COMPARISON THEOREMS FOR CAPUTO–HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950036 ◽  
Author(s):  
LI MA
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Fractional Derivatives ◽  
Comparison Theorems ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential ◽  
The One ◽  
Lipschitz Conditions ◽  
Theoretical Results

The main purpose of this paper is to investigate the comparison theorems for fractional differential equations involving Caputo–Hadamard fractional derivatives. First, we indicate the continuous dependence on parameters of solutions for Caputo–Hadamard fractional differential equations (C-HFDEs). Then, the first and second comparison theorems for C-HFDEs are proposed and proved, respectively. In addition, we establish the generalized comparisons for C-HFDEs under the one-side Lipschitz conditions. At last, the corresponding examples are also provided to verify the theoretical results.

Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

2020 ◽  
pp. 1-10
Author(s):  
Ziqiang Lu ◽  
Yuanguo Zhu ◽  
Jiayu Shen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Liu Process ◽  
Uncertain Dynamic Systems ◽  
Caputo’S Derivative ◽  
Fractional Differential ◽  
Lipschitz Conditions

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

COMPARISON PRINCIPLES FOR HADAMARD-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850056 ◽  
Author(s):  
CHUNTAO YIN ◽  
LI MA ◽  
CHANGPIN LI
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Fractional Derivatives ◽  
Comparison Principles ◽  
Right Hand ◽  
Fractional Differential ◽  
The Right ◽  
Continuous Dependence Of Solutions

The aim of this paper is to establish the comparison principles for differential equations involving Hadamard-type fractional derivatives. First, the continuous dependence of solutions on the right-hand side functions of Hadamard-type fractional differential equations (HTFDEs) is proposed. Then, we state and prove the first and second comparison principles for HTFDEs, respectively. The corresponding examples are provided as well.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

scholarly journals Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 701 ◽  
Author(s):  
Suphawat Asawasamrit ◽  
Sotiris Ntouyas ◽  
Jessada Tariboon ◽  
Woraphak Nithiarayaphaks
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Coupled Systems ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Hadamard Fractional Differential Equations ◽  
Special Cases ◽  
Separated Boundary Conditions ◽  
Fractional Differential

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.

Sign in / Sign up

Export Citation Format

Share Document