scholarly journals Positive and Nondecreasing Solutions to anm-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
I. J. Cabrera ◽  
J. Harjani ◽  
K. B. Sadarangani
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Partially Ordered Sets ◽  
Point Boundary ◽  
Ordered Sets ◽  
Nonlinear Fractional Differential Equation ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractionalm-point boundary value problem:D0+αu(t)+f(t,u(t))=0,  0<t<1,  2<α≤3,  u(0)=u'(0)=0,  u'(1)=∑i=1m-2aiu'(ξi), whereD0+αdenotes the standard Riemann-Liouville fractional derivative,f:[0,1]×[0,∞)→[0,∞)is a continuous function,ai≥0fori=1,2,…,m-2, and0<ξ1<ξ2<⋯<ξm-2<1. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also presented to illustrate the main results.

scholarly journals Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. Caballero ◽  
I. Cabrera ◽  
K. Sadarangani
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Nonlinear Fractional Differential Equation ◽  
Integral Boundary ◽  
Fractional Differential

We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions, , , where , and is the Caputo fractional derivative and is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.

scholarly journals Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Chen Yang ◽  
Jieming Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Point Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0<t<1, n-1<α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

scholarly journals Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-8 ◽  
Author(s):  
Moustafa El-Shahed
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Nonexistence ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem:D0+αu(t)+λa(t) f(u(t))=0, 0<t<1, u(0)=u′(0)=u′(1)=0,where2<α<3is a real number andD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.

scholarly journals Some existence results on nonlinear fractional differential equations

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120144 ◽  
Author(s):  
Dumitru Baleanu ◽  
Shahram Rezapour ◽  
Hakimeh Mohammadi
Keyword(s):  
Existence And Uniqueness ◽  
Boundary Value ◽  
Existence Results ◽  
Point Boundary ◽  
Nonlinear Fractional Differential Equation ◽  
Equation Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Fixed Point Methods ◽  
Uniqueness Of A Solution

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D α u ( t )= f ( t , u ( t )) with a Riemann–Liouville fractional derivative via the different boundary-value problems u (0)= u ( T ), and the three-point boundary condition u (0)= β 1 u ( η ) and u ( T )= β 2 u ( η ), where T >0, t ∈ I =[0, T ], 0< α <1, 0< η < T , 0< β 1 < β 2 <1.

scholarly journals Existence and Uniqueness of Positive Solution for a Boundary Value Problem of Fractional Order

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
J. Caballero ◽  
J. Harjani ◽  
K. Sadarangani
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Partially Ordered Sets ◽  
Fractional Boundary Value Problem ◽  
Ordered Sets ◽  
Liouville Fractional Derivative

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem:D0+αu(t)+f(t,u(t))=0,0≤t≤1,3<α≤4,u(0)=u′(0)=u″(0)=u″(1)=0, whereD0+αdenotes the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results.

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