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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]

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]

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 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]

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 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]

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]

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]