scholarly journals Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3031
Author(s):  
Weiwei Liu ◽  
Lishan Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Lower Order ◽  
Higher Order ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Weaker Conditions

This paper deals with the study of the existence of positive solutions for a class of nonlinear higher-order fractional differential equations in which the nonlinear term contains multi-term lower-order derivatives. By reducing the order of the highest derivative, the higher-order fractional differential equation is transformed into a lower-order fractional differential equation. Then, combining with the properties of left-sided Riemann–Liouville integral operators, we obtain the existence of the positive solutions of fractional differential equations utilizing some weaker conditions. Furthermore, some examples are given to demonstrate the validity of our main results.

scholarly journals Eigenvalue of Fractional Differential Equations withp-Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Wenquan Wu ◽  
Xiangbing Zhou
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Upper And Lower Solutions ◽  
Laplacian Operator ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

We investigate the existence of positive solutions for the fractional order eigenvalue problem withp-Laplacian operator-𝒟tβ(φp(𝒟tαx))(t)=λf(t,x(t)),  t∈(0,1),  x(0)=0,  𝒟tαx(0)=0,  𝒟tγx(1)=∑j=1m-2‍aj𝒟tγx(ξj), where𝒟tβ,  𝒟tα,  𝒟tγare the standard Riemann-Liouville derivatives andp-Laplacian operator is defined asφp(s)=|s|p-2s,  p>1.f:(0,1)×(0,+∞)→[0,+∞)is continuous andfcan be singular att=0,1andx=0.By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established.

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 Positive Solutions for Nonlinear Fractional Differential Equations with Boundary Conditions Involving Riemann-Stieltjes Integrals

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Jiqiang Jiang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Integral Boundary Value Problems

We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.

scholarly journals Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Hongliang Gao ◽  
Xiaoling Han
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary conditionD0+αu(t)+a(t)f(t,u(t))=0,0<t<1,u(0)=0,u(1)=∑i=1∞αiu(ξi)is considered, where1<α≤2is a real number,D0+αis the standard Riemann-Liouville differentiation, andξi∈(0,1),  αi∈[0,∞)with∑i=1∞αiξiα-1<1,a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞)).

scholarly journals Uniqueness of Iterative Positive Solution to Nonlinear Fractional Differential Equations with Negatively Perturbed Term

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Jingjing Tan ◽  
Meixia Li ◽  
Aixia Pan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Convergence Rate ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Iterative Scheme ◽  
Approximation Error ◽  
Fractional Differential

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error.

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