Riemann-Liouville Fractional Differential Equations Research Articles

On Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

A Signed Maximum Principle for Boundary Value Problems for Riemann–Liouville Fractional Differential Equations with Analogues of Neumann or Periodic Boundary Conditions

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for which the sufficient conditions can be verified. To show an application of the signed maximum principle, a method of upper and lower solutions coupled with monotone methods is developed to obtain sufficient conditions for the existence of a maximal solution and a minimal solution of a nonlinear boundary value problem. A specific example is provided to show that sufficient conditions for the nonlinear problem can be realized.

Lipschitz Stability for Impulsive Riemann–Liouville Fractional Differential Equations

Initial and impulsive conditions for initial value problems of systems of nonlinear impulsive Riemann–Liouville fractional differential equations are introduced. The case when the lower limit of the fractional derivative is changed at each time point of the impulses is studied. In the case studied, the solution has a singularity at the initial time and at any point of the impulses. This leads to the need to appropriately generalize the classical concept of Lipschitz stability. Two derivative types of Lyapunov functions are utilized in order to deduce sufficient conditions for the new stability concept. Three examples are provided for illustration purpose of the theoretical results.

A study for a higher order Riemann-Liouville fractional differential equation with weakly singularity

<abstract><p>In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems.</p></abstract>

Approximate controllability of higher order Riemann–Liouville fractional systems via cosine family

This work studies the controllability of a class of Riemann–Liouville fractional differential equations of order in a new Banach space using fractional cosine family. To derive the existence of solutions, we construct the definition of mild solutions of the system using Laplace transform theory. Then, by assuming the approximate controllability of corresponding linear equation, we prove that the nonlinear dynamical system is approximately controllable using an iterative technique. Lastly, we give an example to illustrate the established theory.

Maximum, anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $1<\alpha \le 2$, and $BC$ denotes a linear boundary operator, then there exists $\mathcal{B} >0$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

Lyapunov-Type Inequalities for Systems of Riemann-Liouville Fractional Differential Equations with Multi-Point Coupled Boundary Conditions

We consider a system of Riemann–Liouville fractional differential equations with multi-point coupled boundary conditions. Using some techniques from matrix analysis and the properties of the integral operator defined on two Banach spaces, we establish some Lyapunov-type inequalities for the problem considered. Moreover, the comparison between two Lyapunov-type inequalities is given under certain special conditions. The inequalities obtained compliment the existing results in the literature.

Existence, stability and global attractivity results for nonlinear Riemann-Liouville fractional differential equations

Existence, attractivity, and stability of solutions of a non-linear fractional differential equation of Riemann-Liouville type are proved using the classical Schauder fixed point theorem and a fixed point result due to Dhage. The results are illustrated with examples.

Refinable Trapezoidal Method on Riemann–Stieltjes Integral and Caputo Fractional Derivatives for Non-Smooth Functions

The Caputo fractional α-derivative, 0<α<1, for non-smooth functions with 1+α regularity is calculated by numerical computation. Let I be an interval and Dα(I) be the set of all functions f(x) which satisfy f(x)=f(c)+f′(c)(x−a)+gc(x)(x−c)|(x−c)|α, where x,c∈I and gc(x) is a continuous function for each c. We first extend the trapezoidal method on the set Dα(I) and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is kα, where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown.

$\alpha$-$(h,e)$-convex operators and applications for Riemann-Liouville fractional differential equations

In this paper, we consider a class of $\alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $\alpha> 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $\alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $\alpha$-$(h,e)$-convex operators fixed point theorem.

On Multiple Positive Solutions for Singular Fractional Boundary Value Problems with Riemann-Stieltjes Integrals

In this paper, the existence result of at least two positive solutions is obtained for a nonlinear Riemann-Liouville fractional differential equation subject to nonlocal boundary conditions, where fractional derivatives and Riemann-Stieltjes integrals are involved. The nonlinearity possesses singularities on both its time and space variables. The discussion is based on the fixed point index theory on cones.

Measure of non-compactness for nonlocal boundary value problems via $ (k, \psi) $-Riemann-Liouville derivative on unbounded domain

<abstract><p>In this paper, we investigate the existence result for $ (k, \psi) $-Riemann-Liouville fractional differential equations via nonlocal conditions on unbounded domain. The main result is proved by applying a fixed-point theorem for Meir-Keeler condensing operators with a measure of noncompactness. Finally, two numerical examples are also demonstrated to confirm the usefulness and applicability of our theoretical results.</p></abstract>

ON A SYSTEM OF COUPLED NONLOCAL SINGULAR FRACTIONAL BOUNDARY VALUE PROBLEMS WITH <i>δ</i>-LAPLACIAN OPERATORS

We investigate the existence of at least one or two positive solutions of a system of two Riemann-Liouville fractional differential equations with $ \delta $-Laplacian operators and singular nonlinearities, supplemented with coupled nonlocal boundary conditions which contain Riemann-Stieltjes integrals and several fractional derivatives of different orders. We apply the Guo-Krasnosel'skii fixed point theorem in the proof of our main existence results.

The existence and stability results of multi-order boundary value problems involving Riemann-Liouville fractional operators

<abstract><p>In this paper, a general framework for the fractional boundary value problems is presented. The problem is created by Riemann-Liouville type two-term fractional differential equations with a fractional bi-order setup. Moreover, the boundary conditions of the suggested system are considered as mixed Riemann-Liouville integro-derivative conditions with four different orders, which it cover a variety of specific instances previously researched. Further, the provided problem's Hyers-Ulam stability and the possibility of a fixed-point approach solution are both investigated. Finally, to support our theoretical findings, an example is developed.</p></abstract>

Existence of solutions for a higher order Riemann–Liouville fractional differential equation by Mawhin's coincidence degree theory

In this paper, we investigate the existence of at least one solution to the following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition at resonance: by using Mawhin's coincidence degree theory. Here, is the standard Riemann–Liouville fractional derivative of order , and is the Riemann–Stieltjes integral of with respect to . Our choice of in the boundary condition can be any integer between 0 and , which supplements many boundary conditions assumed in the literature. Several examples are given to strengthen our result.

Lyapunov-type inequalities for $(\mathtt{n},\mathtt{p})$-type nonlinear fractional boundary value problems

This paper establishes Lyapunov-type inequalities for a family of two-point $(\mathtt{n},\mathtt{p})$-type boundary value problems for Riemann-Liouville fractional differential equations. To demonstrate how the findings can be applied, we provide a few examples, one of which is a fractional differential equation with delay.

Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Boundary Value Problems

This paper investigates the sufficient conditions for the existence and uniqueness of a class of Riemann-Liouville fractional differential equations of variable order with fractional boundary conditions. The problem is converted into differential equations of constant orders by combining the concepts of generalized intervals and piecewise constant functions. We derive the required conditions for ensuring the uniqueness of the problem in order to utilize the Banach fixed point theorem. The stability of the obtained solution in the Ulam-Hyers-Rassias (UHR) sense is also investigated, and we finally provide an illustrative example.

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

In this paper, we study the existence of positive solutions for a system of fractional differential equations with ρ-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by the presence of a key parameter p. This power parameter p is important to enable researchers to choose an adequate notion of the derivative that properly represents the reality under study, to provide good mathematical models, and to predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, we solve a Caputo and a Riemann–Liouville fractional differential equation.

On a Fractional Differential Equation with r-Laplacian Operator and Nonlocal Boundary Conditions

We study the existence and multiplicity of positive solutions of a Riemann-Liouville fractional differential equation with r-Laplacian operator and a singular nonnegative nonlinearity dependent on fractional integrals, subject to nonlocal boundary conditions containing various fractional derivatives and Riemann-Stieltjes integrals. We use the Guo–Krasnosel’skii fixed point theorem in the proof of our main results.