scholarly journals Maximal and minimal iterative positive solutions for $ p $-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

2021 ◽  
Vol 6 (11) ◽  
pp. 12583-12598
Author(s):  
Limin Guo ◽  
◽  
Lishan Liu ◽  
Ying Wang ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Nonlinear Term ◽  
Iterative Schemes ◽  
Hadamard Fractional Differential Equations ◽  
Equation Boundary ◽  
Derivative Term ◽  
Fractional Differential

<abstract><p>In this paper, the maximal and minimal iterative positive solutions are investigated for a singular Hadamard fractional differential equation boundary value problem with a boundary condition involving values at infinite number of points. Green's function is deduced and some properties of Green's function are given. Based upon these properties, iterative schemes are established for approximating the maximal and minimal positive solutions.</p></abstract>

scholarly journals Maximal and minimal iterative positive solutions for singular infinite-point p-Laplacian fractional differential equations

2018 ◽  
Vol 23 (6) ◽  
pp. 851-865 ◽  
Author(s):  
Limin Guo ◽  
Lishan Liub
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Green’S Function ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Green's Function ◽  
Fractional Differential Equations ◽  
Iterative Schemes ◽  
Infinite Point ◽  
Fractional Differential

The existence of maximal and minimal positive solutions for singular infinite-point p-Laplacian fractional differential equation is investigated in this paper. Green's function is derived, and some properties of Green's function are obtained. Based upon these properties of Green's function, the existence of maximal and minimal positive solutions is obtained, and iterative schemes are established for approximating the maximal and minimal positive solutions.

scholarly journals The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Fuquan Jiang ◽  
Xiaojie Xu ◽  
Zhongwei Cao
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Boundary Value ◽  
Singular Boundary ◽  
Multiple Positive Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Equation Boundary ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Positive Properties

We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem:D0+αu(t)+f(t,u(t))+e(t)=0,0<t<1,u(0)=u'(0)=⋯=u(n-2)(0)=0,u(1)=βu(η), wheren-1<α≤n,n≥3,0<β≤1,0≤η≤1,D0+αis the standard Riemann-Liouville derivative. Here our nonlinearityfmay be singular atu=0. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.

scholarly journals The Green's function for Caputo fractional boundary value problem with a convection term

2022 ◽  
Vol 7 (4) ◽  
pp. 4887-4897
Author(s):  
Youyu Wang ◽  
◽  
Xianfei Li ◽  
Yue Huang
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Green's Function ◽  
Operator Theory ◽  
Fractional Boundary Value Problem ◽  
Convection Term ◽  
New Ideas ◽  
Sturm Liouville ◽  
Fractional Differential

<abstract><p>By using the operator theory, we establish the Green's function for Caputo fractional differential equation under Sturm-Liouville boundary conditions. The results are new, the method used in this paper will provide some new ideas for the study of this kind of problems and easy to be generalized to solving other problems.</p></abstract>

scholarly journals Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1944
Author(s):  
Tohru Morita ◽  
Ken-ichi Sato
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Fractional Differential Equation ◽  
Green's Function ◽  
Nonstandard Analysis ◽  
Polynomial Coefficients ◽  
Particular Solution ◽  
Fractional Differential ◽  
Detailed Derivation

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

scholarly journals Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xinguang Zhang ◽  
Lixin Yu ◽  
Jiqiang Jiang ◽  
Yonghong Wu ◽  
Yujun Cui
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Operator ◽  
Nonlinear Term ◽  
Point Index ◽  
Interesting Point ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.

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