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 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 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 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 On multi-term proportional fractional differential equations and inclusions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Wafa Shammakh ◽  
Hadeel Z. Alzumi ◽  
Zahra Albarqi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Existence Results ◽  
Integral Boundary ◽  
Fractional Differential ◽  
Differential Equations And Inclusions

AbstractThe aim of this paper is to study new nonlocal boundary value problems of fractional differential equations and inclusions supplemented with slit-strips integral boundary conditions. Based on the functional analysis tools, the existence results for a nonlinear boundary value problem involving a proportional fractional derivative are presented. In addition to that, the extension of the problem at hand to its inclusion case is discussed. The obtained results are very interesting and are well illustrated with examples.

scholarly journals Positive Solutions for Boundary Value Problems of Fractional Differential Equation with Integral Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Qiao Sun ◽  
Hongwei Ji ◽  
Yujun Cui
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Existence Results ◽  
Integral Boundary ◽  
Fractional Differential

By using two fixed-point theorems on cone, we discuss the existence results of positive solutions for the following boundary value problem of fractional differential equation with integral boundary conditions: D0+αx(t)+a(t)f(t,x(t))=0, t∈(0,1), x(0)=x′(0)=0, and x(1)=∫01x(t)dA(t).

scholarly journals Existence Results for Fractional Differential Inclusions with Multivalued Term Depending on Lower-Order Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Xiaoyou Liu ◽  
Zhenhai Liu
Keyword(s):  
Boundary Conditions ◽  
Lower Order ◽  
Differential Inclusions ◽  
Integral Boundary Conditions ◽  
Existence Results ◽  
Inclusion Problem ◽  
Fractional Differential Inclusions ◽  
Integral Boundary ◽  
Separated Boundary Conditions ◽  
Fractional Differential

This paper is concerned with a class of fractional differential inclusions whose multivalued term depends on lower-order fractional derivative with fractional (non)separated boundary conditions. The cases of convex-valued and non-convex-valued right-hand sides are considered. Some existence results are obtained by using standard fixed point theorems. A possible generalization for the inclusion problem with integral boundary conditions is also discussed. Examples are given to illustrate the results.

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