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Let 1 < q ≤ p ≤ r ≤ ∞ and τ ∈ (0 , ∞ ] . Besov–Bourgain–Morrey spaces M B ˙ q , r p , τ (R n) in the special case where τ = r , extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent θ ∈ [ 0 , ∞) , the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces M B ˙ q) , r , θ p , τ (R n) . The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of M B ˙ q) , r , θ p , τ (R n) related to Muckenhoupt A 1 (R n) -weights, the authors then obtain an extrapolation theorem of M B ˙ q) , r , θ p , τ (R n) . Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of R n , the authors establish the sharp boundedness on M B ˙ q) , r , θ p , τ (R n) of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance - (D a + α , ρ x) (t) = f (t , x (t) , D a + α - 1 , ρ x (t)) , t ∈ (a , b) , x (a) = 0 , x (b) = ∫ a b x (t) d A (t) , where 1 < α ≤ 2 , and D a + α , ρ is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫ a b x (t) d A (t) denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results. [ABSTRACT FROM AUTHOR]

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 the present article, the problem of construction of the optimal quadrature formula for numerical integration of the right Riemann–Liouville fractional integral in the Hilbert space of real-valued functions is discussed. The error of the quadrature formula is bounded from above by the norm of the error functional. Initially, the norm of the error functional is found using the extremal function of the error functional of the quadrature formula. The norm of the error functional is a multivariate function with respect to the coefficients of the quadrature formula. Then, the Lagrange function is constructed to find the conditional minimum of the norm of the error functional. Thereby, a system of linear equations is obtained for the coefficients of the optimal quadrature formula. The existence and uniqueness of the solution of the obtained system are studied. It is used the discrete analogue of the differential operator to solve the obtained system. The analytical forms of the coefficients of the optimal quadrature formula are obtained. The obtained optimal quadrature formula is used in the numerical calculation for the Riemann–Liouville fractional integral of several functions. The errors in the numerical results are analyzed. [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, the Strichartz’s result of the exponential integrability of fractional integral operators is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that p i ≠ 1 for all i = 1 , … , m and extend some known 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]

This note is devoted to the study of Hytönen's extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calderón-Zygmund operators, fractional integrals and bilinear Calderón-Zygmund operators. [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]

In order to unify the results of boundedness for fractional integral I α and modified fractional integral I ~ α on Morrey space L p , λ (R n) and central Morrey space B p , λ (R n) , we introduced B σ -Morrey space B σ (L p , λ) (R n) . On the other hand, recently, we defined the d-modified fractional integrals I ~ α , d and proved several estimates of boundedness of I ~ α , d on the central Morrey spaces B p , λ (R n) , using the central Campanato spaces Λ p , λ (d) (R n) , the generalized σ -Lipschitz spaces Lip β , σ (d) (R n) and so on. In this paper, we will consider the results of the boundedness for I ~ α , d on Morrey space L p , λ (R n) , by showing the estimates for I ~ α , d on B σ -Morrey space B σ (L p , λ) (R n) . [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 article, we investigate the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo-Fabrizio type. In order to study this problem we used a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo-Fabrizio, therefore, so we transformed the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and taking sufficient conditions existence and uniqueness of this solution is proven based on the results obtained. The analytical study is followed by a complete numerical study. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 , Ω ⊂ R n be a bounded non-tangentially accessible domain, and p (·) : R n → (0 , ∞) a variable exponent function satisfying 0 < p - ≤ p + < ∞ , where p - : = ess inf x ∈ R n p (x) and p + : = ess sup x ∈ R n p (x) . Assume that L D is a second order divergence form elliptic operator having real-valued, bounded, and measurable coefficients on L 2 (Ω) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors establish the molecular characterization for the variable Hardy space H L D p (·) (Ω) associated with L D . In particular, when L D is non-negative and self-adjoint, the authors further obtain the atomic and the maximal function characterization of H L D p (·) (Ω) . Secondly, the authors introduce the “geometrical” variable Hardy space H r p (·) (Ω) and its local version h r p (·) (Ω) by restricting any element of the variable Hardy space H p (·) (R n) and its local version h p (·) (R n) , respectively, to Ω , and then show that, when p - ∈ (n n + θ , 1 ] , H p (·) (Ω) = H r p (·) (Ω) = H L D p (·) (Ω) = h r p (·) (Ω) with equivalent quasi-norms, where H p (·) (Ω) denotes the variable Hardy space on Ω and θ ∈ (0 , 1 ] is the critical index of Hölder continuity for the heat kernels { p t , L D } t > 0 generated by L D . Thirdly, as applications, the authors prove the boundedness of the Riesz transform ∇ L D - 1 / 2 on the variable Lebesgue space L p (·) (Ω) when 1 < p - ≤ p + ≤ 2 , and from H L D p (·) (Ω) to L p (·) (Ω) when 0 < p - ≤ p + ≤ 1 , or to H p (·) (Ω) when n n + 1 < p - ≤ p + ≤ 1 . Meanwhile, the authors also show the boundedness of the fractional integral L D - β from L p (·) (Ω) to L q (·) (Ω) , when 1 < p (·) < q (·) < ∞ , and from H L D p (·) (Ω) to L q (·) (Ω) , when 0 < p (·) ≤ 1 < q (·) < ∞ , or to H L D q (·) (Ω) , when 0 < p (·) < q (·) ≤ 1 , where β ∈ (0 , n 2) and 1 p (·) - 1 q (·) = 2 β n . As a corollary, global gradient estimates, in both L p (·) (Ω) when 1 < p - ≤ p + ≤ 2 and H p (·) (Ω) when n n + 1 < p - ≤ p + ≤ 1 , for the inhomogeneous Dirichlet problem of L D on Ω are obtained. Finally, a div-curl lemma in the space H r p (·) (Ω) is established. [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]

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calderón–Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results for these operators in the usual versions of weighted Morrey spaces, neither restricted to the Sobolev relation of the exponents nor to the one-weighted setting. [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]

We consider the commutators [b, T] and [ b , I ρ ] , where T is a Calderón–Zygmund operator, I ρ is a generalized fractional integral operator and b is a function in the closure of C comp ∞ (R n) with respect to generalized Campanato spaces. We give a necessary and sufficient condition for the compactness of [b, T] and [ b , I ρ ] on Orlicz-Morrey spaces. The Orlicz-Morrey spaces unify Orlicz and Morrey spaces, and the Campanato spaces unify BMO and Lipschitz spaces. Therefore, our results contain many previous results as corollaries. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to proving the existence of solutions to a class of initial value problems for nonlinear fractional Hybrid implicit differential equations with a ψ -Hilfer fractional derivative. The result is based on a fixed point theorem due to Dhage. Examples are provided to demonstrate the result. [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]

The subject of this note is the mixed Katugampola fractional integral of a bivariate function defined on a rectangular region in the Cartesian plane. This is a natural extension of the Katugampola fractional integral of a univariate function—a concept well-received in the recent literature on fractional calculus and its applications. It is shown that the mixed Katugampola fractional integral of a prescribed bivariate function preserves properties such as boundedness, continuity and bounded variation of the function. Furthermore, we estimate fractal dimension of the graph of the mixed Katugampola integral of a continuous bivariate function. Some examples for bivariate functions that are not of bounded variation but with graphs having box dimension 2 are constructed. The findings in the current note may be viewed as a sequel to our work reported in [Appl. Math. Comp., 339, 2018, pp. 220–230]. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new fractional integral (FI) associated with the Caputo–Fabrizio fractional derivative. As a theoretical example, we have solved a fractional boundary value problem (BVP) using the proposed FI. A Matlab script to solve this BVP is also provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new version of the Bihari inequality with singular kernel and give a simple proof of the fractional Gronwall lemma. Our new ideas rest on the use of Young's and Hölder's inequalities to simplify the complex inequalities. Based on this new type of Bihari inequality we can relax many results of fractional differential equations and inclusions and stochastic differential equations. Also, the obtained inequalities can be used to analyze a specific class of fractional differential equations, both linear and nonlinear. Using the Caputo fractional derivative, the study of an initial valued problem for a fractional differential equation provides some topological proprieties for the solution set, and shows it is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser's theorem on the solution structure of the ordinary differential equation and relax some results about the fractional differential equation. Also, we establish existence results for Caputo fractional stochastic differential equations. Finally, we study the existence of solution for fractional differential inclusion in Banach lattice. [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]

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both. [ABSTRACT FROM AUTHOR]