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 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 Existence Results for Singular Boundary Value Problem of Nonlinear Fractional Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yujun Cui
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Existence Results ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem: , , , where , is the Riemann-Liouville fractional derivative.

scholarly journals Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions

2020 ◽  
Author(s):  
Alberto Cabada ◽  
Om Kalthoum Wanassi
Keyword(s):  
Nonlinear Boundary ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Index Theory ◽  
Existence Results ◽  
Nonlinear Fractional Differential Equation ◽  
Integral Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Optimal Set

This paper deals with the study of the existence and non existence of solutions of a three parameter's family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the $\alpha$-Riemann-Liouville fractional derivative, with $\alpha \in (1,2]$. In order to deduce the existence and non existence results, we first study the linear equation, by deducing the main properties of the related Green's functions. We obtain the optimal set of parameters where the Green's function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

scholarly journals Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 255 ◽  
Author(s):  
Alberto Cabada ◽  
Om Kalthoum Wanassi
Keyword(s):  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Index Theory ◽  
Existence Results ◽  
Nonlinear Fractional Differential Equation ◽  
Integral Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
The Green Function ◽  
Optimal Set

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the α -Riemann-Liouville fractional derivative, with α ∈ ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

scholarly journals On Antiperiodic Nonlocal Three-Point Boundary Value Problems for Nonlinear Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Bashir Ahmad ◽  
Jorge Losada ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Results ◽  
Point Boundary ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential ◽  
The Given

We introduce boundary value conditions involving antiperiodic and nonlocal three-point boundary conditions. We solve a nonlinear fractional differential equation supplemented with those conditions. We obtain some existence results for the given problem by applying some standard tools of fixed point theory. These results are well illustrated with the aid of examples.

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.

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