Riemann-Liouville Fractional Boundary Value Problems Research Articles

Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems

In this paper, we provide sufficient conditions for the existence, uniqueness and global behavior of a positive continuous solution to some nonlinear Riemann-Liouville fractional boundary value problems. The nonlinearity is allowed to be singular at the boundary. The proofs are based on perturbation techniques after reducing the considered problem to the equivalent Fredholm integral equation of the second kind. Some examples are given to illustrate our main results.

On a System of Riemann–Liouville Fractional Boundary Value Problems with ϱ-Laplacian Operators and Positive Parameters

In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with ϱ-Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply the Schauder fixed point theorem in the proof of the existence result.

Approximate Solutions of an Extended Multi-Order Boundary Value Problem by Implementing Two Numerical Algorithms

In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann–Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions.

Hartman-Type and Lyapunov-Type Inequalities for a Fractional Differential Equation with Fractional Boundary Conditions

We prove Hartman-type and Lyapunov-type inequalities for a class of Riemann–Liouville fractional boundary value problems with fractional boundary conditions. Some applications including a lower bound for the corresponding eigenvalue problem are obtained.

Existence Result and Uniqueness for Some Fractional Problem

In this article, by the use of the lower and upper solutions method, we prove the existence of a positive solution for a Riemann–Liouville fractional boundary value problem. Furthermore, the uniqueness of the positive solution is given. To demonstrate the serviceability of the main results, some examples are presented.

Lyapunov-type inequalities for Riemann-Liouville type fractional boundary value problems

In this article we establish a Lyapunov-type inequality for two-point Riemann-Liouville fractional boundary value problems associated with well-posed Robin boundary conditions. To illustrate the applicability of established result, we deduce criterion for the nonexistence of real zeros of a Mittag-Leffler function.

Monotone iterative method for two-point fractional boundary value problems

In this work, we deal with two-point Riemann–Liouville fractional boundary value problems. Firstly, we establish a new comparison principle. Then, we show the existence of extremal solutions for the two-point Riemann–Liouville fractional boundary value problems, using the method of upper and lower solutions. The performance of the approach is tested through a numerical example.

Extremal points for fractional boundary value problems

This article is concerned with characterizing the first extremal point, b0, for a Riemann–Liouville fractional boundary value problem, Dα0+y + p(t)y = 0, 0 < t < b, y(0) = y′(0) = y″(b) = 0, 2 < α ≤ 3, by applying the theory of u0-positive operators with respect to a suitable cone in a Banach space. The key argument is that a mapping, which maps a linear, compact operator, depending on b to its spectral radius, is continuous and strictly increasing as a function of b. Furthermore, an application to a nonlinear case is given.

Positive solutions for some Riemann-Liouville fractional boundary value problems

Positive solutions for some Riemann-Liouville fractional boundary value problems

Existence and Nonexistence of Positive Solutions for Coupled Riemann-Liouville Fractional Boundary Value Problems

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.

A Chebyshev spectral method for solving Riemann–Liouville fractional boundary value problems

The authors derive a series of explicit formulas to approximate the Riemann–Liouville derivative and integral of arbitrary order by shifted Chebyshev polynomials. This is then applied to solve boundary value problems involving Riemann–Liouville derivatives.

Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem

By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.