Riemann-Liouville Integral Operator Research Articles

Some integral inequalities for (k, s) – Riemann-Liouville fractional operators

In this paper, by using a new fractional integral approach of M.Z. Sarikaya et al., we establish new generalized fractional integral inequalities involving (k, s) – Riemann-Liouville integral operators. Some classical and non classical integral inequalities can be deduced as some special cases.

English

A new hybrid of block-pulse functions and Boubaker polynomials is constructed to solve the inequality constrained fractional optimal control problems (FOCPs) with quadratic performance index and fractional variational problems (FVPs). First, the general formulation of the Riemann-Liouville integral operator for Boubaker hybrid function is presented for the first time. Then it is applied to reduce the problems to optimization problems, which can be solved by the existing method. In this way we find the extremum value of FOCPs without adding slack variables to inequality trajectories. Also we show that if the number of bases is increased, the used approximations in this method are convergent. The applicability and validity of the method are shown by numerical results of some examples, moreover, a comparison with the existing results shows the preference of this method.

Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Holder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

Fractional Hermite-Hadamard type inequalities for n-times log-convex functions

In this paper, we establish some Hermite-Hadamard type inequalities for function whose n-th derivatives are logarithmically convex by using Riemann-Liouville integral operator.

Generalized integral inequalities for fractional calculus

In this paper, we present a variety of integral inequalities in Lp and Lp,r spaces for the integral operator involving generalized Mittag-Leffler function in its kernel, Hilfer fractional derivative, generalized Riemann-Liouville and Riemann-Liouville k-fractional integral operators.

On the Hermite-Hadamard-Fejer-type inequalities for co-ordinated convex functions via fractional integrals

In this paper, using Riemann-Liouville integral operators, we establish new fractional in- tegral inequalities of Hermite-Hadamard-Fejer type for co-ordinated convex functions on a rectangle of R^2. The results presented here would provide extensions of those given in earlier works.

Which Functions are Fractionally Differentiable?

We examine the existence of fractional derivatives of a function in terms of the pointwise convergence or equiconvergence of certain improper integrals containing this function. The fractional differentiation operator is treated as the inverse to the Riemann-Liouville integral operator. Technically, we give a description of the range of the Riemann-Liouville operator. The results are reformulated also for Riemann-Liouville and Caputo fractional derivatives.

Integral equations of fractional order in Lebesgue spaces

Abstract We discuss solvability of a nonlinear Riemann-Liouville integral equation in Lebesgue spaces. We treat the Volterra equations of the first and the second types by applying boundedness criteria for the Riemann-Liouville integral operator. The existence of a solution to integral equations will follow from the Leray-Schauder Nonlinear Alternative.

Some generalized Riemann-Liouville k-fractional integral inequalities

The focus of the present study is to prove some new Polya-Szego type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operator. These inequalities are used then to establish some fractional integral inequalities of Chebyshev type.

WITHDRAWN: Riemann–Liouville fractional calculus in Morrey spaces and applications

Abstract We study the boundedness and compactness of Riemann–Liouville integral operators on the so-called Morrey spaces which are nonseparable spaces. There are no approximation or contractive skills in this kind of spaces. Moreover, unlike the use of dual or maximal point of view in integrable function spaces, the idea of our proof proceeds from the compactness of the truncated Riemann–Liouville fractional integrals by using a criterion for strongly pre-compact set. Constructing a truncated Marchaud fractional derivative function, we show the characterization of the solution to Abel’s equation in Morrey spaces. With the aid of fixed-point theorem, we establish the existence and uniqueness of solutions to a Cauchy type problem for fractional differential equations. We also give an example to illustrate the sufficiency of the conditions in our main result.

L P -solutions for fractional integral equations

Abstract In this article, we examine L p-solutions of fractional integral equations in Banach spaces involving the Riemann-Liouville integral operator. Using a compactness type condition, we obtain local and global existence of solutions. Also other types of existence and uniqueness results are established. At the end, an application is given to illustrate the main result.

Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski’s

This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.

ON WEIGHTED INEQUALITIES WITH GEOMETRIC MEAN OPERATOR

AbstractWe give a characterization of pairs of weights for the validity of weighted inequalities involving certain generalized geometric mean operators generated by some Volterra integral operators, which include the Hardy averaging operator and the Riemann–Liouville integral operators. The estimations of the constants are also discussed. Our results generalize the work done by J. A. Cochran, C.-S. Lee, H. P. Heinig, B. Opic, P. Gurka, and L. Pick.