Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem

2021 ◽  
Vol 24 (4) ◽  
pp. 1069-1093
Author(s):  
Dandan Min ◽  
Fangqi Chen
Keyword(s):  
Differential Equation ◽  
Variational Methods ◽  
Impulsive Differential Equation ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Laplacian Operator ◽  
Infinitely Many Solutions ◽  
Critical Point Theorem ◽  
Sturm Liouville ◽  
New Criteria

Abstract In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p-Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results.

scholarly journals The Solutions of Sturm-Liouville Boundary-Value Problem for Fourth-Order Impulsive Differential Equation via Variational Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yu Tian ◽  
Dongpo Sun
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Fourth Order ◽  
Multiple Solutions ◽  
Impulsive Differential Equation ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Existence Results ◽  
Sturm Liouville ◽  
Main Ideas

The Sturm-Liouville boundary-value problem for fourth-order impulsive differential equations is studied. The existence results for one solution and multiple solutions are obtained. The main ideas involve variational methods and three critical points theory.

scholarly journals APPLICATIONS OF VARIATIONAL METHODS TO BOUNDARY-VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL EQUATIONS

2008 ◽  
Vol 51 (2) ◽  
pp. 509-527 ◽  
Author(s):  
Yu Tian ◽  
Weigao Ge
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Variational Methods ◽  
Positive Solutions ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Impulsive Effects ◽  
Differential Inequalities ◽  
Existence Of Positive Solutions ◽  
Sturm Liouville

AbstractIn this paper, we investigate the existence of positive solutions to a second-order Sturm–Liouville boundary-value problem with impulsive effects. The ideas involve differential inequalities and variational methods.

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.

scholarly journals MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATION VIA VARIATIONAL METHODS

2010 ◽  
Vol 82 (3) ◽  
pp. 446-458 ◽  
Author(s):  
JUNTAO SUN ◽  
HAIBO CHEN ◽  
TIEJUN ZHOU
Keyword(s):  
Differential Equation ◽  
Variational Methods ◽  
Fourth Order ◽  
Critical Points ◽  
Impulsive Differential Equation ◽  
Classical Solutions ◽  
Multiplicity Of Solutions ◽  
Three Critical Points Theorem ◽  
New Criteria

AbstractIn this paper, we deal with the multiplicity of solutions for a fourth-order impulsive differential equation with a parameter. Using variational methods and a ‘three critical points’ theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. An example is also given in order to illustrate the main results.

scholarly journals Nontrivial Solutions for a Boundary Value Problem of nth-Order Impulsive Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jiafa Xu ◽  
Zhongli Wei
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Impulsive Differential Equation ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Degree Theory ◽  
Impulsive Effects ◽  
Nontrivial Solutions

We study the existence of nontrivial solutions for nth-order boundary value problem with impulsive effects. We utilize Leray-Schauder degree theory to establish our main results. Furthermore, our nonlinear term f is allowed to grow superlinearly and sublinearly.

scholarly journals The existence of solutions for Sturm–Liouville differential equation with random impulses and boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zihan Li ◽  
Xiao-Bao Shu ◽  
Tengyuan Miao
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Lower Solution ◽  
Iterative Sequence ◽  
Random Impulses ◽  
Sturm Liouville

AbstractIn this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

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