The paper presents a new fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as special cases. Conditions are given for such a generalized fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for th...

The authors shall discuss Heinz inequalities involving Riemann-Liouville fractional integrals for certain unitarily invariant norms. By using the convexity of function and fractional Hermit-Hadamard integral inequality, some refinements of Heinz inequalities are derived.

Let D denote the Riemann-Liouville fractional differential operator of order α. Let 1 < α < 2 and 0 < β < α. Define the operator L by L = D − aD where a ∈ R. We give sufficient conditions for the existence of solutions of the nonlinear fractional boundary value problem Lu(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) = 0.

The principal motivation of this paper is to establish a new integral equality related k-Riemann Liouville fractional operator. Employing equality, we present several inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard inequality. Additionally, some novel cases the established results different kinds derived. This sums up Riemann–Liouville and Hermit...

Abstract In this paper, we first provide a short summary of the main properties so-called general fractional derivatives with Sonin kernels introduced so far. These are integro-differential operators defined as compositions order derivative and an integral operator convolution type. Depending on succession these operators, Riemann-Liouville Caputo types were studied. The objective paper is cons...

and Applied Analysis 3 2. Background Materials and Lemmas For the convenience of the readers, in this section, we provide definitions of RiemannLiouville fractional integral and fractional derivative and some of their basic properties which will be helpful in the forth coming investigations. Definition 2.1 see 2, 5 . For a function φ : a,∞ → R, the Riemann-Liouville fractional integral of order...

We aim to introduce extended generalized Mittag-Leffler function (EGMLF) via the Beta and obtain certain integral differential representation of them. Further, we present some formulas Riemann-–Liouville fractional integration differentiation operators. Also, derive various transforms, including Euler transform, Laplace Whittakar transform K-transform. The operator images are expressed in terms...

In this study, we present a new notion of nonlocal closed boundary conditions. Equipped with these conditions, discuss the existence solutions for mixed nonlinear differential equation involving right Caputo fractional derivative operator, and left Riemann–Liouville integral operators different orders. We apply decent fruitful approach fixed point theory to establish desired results. Examples a...

In this paper, we obtained some new estimates on generalization of Hadamard, Ostrowski and Simpson-like type inequalities for harmonically quasi-convex functions via Riemann Liouville fractional integral.

This paper considers the Riemann–Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. ...