scholarly journals Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ya-ling Li ◽  
Shi-you Lin
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Continuous Function ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fixed Point Theorems ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential

We study the following nonlinear fractional differential equation involving thep-Laplacian operatorDβφpDαut=ft,ut,1<t<e,u1=u′1=u′e=0,Dαu1=Dαue=0, where the continuous functionf:1,e×0,+∞→[0,+∞),2<α≤3,1<β≤2.Dαdenotes the standard Hadamard fractional derivative of the orderα, the constantp>1, and thep-Laplacian operatorφps=sp-2s. We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and thep-Laplacian operator.

scholarly journals Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jun-Rui Yue ◽  
Jian-Ping Sun ◽  
Shuqin Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii Fixed Point Theorem ◽  
Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)),  t∈[0,1],  u(0)=0,   u′(0)+u′′(0)=0,  u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

scholarly journals New Fixed Point Theorems in Orthogonal F -Metric Spaces with Application to Fractional Differential Equation

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 832
Author(s):  
Tanzeela Kanwal ◽  
Azhar Hussain ◽  
Hamid Baghani ◽  
Manuel de la Sen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Periodic Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Nonlinear Fractional Differential Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

We present the notion of orthogonal F -metric spaces and prove some fixed and periodic point theorems for orthogonal ⊥ Ω -contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness of the solution of a nonlinear fractional differential equation.

scholarly journals Existence of Concave Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation withp-Laplacian Operator

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Jinhua Wang ◽  
Hongjun Xiang ◽  
ZhiGang Liu
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Multiplicity ◽  
Fractional Differential

We consider the existence and multiplicity of concave positive solutions for boundary value problem of nonlinear fractional differential equation withp-Laplacian operatorD0+γ(ϕp(D0+αu(t)))+f(t,u(t),D0+ρu(t))=0,0<t<1,u(0)=u′(1)=0,u′′(0)=0,D0+αu(t)|t=0=0, where0<γ<1,2<α<3,0<ρ⩽1,D0+αdenotes the Caputo derivative, andf:[0,1]×[0,+∞)×R→[0,+∞)is continuous function,ϕp(s)=|s|p-2s,p>1,  (ϕp)-1=ϕq,  1/p+1/q=1. By using fixed point theorem, the results for existence and multiplicity of concave positive solutions to the above boundary value problem are obtained. Finally, an example is given to show the effectiveness of our works.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Changyou Wang ◽  
Haiqiang Zhang ◽  
Shu Wang
Keyword(s):  
Differential Equation ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Caputo Derivative ◽  
Nonlinear Term ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Control Functions ◽  
Fractional Differential

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.

Existence of positive solutions for systems of nonlinear fractional differential equation with p-Laplacian

2019 ◽  
Vol 13 (05) ◽  
pp. 2050089 ◽  
Author(s):  
S. Nageswara Rao ◽  
Meshari Alesemi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Nonlinear Fractional Differential Equation ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

In this paper, we establish sufficient conditions for the existence of positive solutions for a system of nonlinear fractional [Formula: see text]-Laplacian boundary value problems under different combinations of superlinearity and sublinearity of the nonlinearities via the Guo–Krasnosel’skii fixed point theorem. Moreover, an example is given to illustrate our results.

scholarly journals Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2231
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Power Law ◽  
Dynamic Processes ◽  
Nonlinear Fractional Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

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