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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]

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]

This survey paper complements to our previous review papers on Lyapunov-type inequalities and contains some of the most recent results on these inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. In precise terms, we have included the results related to Riemann-Liouville, Caputo, mixed Riemann-Liouville and Caputo, Riesz-Caputo, ψ -Caputo, Hadamard, Katugampola, Hilfer, ψ - Hilfer, proportional, variable order Hadamard, partial, systems of Riemann-Liouville, bi-ordinal Hilfer-Katugampola, and ψ -Hilfer fractional derivative operators. The Lyapunov-type inequalities for discrete fractional boundary value problems are also presented. [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]

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]

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

In the present article, we prove some results concerning the existence of solutions for a class of initial value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivative. The results are based on Banach contraction principle and Schaefer's fixed point theorem. Examples are included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

Fractional q -calculus is considered to be the fractional analogs of q -calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) q -integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving ℏ -convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF q -integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided. [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 paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0

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.

Using the coincidence degree theory of Mawhin and constructing appropriate operators, we investigate the existence of solutions to Hadamard fractional differential equations (FRDEs) at resonance ... where 1 < γ < 2, f: [1, e]×R² → R satisfies Carathéodory conditions, ∫e1 u(t)dA(t) is the Riemann-Stieltjes integration, and (HDγu) is the Hadamard fractional derivation of u of order γ. An example is included to illustrate our result. [ABSTRACT FROM AUTHOR]

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 λ 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 FROM AUTHOR]

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 λ-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 λ-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 ≤ 2. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result. [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]

Let (Ω,F,P)$(\Omega ,\mathcal {F},\mathbb {P})$ be a complete probability space and let p(·),q(·):[0,1]→(0,∞)$p(\cdot),q(\cdot):[0,1]\rightarrow (0,\infty)$ be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces Lp(·),q(·)(Ω)$\mathcal {L}_{p(\cdot),q(\cdot)}(\Omega)$, which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces Hp(·),q(·)(Ω)$\mathcal {H}_{p(\cdot),q(\cdot)}(\Omega)$. [ABSTRACT FROM AUTHOR]

In this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay principles combined with a technique based on vector-valued metrics and convergent to zero matrices. [ABSTRACT FROM AUTHOR]

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

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.

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.

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order α where $0 \lt \alpha\unicode{x2A7D}1$ . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as $n_{k} = \frac{p(2p-1)(2-\alpha)}{p(1-\alpha)+1}\Gamma\left(\alpha+\frac{2-\alpha}{1-p}\right)\frac{\Gamma(k)}{\Gamma\left(k-1+\alpha+\frac{2-\alpha}{1-p}\right)}t$ for $\alpha \in (0,1]$ , where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for α = 1. The results corresponding to various values of α and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p , thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.