scholarly journals Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Wang ◽  
Yansheng Liu ◽  
Yujun Cui
Keyword(s):  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Point Theory ◽  
Fractional Boundary Value Problem ◽  
Fountain Theorems ◽  
Left And Right ◽  
Variant Fountain Theorems ◽  
Derivatives Of

This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: DT-αaxD0+αux=fx,ux,  x∈0,T, u0=uT=0, where α∈1/2,1, ax∈L∞0,T with a0=ess  infx∈0,Tax>0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.

EXISTENCE RESULTS FOR FRACTIONAL BOUNDARY VALUE PROBLEM VIA CRITICAL POINT THEORY

2012 ◽  
Vol 22 (04) ◽  
pp. 1250086 ◽  
Author(s):  
FENG JIAO ◽  
YONG ZHOU
Keyword(s):  
Boundary Value Problem ◽  
Critical Point ◽  
Critical Point Theory ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Point Theory ◽  
Fractional Boundary Value Problem ◽  
Fractional Differential ◽  
Left And Right ◽  
First Time

In this paper, by the critical point theory, the boundary value problem is discussed for a fractional differential equation containing the left and right fractional derivative operators, and various criteria on the existence of solutions are obtained. To the authors' knowledge, this is the first time, the existence of solutions to the fractional boundary value problem is dealt with by using critical point theory.

scholarly journals Nontrivial Solutions for a Higher Order Nonlinear Fractional Boundary Value Problem Involving Riemann-Liouville Fractional Derivatives

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Keyu Zhang ◽  
Donal O’Regan ◽  
Jiafa Xu ◽  
Zhengqing Fu
Keyword(s):  
Boundary Value Problem ◽  
Topological Degree ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Higher Order ◽  
Fractional Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Derivatives Of

In this paper using topological degree we study the existence of nontrivial solutions for a higher order nonlinear fractional boundary value problem involving Riemann-Liouville fractional derivatives. Here, the nonlinearity can be sign-changing and can also depend on the derivatives of unknown functions.

scholarly journals Existence of Solutions for Fractional Boundary Value Problem with Nonlinear Derivative Dependence

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wenzhe Xie ◽  
Jing Xiao ◽  
Zhiguo Luo
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Iterative Technique ◽  
Fractional Boundary Value Problem ◽  
Left And Right

We investigate the existence of solutions for fractional boundary value problem including both left and right fractional derivatives by using variational methods and iterative technique.

scholarly journals A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Nouara ◽  
Abdelkader Amara ◽  
Eva Kaslik ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Research Work ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Derivatives And Integrals ◽  
Fractional Differential

AbstractIn this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

scholarly journals Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuqi Wang ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Novel Approach ◽  
Banach Contraction ◽  
Left And Right ◽  
Banach Contraction Mapping Principle

AbstractIn this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

scholarly journals Existence and uniqueness of solutions for a fractional boundary value problem on a graph

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
John Graef ◽  
Lingju Kong ◽  
Min Wang
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Conditions ◽  
Point Theory ◽  
Star Graph ◽  
Fractional Boundary Value Problem ◽  
Uniqueness Of Solutions

AbstractIn this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory.

scholarly journals Research on Sturm–Liouville boundary value problems of fractional p-Laplacian equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tingting Xue ◽  
Fanliang Kong ◽  
Long Zhang
Keyword(s):  
Differential Equation ◽  
Variational Methods ◽  
Critical Point Theory ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Point Theory ◽  
Laplacian Equation ◽  
Sturm Liouville ◽  
Beta 2 ◽  
Derivatives Of

AbstractIn this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$ { D T α t ( 1 ( h ( t ) ) p − 2 ϕ p ( h ( t ) 0 C D t α u ( t ) ) ) + a ( t ) ϕ p ( u ( t ) ) = λ f ( t , u ( t ) ) , a.e.  t ∈ [ 0 , T ] , α 1 ϕ p ( u ( 0 ) ) − α 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( 0 ) ) ) = 0 , β 1 ϕ p ( u ( T ) ) + β 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( T ) ) ) = 0 , where ${}_{0}^{C}D_{t}^{\alpha }$ D t α 0 C , ${}_{t}D_{T}^{\alpha }$ D T α t are the left Caputo and right Riemann–Liouville fractional derivatives of order $\alpha \in ( {\frac{1}{2},1} ]$ α ∈ ( 1 2 , 1 ] , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.

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