scholarly journals Existence and continuous dependence for weighted fractional differential equations with infinite delay

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Qixiang Dong
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Infinite Delay ◽  
Fractional Differential

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.

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.

scholarly journals Existence and Compactness Results for a System of Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Cheikh Guendouz ◽  
Jamal Eddine Lazreg ◽  
Juan J. Nieto ◽  
Abdelghani Ouahab
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Solution Sets ◽  
Fractional Differential ◽  
Continuous Dependence Of Solutions

The existence and uniqueness, boundedness, and continuous dependence of solutions for fractional differential equations with Caputo fractional derivative is proven by Perov’s fixed point theorem in vector Banach spaces. We study the existence and compactness of solution sets and the u.s.c. of operator solutions.

scholarly journals On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Maryam Aleem ◽  
Mujeeb Ur Rehman ◽  
Jehad Alzabut ◽  
Sina Etemad ◽  
Shahram Rezapour
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Existence Results ◽  
Existence Theory ◽  
Fractional Operator ◽  
General Boundary Conditions ◽  
Special Cases ◽  
Fractional Differential ◽  
Continuous Dependence Of Solutions

AbstractIn this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

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.

scholarly journals Weighted fractional differential equations with infinite delay in Banach spaces

2016 ◽  
Vol 14 (1) ◽  
pp. 370-383 ◽  
Author(s):  
Qixiang Dong ◽  
Can Liu ◽  
Zhenbin Fan
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Infinite Delay ◽  
Type Inequality ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential

AbstractThis paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

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