We primarily focus on characterizing some of the qualitative properties of convex functions with the assistance of fractional order operators. Also, we discuss the connection of optimality, monotonicity, and convexity of a function in the sense of fractional calculus. Several examples are provided to support the proposed formulation, and one can establish the fractional convexity of a non-convex function. [ABSTRACT FROM AUTHOR]

Introducing the fractional Λ -derivative, with the corresponding Λ -fractional spaces, the fractional beam bending problem is presented. In fact, non-local derivatives govern the beam bending problem that accounts for the interaction of microcracks or materials non-homogeneities, such as composite materials or materials with fractal geometries. The proposed theory is implemented to the fractional bending deformation of a simply supported beam and a cantilever beam under continuously distributed loading. [ABSTRACT FROM AUTHOR]

In the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO. [ABSTRACT FROM AUTHOR]

We use a recently proposed fractional integral to establish a generalization of Grüss-type integral inequalities. We prove two theorems about these inequalities and enunciate and prove other inequalities associated with this fractional operator. [ABSTRACT FROM AUTHOR]

Vallis proposed a simple model for El-Niño weather phenomenon (referred as Vallis system) by adding an additional parameter p to the Lorenz system. He showed that the chaotic behavior of the Vallis system is related to the El-Niño effect. In the present article we study fractional version of Vallis system in detail. We investigate bifurcations and chaos present in the fractional Vallis system and the effect of variation of system parameter p. It is observed that the range of values of parameter p for which the Vallis system is chaotic, reduces with the reduction of the fractional order. Further we analyze the incommensurate fractional Vallis system and find the critical value below which the system loses chaos. We also synchronize Vallis system with Bhalekar-Gejji system. [ABSTRACT FROM AUTHOR]

During formation of echocardiographic images, speckle noise is introduced, which diminishes important information present in an image and effects physician’s capability to interpret image correctly. In the literature, many techniques have been proposed to remove unwanted noise from the image. In this paper, an intelligent denoising algorithm for echocardiographic images has been proposed, which first divides input image into different regions, namely smooth, texture and edge, using coefficient of variation. Fuzzy logic is used to draw boundaries between these image regions. Average filter and fractional integral filters are deployed to denoise pixels of various regions. Selection of filter depends on the characteristics of a region. The proposed technique improves quality of denoised image by suppressing maximum noise and producing no artifacts. Simulation results show superiority of proposed methodology over state-of-the-art existing methodologies, visually and using quantitative measures i.e. mean square error, peak signal to noise, edge preservation index, correlation coefficient and structure similarity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate four different types of Ulam stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of nonlinear implicit fractional differential equations with non-instantaneous integral impulses and nonlinear integral boundary condition. We also establish certain conditions for the existence and uniqueness of solutions for such a class of fractional differential equations using Caputo fractional derivative. The arguments are based on generalized Diaz-Margolis's fixed point theorem. We provide two examples, which shows the validity of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the $(k, s)$ -fractional integral and differential operators involving k-Mittag-Leffler function $E_{k,\rho,\beta}^{\delta}(z)$ as its kernel. Also, we establish various properties of these operators. Further, we consider a number of certain consequences of the main results. [ABSTRACT FROM AUTHOR]

We characterize the dual spaces of weak martingale Hardy-Lorentz-Karamata spaces. As an application, we obtain a weak type John-Nirenberg theorem when the stochastic basis is regular. We also extend the boundedness of fractional integrals to the setting of martingale Hardy-Lorentz-Karamata spaces. [ABSTRACT FROM AUTHOR]

We present a new way of constructing a fractional-based convolution mask with an application to image edge analysis. The mask was constructed based on the Riemann-Liouville fractional derivative which is a special form of the Srivastava-Owa operator. This operator is generally known to be robust in solving a range of differential equations due to its inherent property of memory effect. However, its application in constructing a convolution mask can be devastating if not carefully constructed. In this paper, we show another effective way of constructing a fractional-based convolution mask that is able to find edges in detail quite significantly. The resulting mask can trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges. The experiments conducted on the mask were done using some selected well known synthetic and Medical images with realistic geometry. Using visual perception and performing both mean square error and peak signal-to-noise ratios analysis, the method demonstrated significant advantages over other known methods. [ABSTRACT FROM AUTHOR]

Let $$T_1$$ be a generalized Calderón-Zygmund operator or $$\pm I$$ ( the identity operator), let $$T_2$$ and $$T_3$$ be the linear operators, and let $$T_3=\pm I$$ . Denote the Toeplitz type operator by where $$M^bf=bf,$$ and $$I_\alpha $$ is fractional integral operator. In this paper, we establish the sharp maximal function estimates for $$T^b$$ when b belongs to weighted Lipschitz function space, and the weighted norm inequalities of $$T^b$$ on weighted Lebesgue space are obtained. [ABSTRACT FROM AUTHOR]

A formula relating the Radon transform of functions of spherical symmetries to the Radon-Kipriyanov transform K for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K -transform of a radial function as fractional integrals of Erdelyi-Kober integral type and of Riemann-Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon-Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables. [ABSTRACT FROM AUTHOR]

In this paper we prove several fractional quantum integral inequalities for the new q-shifting operator ${_{a}}\Phi_{q}(m) = qm + (1-q)a$ introduced in Tariboon et al. (Adv. Differ. Equ. 2015:18, ), such as: the q-Hölder inequality, the q-Hermite-Hadamard inequality, the q-Cauchy-Bunyakovsky-Schawrz integral inequality, the q-Grüss integral inequality, the q-Grüss-Čebyšev integral inequality, and the q-Pólya-Szegö integral inequality. [ABSTRACT FROM AUTHOR]

We prove the boundedness of fractional integral with a rough kernel on Triebel-Lizorkin spaces, where the rough kernel belongs to the block space and does not need to satisfy any moment conditions on the unit sphere. [ABSTRACT FROM AUTHOR]

Vector-valued fractional maximal inequalities on variable Morrey spaces are proved. Applying atomic decomposition of variable Hardy-Morrey spaces, we obtain the boundedness of fractional integrals on variable Hardy-Morrey spaces, which extends the Taibleson-Weiss's results for the boundedness of fractional integrals on Hardy spaces. The corresponding boundedness for the fractional type integrals is also considered. [ABSTRACT FROM AUTHOR]

In this note, we prove that the fractional integral operators from Herz spaces with variable exponent $\dot{K}^{\alpha}_{p(\cdot), q}$ to Lipschitz-type spaces are bounded provided $p(\cdot)$ is locally log-Hölder continuous and log-Hölder continuous at infinity. [ABSTRACT FROM AUTHOR]

This work is concerned with the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a class of semilinear fractional differential equations, $D_{t}^{\alpha}x(t)=Ax(t)+D_{t}^{\alpha-1}F(t,x(t))$, $t\in \mathbb{R}$, where $1<\alpha<2$, A is a linear densely defined operator of sectorial type of $\omega<0$ on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a fractional relaxation-oscillation equation. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of fractional neutral stochastic functional differential systems. We obtain the controllability of the stochastic functional differential systems by the Sadovskii's fixed point theorem under some suitable assumptions. An example is given to illustrate the theory. [ABSTRACT FROM AUTHOR]

The article presents mathematical analysis of fractional integrals and prove the analog of inequality for measures of polynomials.

A one-dimensional continuous function of unbounded variation on [0, 1] has been constructed. The length of its graph is infinite, while part of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated. [ABSTRACT FROM AUTHOR]

To obtain the solution of nonlinear sequential fractional differential equations for Caputo operator two methods namely the Adomian decomposition method and Daftardar-Gejji and Jafari iterative method are applied in this paper. Finally some examples are presented to illustrate the efficiency of these methods. [ABSTRACT FROM AUTHOR]

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered. [ABSTRACT FROM AUTHOR]

In this paper, the weak type Llog L estimate for the multilinear fractional commutator is obtained by introducing a new kind of maximal operator of the multilinear fractional order associated with the mean Luxumburg norm and using the technique of sharp function. [ABSTRACT FROM AUTHOR]

We study the relationship between the solutions of abstract differential equations with fractional derivatives and their stability with respect to the perturbation by a bounded operator. Besides, we obtain representations for the solution of an inhomogeneous equation and for an equation containing a fractional power of the generator of a cosine operator function. [ABSTRACT FROM AUTHOR]

In this paper, two classes of closely related multilinear singular and fractional integrals, which include the commutators as special cases, are studied and their boundedness on Herz type spaces is discussed. In fact, it is proved that these operators are actually not bounded in certain extreme cases. [ABSTRACT FROM AUTHOR]

The authors define the directional hyper Hilbert transform and give its mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the L[sup p]-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The present paper aims to establish certain new classes of integral inequalities for a class of n (n ∈ 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 FROM AUTHOR]

This is an erratum to the paper “Fractional integral on martingale Hardy spaces with variable exponents”, Fract. Calc. Appl. Anal. 18, No 5 (2015), 1128–1145; DOI: 10.1515/fca-2015-0065. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let L=−ΔHn+V be a Schrödinger operator on the Heisenberg group Hn, where the nonnegative potential V belongs to the reverse Hölder class RHq1 for some q1≥Q/2, and Q is the homogeneous dimension of Hn. Let b belong to a new Campanato space Λνθ(ρ), and let IβL be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b,IβL] with b∈Λνθ(ρ) on central generalized Morrey spaces LMp,φα,V(Hn), generalized Morrey spaces Mp,φα,V(Hn), and vanishing generalized Morrey spaces VMp,φα,V(Hn) associated with Schrödinger operator, respectively. When b belongs to Λνθ(ρ) with θ>0, 0<ν<1 and (φ1,φ2) satisfies some conditions, we show that the commutator operator [b,IβL] is bounded from LMp,φ1α,V(Hn) to LMq,φ2α,V(Hn), from Mp,φ1α,V(Hn) to Mq,φ2α,V(Hn), and from VMp,φ1α,V(Hn) to VMq,φ2α,V(Hn), 1/p−1/q=(β+ν)/Q. [ABSTRACT FROM AUTHOR]