# 分数阶微分方程积分边值问题正解的唯一性Uniqueness of Positive Solutions for the Fractional Differential Equation with Integral Boundary Value Problem

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Fractional calculus is a kind of differential and integral which can deal with any order. It is a gen-eralization of integral calculus. In this paper, we mainly study the positive solutions for the frac-tional differential equation with integral boundary value problem. By using the Banach fixed point theorem, we obtain the uniqueness of the positive solutions of the equation.

1. 引言

$\left\{\begin{array}{l}{D}_{{0}^{+}}^{\alpha }x\left(t\right)+\lambda f\left(t,x\left(t\right)\right)=0,0 (1.1)

2. 预备知识

${I}_{{0}^{+}}^{\alpha }x\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}x\left(s\right)\text{d}s,$

${D}_{{0}^{+}}^{\alpha }x\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{\left(\frac{\text{d}}{\text{d}t}\right)}^{n}{\int }_{0}^{t}{\left(t-s\right)}^{n-\alpha -1}x\left(s\right)\text{d}s,$

${I}_{{0}^{+}}^{\alpha }{D}_{{0}^{+}}^{\alpha }x\left(t\right)=x\left(t\right)+{c}_{1}{t}^{\alpha -1}+{c}_{2}{t}^{\alpha -2}+\cdots +{c}_{n}{t}^{\alpha -n},$

$\left\{\begin{array}{l}{D}_{{0}^{+}}^{\alpha }x\left(t\right)+y\left(t\right)=0,t\in \left(0,1\right),n-1<\alpha \le n,n\ge 2,\\ x\left(0\right)={x}^{\prime }\left(0\right)=\cdots ={x}^{\left(n-2\right)}=0,{D}_{{0}^{+}}^{\alpha -1}x\left(1\right)={\int }_{0}^{1}h\left(t\right)x\left(t\right)\text{d}A\left(t\right)\end{array}$ (2.1)

$x\left(t\right)={\int }_{0}^{\infty }G\left(t,s\right)y\left(s\right)\text{d}s,$

$G\left(t,s\right)={G}_{0}\left(t,s\right)+{G}_{1}\left(t,s\right),$ (2.2)

${G}_{0}\left(t,s\right)=\frac{1}{\Gamma \left(\alpha \right)}\left\{\begin{array}{l}{t}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le s\le t\le 1,\\ {t}^{\alpha -1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le s\le 1,\end{array}$

${G}_{1}\left(t,s\right)=\frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)}{\int }_{0}^{1}h\left(t\right){G}_{0}\left(t,s\right)\text{d}A\left(t\right)\text{ }\text{ }.$

$x\left(t\right)=-{I}_{{0}^{+}}^{\alpha }y\left(t\right)+{c}_{1}{t}^{\alpha -1}+{c}_{2}{t}^{\alpha -2}+\cdots +{c}_{n}{t}^{\alpha -n},\text{\hspace{0.17em}}{c}_{i}\in \left(-\infty ,+\infty \right),i=1,2,\cdots ,n,$

$x\left(t\right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\text{d}s+{c}_{1}{t}^{\alpha -1}+{c}_{2}{t}^{\alpha -2}+\cdots +{c}_{n}{t}^{\alpha -n},\text{\hspace{0.17em}}{c}_{i}\in \left(-\infty ,+\infty \right),i=1,2,\cdots ,n,$

$x\left(t\right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\text{d}s+{c}_{1}{t}^{\alpha -1},$

${D}_{{0}^{+}}^{\alpha -1}x\left(t\right)={c}_{1}\Gamma \left(\alpha \right)-{\int }_{0}^{t}y\left(s\right)\text{d}s.$

${D}_{{0}^{+}}^{\alpha -1}x\left(1\right)={c}_{1}\Gamma \left(\alpha \right)-{\int }_{0}^{1}y\left(s\right)\text{d}s.$

${c}_{1}=\frac{1}{\Gamma \left(\alpha \right)}\left({\int }_{0}^{1}h\left(t\right)x\left(t\right)\text{d}A\left(t\right)+{\int }_{0}^{1}y\left(s\right)\text{d}s\right),$

$\begin{array}{c}x\left(t\right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}y\left(s\right)\text{d}s+{t}^{\alpha -1}\frac{1}{\Gamma \left(\alpha \right)}\left({\int }_{0}^{1}h\left(t\right)x\left(t\right)\text{d}A\left(t\right)+{\int }_{0}^{1}y\left(s\right)\text{d}s\right)\\ ={\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s+\frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)}{\int }_{0}^{1}h\left(t\right)x\left(t\right)\text{d}A\left(t\right)\text{ }\text{ }.\end{array}$ (2.3)

${\int }_{0}^{1}h\left(t\right)x\left(t\right)\text{d}A\left(t\right)=\frac{\Gamma \left(\alpha \right)}{\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)}{\int }_{0}^{1}h\left(t\right){\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s\text{d}A\left(t\right)\text{ }\text{ },$

$\begin{array}{c}x\left(t\right)={\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s+\frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)}{\int }_{0}^{1}h\left(t\right)x\left(t\right)\text{d}A\left(t\right)\\ ={\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s+\frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)}\frac{\Gamma \left(\alpha \right)}{\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)}{\int }_{0}^{1}h\left(t\right){\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s\text{d}A\left(t\right)\\ ={\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s+\frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)}{\int }_{0}^{1}h\left(t\right){\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s\text{d}A\left(t\right)\\ ={\int }_{0}^{1}{G}_{0}\left(t,s\right)y\left(s\right)\text{d}s+{\int }_{0}^{1}{G}_{1}\left(t,s\right)y\left(s\right)\text{d}s={\int }_{0}^{1}G\left(t,s\right)y\left(s\right)\text{d}s\text{ }\text{ }.\end{array}$

1) $G\left(t,s\right)\ge 0$$\left(t,s\right)\in \left[0,1\right]×\left[0,1\right]$

2) $G\left(t,s\right)$$\left[0,1\right]×\left[0,1\right]$ 上连续。

3) $G\left(t,s\right)\le \omega$$\omega =\mathrm{max}\left\{\frac{1}{\Gamma \left(\alpha \right)},\frac{{\int }_{0}^{1}h\left(t\right)\text{d}A\left(t\right)}{\Gamma \left(\alpha \right)\left(\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)\right)}\right\}$

${G}_{0}\left(t,s\right)\le \frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)}\le \frac{1}{\Gamma \left(\alpha \right)},$

$\begin{array}{c}{G}_{1}\left(t,s\right)\le \frac{{t}^{\alpha -1}}{\Gamma \left(\alpha \right)\left(\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)\right)}{\int }_{0}^{1}h\left(t\right)\text{d}A\left(t\right)\\ \le \frac{1}{\Gamma \left(\alpha \right)\left(\Gamma \left(\alpha \right)-{\int }_{0}^{1}h\left(t\right){t}^{\alpha -1}\text{d}A\left(t\right)\right)}{\int }_{0}^{1}h\left(t\right)\text{d}A\left(t\right)\text{ }\text{ },\end{array}$

3. 主要结果

$X=C\left[0,1\right]$，定义范数 $‖x‖={\mathrm{max}}_{0\le t\le 1}|x\left(t\right)|$。则X是Banach空间，记

$K=\left\{x\in X:x\left(t\right)\ge 0,t\in \left[0,1\right]\right\},$

(H1) $f:\left[0,1\right]×\left[0,+\infty \right)\to \left[0,+\infty \right)$ 是连续函数。

$\left(Tx\right)\left(t\right)=\lambda {\int }_{0}^{1}G\left(t,s\right)f\left(s,x\left(s\right)\right)\text{d}s,\text{\hspace{0.17em}}t\in \left[0,1\right]$ (3.1)

(H2) $|f\left(t,{x}_{2}\right)-f\left(t,{x}_{1}\right)|\le m\left(t\right)|{x}_{2}-{x}_{1}|,t\in \left[0,1\right],{x}_{1},{x}_{2}\in \left[0,+\infty \right)$

${\int }_{0}^{1}m\left(s\right)\text{d}s<\frac{1}{\lambda \omega },$

$\begin{array}{c}‖T{x}_{2}-T{x}_{1}‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}|T{x}_{2}\left(t\right)-T{x}_{1}\left(t\right)|\\ =\underset{t\in \left[0,1\right]}{\mathrm{max}}\lambda {\int }_{0}^{1}G\left(t,s\right)|f\left(s,{x}_{2}\left(s\right)\right)-f\left(s,{x}_{1}\left(s\right)\right)|\text{d}s\\ \le \lambda {\int }_{0}^{1}\omega m\left(s\right)|{x}_{2}\left(s\right)-{x}_{1}\left(s\right)|\text{d}s\\ \le \lambda \omega {\int }_{0}^{1}m\left(s\right)\text{d}s‖{x}_{2}-{x}_{1}‖\\ <‖{x}_{2}-{x}_{1}‖\end{array}$

 [1] Wang, G., Agarwal, R. and Cabada, A. (2012) Existence Results and Monotone Iterative Technique for Systems of Nonlinear Fractional Differential Equations. Applied Mathematics Letters, 25, 1019-1024. https://doi.org/10.1016/j.aml.2011.09.078 [2] Guo, L., Liu, L. and Wu, Y. (2016) Uniqueness of Iterative Posi-tive Solutions for the Singular Fractional Differential Equations with Integral Boundary Conditions. Boundary Value Problems, 2016, 147.https://doi.org/10.1186/s13661-016-0652-1 [3] Wang, Y. (2016) Positive Solutions for Fractional Differential Equation Involving the Riemann-Stieltjes Integral Conditions with Two Parameters. Journal of Nonlinear Sciences and Applications, 9, 5733-5740.https://doi.org/10.22436/jnsa.009.11.02 [4] Zou, Y. and He, G. (2017) On the Uniqueness of Solutions for a Class of Fractional Differential Equations. Applied Mathematics Letters, 74, 68-73. https://doi.org/10.1016/j.aml.2017.05.011 [5] Jiang, J., Liu, W. and Wang, H. (2018) Positive Solutions to Singular Dirichlet-Type Boundary Value Problems of Nonlinear Fractional Differential Equations. Advances in Difference Equations, 2018, 169. https://doi.org/10.1186/S13662-018-1627-6 [6] Cui, Y., Ma, W., Wang, X. and Su, X. (2018) Uniqueness Theorem of Differential System with Coupled Integral Boundary Conditions. Electronic Journal of Qualitative Theory of Differential Equations, 9, 1-10. [7] Cabada, A. and Wang, G. (2012) Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions. Journal of Mathematical Analysis and Applications, 389, 403-411. https://doi.org/10.1016/j.jmaa.2011.11.065 [8] Wang, Y. (2018) Existence and Nonexistence of Positive Solutions for Mixed Fractional Boundary Value Problem with Parameter and p-Laplacian Operator. Journal of Function Spaces, 2018, Article ID: 1462825. [9] Hao, X., Zhang, L. and Liu, L. (2019) Positive Solutions of Higher Order Fractional Integral Boundary Value Problem with a Parameter. Nonlinear Analysis-Modelling and Control, 24, 210-223. https://doi.org/10.15388/NA.2019.2.4 [10] Podlubny, I. (1999) Fractional Differential Equations, Volume 198. Academic Press, San Diego, CA. [11] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.