## Existence of Solutions for Anti-Periodic Boundary Value Problems of Fractional Langevin Equation with p(t)-Laplacian Operator

Zhang Jifeng, Zhang Wei,, Ni Jinbo, Ren Dandan

 基金资助: 国家自然科学基金.  11801008安徽高校自然科学研究项目.  KJ2020A0291安徽理工大学研究生创新基金项目.  2021CX2117

 Fund supported: the NSFC.  11801008the Key Program of University Natural Science Research Fund of Anhui Province.  KJ2020A0291the Postgraduate Innovation Fund Project of Anhui University of Science and Technology.  2021CX2117

Abstract

This paper studies the anti-periodic boundary value problems of fractional Langevin equation with p (t)-Laplacian operator. The sufficient conditions for the existence of solutions are obtained by using Schaefer fixed point theorem, and the main result is well illustrated with the aid of an example. The results obtained in this paper extend and enrich the existing related works.

Keywords： Fractional differential equation ; Anti-periodic boundary value problem ; p (t)-Laplacian operator ; Schaefer fixed point theorem

Zhang Jifeng, Zhang Wei, Ni Jinbo, Ren Dandan. Existence of Solutions for Anti-Periodic Boundary Value Problems of Fractional Langevin Equation with p(t)-Laplacian Operator. Acta Mathematica Scientia[J], 2021, 41(4): 1024-1032 doi:

## 1 引言

Langevin方程描述了布朗运动由于流体分子的碰撞, 颗粒在流体中做无规则运动的动力学行为, 具体模型如下

$$$\frac{{{\rm d}^2}x}{{\rm d}{t^2}} + \gamma \frac{{{\rm d}x}}{{{\rm d}t}} = \Gamma (t),$$$

$$$x'' + \gamma {}^CD_{0 + }^\alpha x = \Gamma (t), \;0 < \alpha < 1,$$$

2017年, Zhou等[8]讨论了带$p$-Laplacian算子的分数阶Langevin方程反周期边值问题

$$$\left\{ \begin{array}{l} {}^CD_{0{\rm{ + }}}^\beta {\varphi _p}[({}^CD_{0{\rm{ + }}}^\alpha + \lambda )x(t)] = f(t, x(t), {}^CD_{0{\rm{ + }}}^\alpha x(t)), \;t \in (0, 1), \\ x(0) = - x(1), \;\;{}^CD_{0{\rm{ + }}}^\alpha x(0) = - {}^CD_{0{\rm{ + }}}^\alpha x(1), \end{array} \right.$$$

$$$\left\{ \begin{array}{l} D_{0{\rm{ + }}}^\beta {\varphi _{p(t)}}(D_{0{\rm{ + }}}^\alpha x(t)) + f(t, x(t)) = 0, \;t \in (0, 1), \\ x(0) = 0, \;D_{0{\rm{ + }}}^{\alpha - 1}x(1) = \gamma I_{0{\rm{ + }}}^{\alpha - 1}x(\eta ), \;D_{0{\rm{ + }}}^\alpha x(0) = 0, \end{array} \right.$$$

$$$\left\{ \begin{array}{l} {}^CD_{0 + }^\beta {\varphi _{p(t)}}[({}^CD_{0 + }^\alpha + \lambda )x(t)] = f(t, x(t), {}^CD_{0 + }^\alpha x(t)), \;t \in (0, 1), \\ x(0) = - x(1), \;\;{}^CD_{0 + }^\alpha x(0) = - {}^CD_{0 + }^\alpha x(1), \end{array} \right.$$$

## 3 主要结果

$$${}^CD_{0 + }^\beta {\varphi _{p(t)}}[({}^CD_{0 + }^\alpha + \lambda )x(t)] = h(t), \;\;t \in (0, 1),$$$

$$$x(0) = - x(1), \;{}^CD_{0 + }^\alpha x(0) = - {}^CD_{0 + }^\alpha x(1)$$$

利用算子$I_{0 + }^\beta$作用到(3.1) 式的两端, 则由引理2.1以及定义2.3知

$$$({}^CD_{0 + }^\alpha + \lambda )x(t) = \varphi _{p(t)}^{ - 1}\left( {{c_0} + I_{0 + }^\beta h(t)} \right), \;{c_0} \in {{{\Bbb R}} },$$$

$$$x(t) = {c_1} + I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}(Ah(t) + I_{0 + }^\beta h(t)) - \lambda I_{0 + }^\alpha x(t), \;{c_1} \in {{{\Bbb R}} },$$$

$$${c_1} = - \frac{1}{2}I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}{\left. {(Ah(t) + I_{0 + }^\beta h(t))} \right|_{t = 1}} + \frac{\lambda }{2}{\left. {I_{0 + }^\alpha x(t)} \right|_{t = 1}} = Bh(t),$$$

(H)     存在非负函数$\zeta , \phi , \psi \in C[0, 1]$使得

$$$\ell \left( {\frac{{3Q}}{{2\alpha }} + Q\Gamma (\alpha)}\right) + \frac{{3\lambda }}{{2\Gamma (\alpha + 1)}} + \lambda < 1$$$

关于定理的证明分为两步完成. 第一步. 证明$T$是全连续算子. 事实上, 对任意的$\delta > 0$, 定义$X$上的有界开集$\Omega = \left\{ {x \in X:{\rm{||}}x{\rm{|}}{{\rm{|}}_X} < \delta } \right\}$.$f $$\varphi _{p(t)}^{ - 1}( \cdot ) 的连续性易证 T$$ [0, 1]$上是连续的, 且存在常数$M > 0$使得

## 4 例子

$$$\left\{\begin{array}{ll} {}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}{\varphi _{( - {t^2} {+} t {+} 2)}}(({}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}{\rm{ {+} (}}{1 \mathord{\left/ {\vphantom {1 {10}}} \right. } {10}}{\rm{)}})x(t)) { = } \sin t {+} \frac{3}{{200}}x {+} \frac{1}{{100}}{}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}x, t \in (0, 1), \\ x(0) { = } - x(1), \; {}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}x(0) { = } - {}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}x(1), \end{array}\right.$$$

$r = 2, \;\zeta (t) = 1, \;\phi (t) = \frac{3}{{200}}, \;\psi (t) = \frac{1}{{100}}.$易验证

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Lin Z H . Thermodynamics and Statistical Physics. Beijing: Peking University Press, 2007

Chen W , Sun H G , Li X C . Fractional Derivative Modeling of Mechanical and Engineering Problems. Beijing: Science Press, 2010

Lutz E .

Fractional Langevin equation

Phys Rev E, 2001, 64 (5): 051106

Burov S, Barkai E. The critical exponent of the fractional Langevin equation is ac ≈ 0.402. 2007, arXiv: 0712.3407

Liu Y J .

Solvability of BVPs for impulsive fractional Langevin type equations involving three Riemann-Liouville fractional derivatives

Acta Math Sci, 2020, 40A (1): 103- 131

Pu L J , Yang X Z , Sun S Z .

Numerical analysis of a class of fractional Langevin equation by predictor-corrector method

Acta Math Sci, 2020, 40A (4): 1018- 1028

Ahmad B , Nieto J J , Alsaedi A , El-Shahed M .

A study of nonlinear Langevin equation involving two fractional orders in different intervals

Nonlinear Anal Real World Appl, 2012, 13 (2): 599- 606

Zhou H , Alzabut J , Yang L .

On fractional Langevin differential equations with anti-periodic boundary conditions

Eur Phys J Special Topics, 2017, 226 (16-18): 3577- 3590

Fazli H , Nieto J J .

Fractional Langevin equation with anti-periodic boundary conditions

Chaos Solitons Fractals, 2018, 114, 332- 337

Wang G T , Qin J F , Zhang L H , Baleanu D .

Explicit iteration to a nonlinear fractional Langevin equation with non-separated integro-differential strip-multi-point boundary conditions

Chaos Solitons Fractals, 2020, 131, 109476

Zhai C B , Li P P .

Nonnegative solutions of initial value problems for Langevin equations involving two fractional orders

Mediterr J Math, 2018, 15, 164

Baghani O .

On fractional Langevin equation involving two fractional orders

Commun Nonlinear Sci Numer Simul, 2017, 42, 675- 681

Wang J R , Li X Z .

Ulam-Hyers stability of fractional Langevin equations

Appl Math Comput, 2015, 258, 72- 83

Baghani H .

An analytical improvement of a study of nonlinear Langevin equation involving two fractional orders in different intervals

J Fixed Point Theory Appl, 2019, 21 (4): 95

Zhou Z F , Qiao Y .

Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions

Bound Value Probl, 2018, 2018, 152

Salem A , Alzahrani F , Alghamdi B .

Langevin equation involving two fractional orders with three-point boundary conditions

Differential Integral Equations, 2020, 33 (3/4): 163- 180

Shen T F , Liu W B .

Existence of solutions for fractional integral boundary value problems with p(t)-Laplacian operator

J Nonlinear Sci Appl, 2016, 9 (7): 5000- 5010

Xue T T , Liu W B , Shen T F .

Existence of solutions for fractional Sturm-Liouville boundary value problems with p(t)-Laplacian operator

Bound Value Probl, 2017, 2017, 169

Shen X H , Shen T F .

Existence of solutions for Erdélyi-Kober fractional integral boundary value problems with p(t)-Laplacian operator

Adv Difference Equa, 2020, 2020, 565

Tang X S , Wang X C , Wang Z W , Ouyang P C .

The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator

J Appl Math Comput, 2019, 61 (1/2): 559- 572

Tang X S , Luo J Y , Zhou S , Yan C Y .

Existence of positive solutions of mixed fractional integral boundary value problem with p(t)-Laplacian operator

Ric Mat, 2020,

Wang G T , Ren X Y , Zhang L H , Ahmad B .

Explicit iteration and unique positive solution for a Caputo-Hadamard fractional turbulent flow model

IEEE Access, 2019, 7, 109833- 109839

Chen Y M , Levine S , Rao M .

Variable exponent, linear growth functionals in image restoration

SIAM J Appl Math, 2006, 66 (4): 1383- 1406

Růžička M . Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer-Verlag, 2000

Zhikov V V .

Averaging of functionals of the calculus of variations and elasticity theory

Izv Akad Nauk SSSR Ser Mat, 1986, 50 (4): 675- 710

Zhang Q H .

Existence of solutions for weighted p(r)-Laplacian system boundary value problems

J Math Anal Appl, 2007, 327 (1): 127- 141

Kilbas A A , Srivastava H M , Trujillo J J .

Theory and Applications of Fractional Differential Equations

Amsterdam: Elsevier, 2006,

Zhou Y , Wang J R , Zhang L . Basic Theory of Fractional Differential Equations. Second edition. Singapore: World Scientific, 2017

/

 〈 〉