#### 期刊菜单

The Uniform q-Order Growth Condition
DOI: 10.12677/AAM.2019.81014, PDF, HTML, XML, 下载: 1,225  浏览: 2,296

Abstract: Uniform second order growth condition is an important notion in optimization and has been studied extensively. Recently, as a natural extension of the uniform second order growth condition, with a positive number q replacing 2, the uniform q-order growth condition was introduced and studied in [1]. Motivated by [1], this thesis further studies the uniform q-order growth condition of a q-order regular real-valued function f. In terms of the Hölder metric regularity of the subdifferential mapping ∂f, we provide sufficient and necessary conditions for f to have the uniform q-order growth condition. In particular, using the modulus and radius appearing in the Hölder metric regularity of the subdifferential mapping ∂f, we give an exact quantitative formula of the radius appearing in the uniform q-order growth condition, which improves some existing results on the uniform second order growth condition and uniform q-order growth condition.

1. 引言

$f:{R}^{n}\to R\cup \left\{+\infty \right\}$ 是一个真下半连续泛函且 $\left(\stackrel{¯}{x},{\stackrel{¯}{x}}^{\ast }\right)\in gph\left(\partial f\right)$ ，若存在 $\rho ,\delta \in \left(0,+\infty \right)$ 使得对于任意的 $y\in B\left(\stackrel{¯}{x},\delta \right)$$\left(x,f\left(x\right)\right)\in B\left(\left(\stackrel{¯}{x},f\left(\stackrel{¯}{x}\right)\right),\delta \right)$${x}^{\ast }\in gph\left(\partial f\right)\cap B\left({\stackrel{¯}{x}}^{\ast },\delta \right)$ 都有

$〈{x}^{\ast },y-x〉\le f\left(y\right)-f\left(x\right)+\rho {‖y-x‖}^{2},$ (1.1)

$M:v\to \underset{x\in {B}_{X}\left[\stackrel{¯}{x},\delta \right]}{\mathrm{arg}\mathrm{min}}\left\{f\left(x\right)-f\left(\stackrel{¯}{x}\right)-〈v,x-\stackrel{¯}{x}〉\right\}$

$\delta ,\kappa ,r\in \left(0,+\infty \right)$ 及映射 $\vartheta :B\left(0,\delta \right)\to B\left(\stackrel{¯}{x},r\right)$ 使得对于任意的 $\left(x,{u}^{\ast }\right)\in B\left(\stackrel{¯}{x},r\right)×B\left(0,\delta \right)$$\vartheta \left(0\right)=\stackrel{¯}{x}$

$\kappa {‖x-\vartheta \left({u}^{\ast }\right)‖}^{2}\le f\left(x\right)-f\left(\vartheta \left({u}^{\ast }\right)\right)-〈{u}^{\ast },x-\vartheta \left({u}^{\ast }\right)〉.$ (1.2)

$\phi \left(‖x-\stackrel{¯}{x}‖\right)\le f\left(x\right)-f\left(\stackrel{¯}{x}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in X.$

$\vartheta :{B}_{{X}^{\ast }}\left(0,\delta \right)\to {B}_{X}\left(\stackrel{¯}{x},r\right)$ 使得 $\vartheta \left(0\right)=\stackrel{¯}{x}$ 且对于任意的 $\left(x,{u}^{\ast }\right)\in {B}_{X}\left(\stackrel{¯}{x},r\right)×{B}_{{X}^{\ast }}\left(0,\delta \right)$ 都有

$\kappa \phi \left(\tau ‖x-\vartheta \left({u}^{\ast }\right)‖\right)\le f\left(x\right)-f\left(\vartheta \left({u}^{\ast }\right)\right)-〈{u}^{\ast },x-\vartheta \left({u}^{\ast }\right)〉,$ (1.3)

$f:X\to R\cup \left\{+\infty \right\}$ 是一个真的下半连续泛函，若存在 $\delta ,\rho ,\tau \in \left(0,+\infty \right)$ 使得对于任意 $x\in {B}_{X}\left(\stackrel{¯}{x},\delta \right)$$\left(z,f\left(z\right)\right)\in B\left(\left(\stackrel{¯}{x},f\left(\stackrel{¯}{x}\right)\right),\delta \right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left({\stackrel{¯}{x}}^{\ast },\delta \right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+\rho \phi \left(\tau ‖x-z‖\right),$

2. 预备知识

$dom\left(f\right):=\left\{x\in X:f\left(x\right)<+\infty \right\}$

$epi\left(f\right):=\left\{\left(x,\alpha \right)\in X×R:f\left(x\right)<\alpha \right\}$

${f}^{↑}\left(x,h\right):=\underset{\epsilon ↓0}{\mathrm{lim}}\underset{u\stackrel{f}{\to }x,t↓0}{\mathrm{lim}\mathrm{sup}}\underset{\omega \in h+\epsilon {B}_{X}}{\mathrm{inf}}\frac{f\left(u+t\omega \right)-f\left(u\right)}{t},$

${f}^{↑}\left(x,h\right):=\underset{u\to x}{\mathrm{lim}}\underset{t↓0}{\mathrm{sup}}\frac{f\left(u+th\right)-f\left(u\right)}{t},$

${\kappa }_{0}:=\kappa$

$\begin{array}{c}\partial f\left(x\right)=\left\{{x}^{\ast }\in {X}^{\ast }:〈{x}^{\ast },y-x〉\le f\left(y\right)-f\left(x\right),\forall y\in X\right\}\\ =\left\{{x}^{\ast }\in {X}^{\ast }:〈{x}^{\ast },h〉\le \underset{t\to {0}^{+}}{\mathrm{lim}}\frac{f\left(x+th\right)-f\left(x\right)}{h},\forall y\in X\right\}\end{array}$

$\underset{\left(x,{x}^{\ast }\right)\stackrel{gph\left(\partial f\right)}{\to }\left(\stackrel{¯}{x},{\stackrel{¯}{x}}^{\ast }\right)}{\mathrm{lim}}f\left(x\right)=f\left(\stackrel{¯}{x}\right),$

$x\in dom\left({f}_{1}\right)\cap dom\left({f}_{2}\right)$${f}_{1}$ 在x附近是Lipschitz的，则

$\partial \left({f}_{1}+{f}_{2}\right)\left(x\right)\subset \partial {f}_{1}\left(x\right)+\partial {f}_{2}\left(x\right)$

$\partial \left(\alpha {f}_{1}\right)\left(x\right)=\alpha \partial {f}_{1}\left(x\right),\forall \alpha \in R.$

$\kappa {‖x-\vartheta \left({u}^{\ast }\right)‖}^{q}\le f\left(x\right)-f\left(\vartheta \left({u}^{\ast }\right)\right)-〈{u}^{\ast },x-\vartheta \left({u}^{\ast }\right)〉,\text{\hspace{0.17em}}\forall \left(x,{u}^{\ast }\right)\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{2}\right)×{B}_{{X}^{\ast }}\left(0,{\delta }_{1}\right),$

1) 若存在 $\kappa ,{\delta }_{1},{\delta }_{2}\in \left(0,+\infty \right)$ 使得

$d{\left(x,{F}^{-1}\left(y\right)\right)}^{p}\le \kappa d\left(y,F\left(x\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \left(x,y\right)\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{1}\right)×{B}_{Y}\left(\stackrel{¯}{y},{\delta }_{2}\right),$ (2.1)

2) 若存在 $\kappa ,{\delta }_{1},{\delta }_{2},{\delta }_{3}\in \left(0,+\infty \right)$ 使得(2.1)成立且

${F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{\delta }_{3}\right)=\left\{{z}_{y}\right\},\forall y\in {B}_{Y}\left(\stackrel{¯}{y},{\delta }_{2}\right),$

${‖x-\theta \left(y\right)‖}^{p}\le {\kappa }_{0}d\left(y,F\left(x\right)\right),\text{\hspace{0.17em}}\forall \left(x,y\right)\in {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{1}\right)×{B}_{Y}\left(\stackrel{¯}{y},{{\delta }^{\prime }}_{1}\right)$ (2.2)

${F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)=\left\{\theta \left(y\right)\right\},\forall y\in {B}_{Y}\left(\stackrel{¯}{y},{{\delta }^{\prime }}_{1}\right).$ (2.3)

${F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{\delta }_{3}\right)=\left\{{z}_{y}\right\},\forall y\in {B}_{Y}\left(\stackrel{¯}{y},{\delta }_{2}\right).$ (2.4)

$d{\left(x,{F}^{-1}\left(y\right)\right)}^{p}\le \kappa d\left(y,F\left(x\right)\right)\le \kappa ‖y-\stackrel{¯}{y}‖,\text{\hspace{0.17em}}\forall y\in {B}_{Y}\left(\stackrel{¯}{y},{\delta }_{2}\right)$

$\underset{y\to \stackrel{¯}{y}}{\mathrm{lim}}d\left(\stackrel{¯}{x},{F}^{-1}\left(y\right)\right)=0.$

$‖{x}_{y}-\stackrel{¯}{x}‖<\mathrm{min}\left\{{\delta }_{1},{\delta }_{2},\frac{{\delta }_{3}}{4}\right\}.$ (2.5)

${{\delta }^{\prime }}_{1}:=\mathrm{min}\left\{{\delta }_{0},{\delta }_{1},{\delta }_{2},\frac{{\delta }_{3}}{4}\right\}$${\kappa }_{0}:=\kappa$${{\delta }^{\prime }}_{2}:={\delta }_{3}$ 。并定义 $\theta :{B}_{Y}\left(\stackrel{¯}{y},{{\delta }^{\prime }}_{1}\right)\to {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)$ 使得 $\theta \left(y\right)={z}_{y},\forall y\in {B}_{Y}\left(\stackrel{¯}{y},{{\delta }^{\prime }}_{1}\right)$

${x}_{y}\in \left({F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right)=\left({F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{\delta }_{3}\right)\right)=\left\{{z}_{y}\right\}=\left(\theta \left(y\right)\right),$

$\begin{array}{c}d\left(x,{F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right)\le ‖x-{x}_{y}‖\le ‖x-\stackrel{¯}{x}‖+‖\stackrel{¯}{x}-{x}_{y}‖\\ <{{\delta }^{\prime }}_{1}+\frac{{\delta }_{3}}{4}\le \frac{{\delta }_{3}}{2}=\frac{{{\delta }^{\prime }}_{2}}{2}\end{array}$

$\begin{array}{c}d\left(x,{F}^{-1}\left(y\right)\{B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right)\ge d\left(\stackrel{¯}{x},{F}^{-1}\left(y\right)\{B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right)-‖x-\stackrel{¯}{x}‖\\ >{{\delta }^{\prime }}_{2}-{{\delta }^{\prime }}_{1}\ge {{\delta }^{\prime }}_{2}-\frac{{\delta }_{3}}{4}=\frac{3{{\delta }^{\prime }}_{2}}{4}\end{array}$

$\begin{array}{c}d\left(x,{F}^{-1}\left(y\right)\right)=\mathrm{min}\left\{d\left(x,{F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right),d\left(x,{F}^{-1}\left(y\right)\{B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right)\right\}\\ =d\left(x,{F}^{-1}\left(y\right)\cap {B}_{X}\left(\stackrel{¯}{x},{{\delta }^{\prime }}_{2}\right)\right)=‖x-\theta \left(y\right)‖\end{array}$ (2.6)

(最后一个等式成立是因为(2.3))，从而由(2.1)和(2.6)可得到(2.2)，证毕。

Prox-正则体现了二阶变分性质，在变分分析中被广泛使用，本文采用更一般的Hölder正则性(参见文献 [1] ，定义3.1，取 $\phi \left(t\right)={t}^{p}$ )。

1) 若存在 $\delta ,\rho \in \left(0,+\infty \right)$ 使得对于任意的 $x\in {B}_{X}\left(\stackrel{¯}{x},\delta \right),\left(z,f\left(z\right)\right)\in B\left(\left(\stackrel{¯}{x},f\left(\stackrel{¯}{x}\right)\right),\delta \right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left({\stackrel{¯}{x}}^{\ast },\delta \right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+\rho {‖x-z‖}^{s},$

2) 若存在 $\delta ,\rho \in \left(0,+\infty \right)$ 使得对于任意的 $x\in {B}_{X}\left(\stackrel{¯}{x},\delta \right),\left(z,f\left(z\right)\right)\in B\left(\left(\stackrel{¯}{x},f\left(\stackrel{¯}{x}\right)\right),\delta \right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left({\stackrel{¯}{x}}^{\ast },\delta \right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+\rho d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{s},$

3) 若存在 $\delta ,\rho \in \left(0,+\infty \right)$ 使得对于任意的 $x\in {B}_{X}\left(\stackrel{¯}{x},\delta \right)$ 都有

$〈{\stackrel{¯}{x}}^{\ast },x-\stackrel{¯}{x}〉\le f\left(x\right)-f\left(\stackrel{¯}{x}\right)+\rho {‖x-\stackrel{¯}{x}‖}^{s},$

4) 若存在 $\delta ,\rho \in \left(0,+\infty \right)$ 使得对于任意的 $x\in {B}_{X}\left(\stackrel{¯}{x},\delta \right)$ 都有

$〈{\stackrel{¯}{x}}^{\ast },x-\stackrel{¯}{x}〉\le f\left(x\right)-f\left(\stackrel{¯}{x}\right)+\rho d{\left(x,{\left(\partial f\right)}^{-1}\left({\stackrel{¯}{x}}^{\ast }\right)\right)}^{s},$

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+\rho {‖x-z‖}^{s}.$

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+{\rho }_{0}{‖x-z‖}^{s}.$ (2.7)

$|f\left(x\right)-f\left(\stackrel{¯}{x}\right)|<{\delta }_{0}.$ (2.8)

$\rho :={\rho }_{0}$ ，由(2.7)和(2.8)可知对于任意的 $x,z\in {B}_{X}\left(\stackrel{¯}{x},\delta \right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left({\stackrel{¯}{x}}^{\ast },\delta \right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+\rho {‖x-z‖}^{s},$

${f}^{\ast }\left({x}^{\ast }\right):=\underset{x\in X}{\mathrm{sup}}\left(〈{x}^{\ast },x〉-f\left(x\right)\right),\forall {x}^{\ast }\in {X}^{\ast }.$

${f}^{\ast }\left({x}^{\ast }\right)=〈{x}^{\ast },x〉-f\left(x\right)⇒x\in \partial {f}^{\ast }\left({x}^{\ast }\right).$

${x}^{\ast }\in \partial f\left(x\right)⇔x\in \partial {f}^{\ast }\left({x}^{\ast }\right).$

3. 一致q阶增长条件

$d{\left(x,A\right)}^{q-1}\le {\kappa }_{0}d\left({\stackrel{¯}{x}}^{\ast },\partial \left(x\right)\right),\text{\hspace{0.17em}}\forall x\in {B}_{X}\left(\stackrel{¯}{x},\delta \right)$

$f\left(x\right)\ge f\left(\stackrel{¯}{x}\right)+〈{\stackrel{¯}{x}}^{\ast },x-\stackrel{¯}{x}〉-{\rho }_{0}d{\left(x,A\right)}^{q},\text{\hspace{0.17em}}\forall x\in {B}_{X}\left[\stackrel{¯}{x},r\right].$

$\nu :=\mathrm{max}\left\{n\in N:n\le 1-{\mathrm{log}}_{\left(1+q\left(1-\alpha \right){\alpha }^{q-1}\right)}\left(1-q{\rho }_{0}{\kappa }_{0}\right)\right\},$ (3.1)

$\lambda :=\left(\frac{1}{q{\alpha }^{q}{\kappa }_{0}}-\frac{{\rho }_{0}}{{\alpha }^{q}}\right){\left(1+q\left(1-\alpha \right){\alpha }^{q-1}\right)}^{\nu }-\frac{1}{q{\alpha }^{q}{\kappa }_{0}}$ (3.2)

${\eta }_{n}:=-\left(\frac{1}{q{\alpha }^{q}{\kappa }_{0}}-\lambda \right){\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)}^{n-1}+\frac{1}{q{\alpha }^{q}{\kappa }_{0}},\forall n\in N.$ (3.3)

a) $\nu \ge 1$$\frac{1}{q{\alpha }^{q}{\kappa }_{0}}>\lambda >0$$\underset{n\to \infty }{\mathrm{lim}}{\eta }_{n}=\frac{1}{q{\alpha }^{q}{\kappa }_{0}}$

b) $f\left(x\right)\ge f\left(\stackrel{¯}{x}\right)+〈{\stackrel{¯}{x}}^{\ast },x-\stackrel{¯}{x}〉+{\eta }_{n}{\alpha }^{q}d{\left(x,A\right)}^{q}$$\forall n\in N$$x\in {B}_{X}\left[\stackrel{¯}{x},\frac{\mathrm{min}\left\{r,\delta \right\}}{{\left(2-\alpha \right)}^{\nu +n-1}}\right]$

$\varphi \left(t\right):={t}^{q},\forall t\in \left(0,1\right).$

$\frac{1}{q}=\frac{\varphi \left(t\right)}{t{\varphi }^{\prime }\left(t\right)}>\left(1-\alpha \right)\frac{\varphi \left(t\right)-\varphi \left(\alpha t\right)}{\left(t-\alpha t\right){\varphi }^{\prime }\left(t\right)}\ge \left(1-\alpha \right)\frac{{\varphi }^{\prime }\left(\alpha t\right)}{{\varphi }^{\prime }\left(t\right)}=\left(1-\alpha \right){\alpha }^{q-1},$

1) 对于任意的 $\left(x,{u}^{\ast }\right)\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{1}\right)×{B}_{{X}^{\ast }}\left(0,{\delta }_{1}\right)$ 都有

${\phi }^{\prime }\left(\tau d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)\right)\le \kappa d\left({u}^{\ast },\partial f\left(x\right)\right);$

2) 对于任意的 $x\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{2}\right)$$\left(z,f\left(z\right)\right)\in {B}_{{X}^{\ast }}\left(\left(\stackrel{¯}{x},f\left(\stackrel{¯}{x}\right)\right),{\delta }_{2}\right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left(0,{\delta }_{2}\right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+\rho \phi \left(\alpha \tau d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)\right).$

$\beta \phi \left(\alpha \tau ‖x-{x}_{{u}^{\ast }}‖\right)\le f\left(x\right)-f\left({x}_{{u}^{\ast }}\right)-〈{u}^{\ast },x-{x}_{{u}^{\ast }}〉,\forall x\in {B}_{X}\left(\stackrel{¯}{x},r\right),$ (3.4)

${\mu }_{\alpha }:=\underset{t>0}{\mathrm{sup}}\frac{\alpha {{\phi }^{\prime }}_{+}\left(\alpha t\right)}{{{\phi }^{\prime }}_{+}\left(t\right)}$ (3.5)

${{\phi }^{\prime }}_{\text{+}}$ 表示 $\phi$ 的右方向导数，故f在 $\stackrel{¯}{x}$ 处满足一致φ-增长条件。

$\phi \left(t\right)={t}^{q}$ 的特殊情况下， ${\mu }_{\alpha }={t}^{q}$ (见(3.5))，由定理I和命题2.2可得以下推论。

1) 对于任意的 $\left(x,{u}^{\ast }\right)\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{1}\right)×{B}_{{X}^{\ast }}\left(0,{\delta }_{1}\right)$ 都有

$d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q-1}\le {\kappa }_{0}d\left({u}^{\ast },\partial f\left(x\right)\right);$

2) 对于任意的 $x,z\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{2}\right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left(0,{\delta }_{2}\right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+{\rho }_{0}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q}.$

$\beta {\alpha }^{q}{‖x-\vartheta \left({u}^{\ast }\right)‖}^{q}\le f\left(x\right)-f\left(\vartheta \left({u}^{\ast }\right)\right)-〈{u}^{\ast },x-\vartheta \left({u}^{\ast }\right)〉,$ (3.6)

1) 对于任意的 $\left(x,{u}^{\ast }\right)\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{1}\right)×{B}_{{X}^{\ast }}\left(0,{\delta }_{1}\right)$ 都有

$d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q-1}\le {\kappa }_{0}d\left({u}^{\ast },\partial f\left(x\right)\right);$ (3.7)

2) 对于任意的 $x,z\in {B}_{X}\left(\stackrel{¯}{x},{\delta }_{2}\right)$${u}^{\ast }\in \partial f\left(z\right)\cap {B}_{{X}^{\ast }}\left(0,{\delta }_{2}\right)$ 都有

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+{\rho }_{0}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q}.$ (3.8)

${N}_{\beta }:=\mathrm{max}\left\{n\in N:{\mathrm{log}}_{\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)}\frac{\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)\left(1-\beta q{\alpha }^{q}{\kappa }_{0}\right)}{1-\lambda q{\alpha }^{q}{\kappa }_{0}}\le n\right\},$

$\gamma \left(\beta \right):=\frac{\mathrm{min}\left\{{\delta }_{1},{\delta }_{2}\right\}}{4{\left(2-\alpha \right)}^{\nu +{N}_{\beta }-1}},$ (3.9)

$r\left(\beta \right):=\mathrm{min}\left\{{\delta }_{1},{\delta }_{2},\frac{\beta {\alpha }^{q}{\gamma }^{q-1}\left(\beta \right)}{{32}^{q-1}},\frac{{\gamma }^{q-1}\left(\beta \right)}{{4}^{q-1}{\kappa }_{0}},\frac{\beta {\alpha }^{q}{\gamma }^{q-1}\left(\beta \right)}{{16}^{q}}\right\},$

$\delta \left(\beta \right):=\frac{\mathrm{min}\left\{{\delta }_{1},{\delta }_{2}\right\}}{64{\left(2-\alpha \right)}^{\nu +{N}_{\beta }-1}}$

$\psi \left({u}^{\ast }\right):=\underset{u\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{arg}\mathrm{min}}\left(f-{u}^{\ast }\right),\forall {u}^{\ast }\in {X}^{\ast },$

$\beta {\alpha }^{q}{‖x-\psi \left({u}^{\ast }\right)‖}^{q}\le f\left(x\right)-f\left(\psi \left({u}^{\ast }\right)\right)-〈{u}^{\ast },x-\psi \left({u}^{\ast }\right)〉,$ (3.10)

$d{\left(\stackrel{¯}{x},{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q-1}\le {\kappa }_{0}d\left({u}^{\ast },\partial f\left(\stackrel{¯}{x}\right)\right)\le {\kappa }_{0}‖{u}^{\ast }‖,\forall {u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{\delta }_{1}\right).$ (3.11)

${N}_{\beta }\ge {\mathrm{log}}_{\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)}\frac{\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)\left(1-\beta q{\alpha }^{q}{\kappa }_{0}\right)}{1-\lambda q{\alpha }^{q}{\kappa }_{0}},$

${\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)}^{{N}_{\beta }}\le \frac{\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)\left(1-\beta q{\alpha }^{q}{\kappa }_{0}\right)}{1-\lambda q{\alpha }^{q}{\kappa }_{0}},$

$\left(1-\lambda q{\alpha }^{q}{\kappa }_{0}\right){\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)}^{{N}_{\beta }-1}\le 1-\beta q{\alpha }^{q}{\kappa }_{0},$

${\eta }_{{N}_{\beta }}:=-\left(\frac{1}{q{\alpha }^{q}{\kappa }_{0}}-\lambda \right){\left(1-q\left(1-\alpha \right){\alpha }^{q-1}\right)}^{{N}_{\beta }-1}+\frac{1}{q{\alpha }^{q}{\kappa }_{0}}\ge \beta .$ (3.12)

$\delta :=\mathrm{min}\left\{{\delta }_{1},{\delta }_{2}\right\}$${r}_{1}\left(\beta \right):=\mathrm{min}\left\{\delta ,\frac{{\gamma }^{q-1}\left(\beta \right)}{{\kappa }_{0}}\right\}$ ，其中 $\gamma \left(\beta \right)$ 是由定义(3.9)定义。由(3.11)， $\delta$${r}_{1}\left(\beta \right)$ 的定义可知对于任意的 ${u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{1}\left(\beta \right)\right)$ 都存在 $z\in {\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)$ 使得 ${‖\stackrel{¯}{x}-z‖}^{q-1}<{\kappa }_{0}{r}_{1}\left(\beta \right)\le {\gamma }^{q-1}\left(\beta \right)$ ，于是 $‖\stackrel{¯}{x}-z‖<\gamma \left(\beta \right)$ ，故

${\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\ne \varnothing .$

${u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{1}\left(\beta \right)\right)$$z\in {\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)$ ，由(3.9)可得

${B}_{X}\left(z,\frac{3\delta }{4}\right)\subset {B}_{X}\left(\stackrel{¯}{x},‖\stackrel{¯}{x}-z‖+\frac{3\delta }{4}\right)\subset {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)+\frac{3\delta }{4}\right)\subset {B}_{X}\left(\stackrel{¯}{x},\delta \right),$

$d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q-1}\le {\kappa }_{0}d\left({u}^{\ast },\partial f\left(x\right)\right),\forall x\in {B}_{X}\left(z,\frac{3\delta }{4}\right)$

$〈{u}^{\ast },x-z〉\le f\left(x\right)-f\left(z\right)+{\rho }_{0}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q},\forall x\in {B}_{X}\left[z,\frac{3\delta }{4}\right],$

$f\left(x\right)\ge f\left(z\right)+〈{u}^{\ast },x-z〉+\eta {N}_{\beta }{\alpha }^{q}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q},\forall x\in {B}_{X}\left[z,\frac{3\delta }{4{\left(2-\alpha \right)}^{\nu +{N}_{\beta }+1}}\right].$

$f\left(x\right)\ge f\left(z\right)+〈{u}^{\ast },x-z〉+\eta {N}_{\beta }{\alpha }^{q}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q},\forall x\in {B}_{X}\left[z,3\gamma \left(\beta \right)\right].$ (3.13)

${f}_{{u}^{\ast }}:=f-{u}^{\ast }$ ，则由(3.12)和(3.13)，有

${f}_{{u}^{\ast }}\left(x\right)\ge {f}_{{u}^{\ast }}\left(z\right)+\beta {\alpha }^{q}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q},\forall x\in {B}_{X}\left[z,3\gamma \left(\beta \right)\right].$

${B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]\subset {B}_{X}\left[z,\gamma \left(\beta \right)+‖\stackrel{¯}{x}-z‖\right]\subset {B}_{X}\left[z,2\gamma \left(\beta \right)\right],$

${f}_{{u}^{\ast }}\left(x\right)\ge {f}_{{u}^{\ast }}\left(z\right)+\beta {\alpha }^{q}d{\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)}^{q},\forall x\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right].$ (3.14)

$\underset{u\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{arg}\mathrm{min}}{f}_{{u}^{\ast }}\left(u\right)\supset {\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right).$ (3.15)

${r}_{2}\left(\beta \right):=\mathrm{min}\left\{{r}_{1}\left(\beta \right),\frac{{\gamma }^{q-1}\left(\beta \right)}{{4}^{q-1}{\kappa }_{0}}\right\}$ ，则由(3.11)可知对于任意的 ${u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{2}\left(\beta \right)\right)\subset {B}_{{X}^{\ast }}\left(0,{\delta }_{1}\right)$ ，都存在 ${x}_{{u}^{\ast }}\in {\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)$ 使得

${‖{x}_{{u}^{\ast }}-\stackrel{¯}{x}‖}^{q-1}<{\kappa }_{0}{r}_{2}\left(\beta \right)\le \frac{{\gamma }^{q-1}\left(\beta \right)}{{4}^{q-1}},$

$‖{x}_{{u}^{\ast }}-\stackrel{¯}{x}‖<\frac{\gamma \left(\beta \right)}{4}$ 。因此，对于任意的 $\left(x,{u}^{\ast }\right)\in {B}_{X}\left(\stackrel{¯}{x},\frac{\gamma \left(\beta \right)}{4}\right)×{B}_{{X}^{\ast }}\left(0,{r}_{2}\left(\beta \right)\right)$ 都有

$d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\le ‖x-{x}_{{u}^{\ast }}‖\le ‖x-\stackrel{¯}{x}‖+‖\stackrel{¯}{x}-{x}_{{u}^{\ast }}‖\le \frac{\gamma \left(\beta \right)}{2}$

$d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\{B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\right)\ge d\left(\stackrel{¯}{x},{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\{B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\right)-‖x-\stackrel{¯}{x}‖\ge \frac{3\gamma \left(\beta \right)}{4}.$

$d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)=\mathrm{min}\left\{d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\right),d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\{B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\right)\right\},$

$d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)=d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)\right).$ (3.16)

$d\left(x,{\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\right)=d\left(x,\underset{u\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{arg}\mathrm{min}}{f}_{{u}^{\ast }}\left(u\right)\right).$

${u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{2}\left(\beta \right)\right)$$z\in {\left(\partial f\right)}^{-1}\left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\gamma \left(\beta \right)\right)$ ，则由(3.14)和(3.16))可得

${f}_{{u}^{\ast }}\left(x\right)\ge {f}_{{u}^{\ast }}\left(z\right)+\beta {\alpha }^{q}d{\left(x,\underset{u\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{arg}\mathrm{min}}{f}_{{u}^{\ast }}\left(u\right)\right)}^{q},\forall x\in {B}_{X}\left(\stackrel{¯}{x},\frac{\gamma \left(\beta \right)}{4}\right).$

$\beta {\alpha }^{q}d{\left(x,\underset{u\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{arg}\mathrm{min}}{f}_{{u}^{\ast }}\left(u\right)\right)}^{q}\le {f}_{{u}^{\ast }}\left(x\right)-{f}_{{u}^{\ast }}\left(z\right)\le {f}_{{u}^{\ast }}\left(x\right)-\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}{f}_{{u}^{\ast }}\left(y\right).$ (3.17)

$\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}{f}_{{u}^{\ast }}\left(y\right)=f\left(\stackrel{¯}{x}\right).$ (3.18)

$\psi \left({B}_{{X}^{\ast }}\left(0,r\left(\beta \right)\right)\right)\subset {B}_{X}\left(\stackrel{¯}{x},\delta \left(\beta \right)\right).$ (3.19)

${r}_{3}\left(\beta \right):=\mathrm{min}\left\{{r}_{2}\left(\beta \right),\frac{\beta {\alpha }^{q}{\gamma }^{q-1}\left(\beta \right)}{{16}^{q}}\right\}$ ，则由(3.17)和(3.18)可知对于任意的 ${u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{3}\left(\beta \right)\right)$ 都有

$\begin{array}{c}\beta {\alpha }^{q}d{\left(\stackrel{¯}{x},\psi \left({u}^{\ast }\right)\right)}^{q}\le {f}_{{u}^{\ast }}\left(\stackrel{¯}{x}\right)-\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}{f}_{{u}^{\ast }}\left(y\right)\\ =\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}f\left(y\right)-\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}\left(f\left(y\right)-〈{u}^{\ast },y-\stackrel{¯}{x}〉\right)\\ \le \underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}f\left(y\right)-\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}\left(f\left(y\right)-\gamma \left(\beta \right)‖{u}^{\ast }‖\right)\\ =\gamma \left(\beta \right)‖{u}^{\ast }‖<\gamma \left(\beta \right){r}_{3}\left(\beta \right).\end{array}$

$d\left(\stackrel{¯}{x},\psi \left({u}^{\ast }\right)\right)<\frac{\gamma \left(\beta \right)}{16},\forall {u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{3}\left(\beta \right)\right),$

$‖{x}_{{u}^{\ast }}-\stackrel{¯}{x}‖<\frac{\gamma \left(\beta \right)}{16}.$ (3.20)

${u}^{\ast },{v}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{3}\left(\beta \right)\right)$$u\in \psi \left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\frac{\gamma \left(\beta \right)}{16}\right)$ ，取 $\left\{{v}_{n}\right\}\subset \psi \left({v}^{\ast }\right)$ 使得

$\underset{n\to \infty }{\mathrm{lim}}‖u-{v}_{n}‖=d\left(u,\psi \left({v}^{\ast }\right)\right).$ (3.21)

$d\left(u,\psi \left({v}^{\ast }\right)\right)\le ‖u-{x}_{{v}^{\ast }}‖\le ‖u-\stackrel{¯}{x}‖+‖\stackrel{¯}{x}-{x}_{{v}^{\ast }}‖<\frac{\gamma \left(\beta \right)}{8}.$

$‖\stackrel{¯}{x}-{v}_{n}‖\le ‖u-{v}_{n}‖+‖\stackrel{¯}{x}-u‖<\frac{3\gamma \left(\beta \right)}{16}<\frac{\gamma \left(\beta \right)}{4},\forall n\in N.$

$\beta {\alpha }^{q}d{\left(u,\psi \left({v}^{\ast }\right)\right)}^{q}\le {f}_{{v}^{\ast }}\left(u\right)-\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}{f}_{{v}^{\ast }}\left(y\right)={f}_{{v}^{\ast }}\left(u\right)-{f}_{{v}^{\ast }}\left(vn\right)$

$\beta {\alpha }^{q}d{\left({v}_{n},\psi \left({u}^{\ast }\right)\right)}^{q}\le {f}_{{u}^{\ast }}\left({v}_{n}\right)-\underset{y\in {B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}{\mathrm{min}}{f}_{{u}^{\ast }}\left(y\right)={f}_{{u}^{\ast }}\left({v}_{n}\right)-{f}_{{u}^{\ast }}\left(u\right).$

$\begin{array}{c}\beta {\alpha }^{q}d{\left(u,\psi \left({v}^{\ast }\right)\right)}^{q}\le \beta {\alpha }^{q}d{\left(u,\psi \left({v}^{\ast }\right)\right)}^{q}+\beta {\alpha }^{q}d{\left({v}_{n},\psi \left({u}^{\ast }\right)\right)}^{q}\\ \le {f}_{{v}^{\ast }}\left(u\right)-{f}_{{v}^{\ast }}\left({v}_{n}\right)+{f}_{{u}^{\ast }}\left({v}_{n}\right)-{f}_{{u}^{\ast }}\left(u\right)\\ =〈{u}^{\ast }-{v}^{\ast },u-{v}_{n}〉\le ‖{u}^{\ast }-{v}^{\ast }‖‖u-{v}_{n}‖.\end{array}$

$\beta {\alpha }^{q}d{\left(u,\psi \left({\nu }^{\ast }\right)\right)}^{q}\le ‖{u}^{\ast }-{\nu }^{\ast }‖d\left(u,\psi \left({\nu }^{\ast }\right)\right),$

$d\left(u,\psi \left({\nu }^{\ast }\right)\right)\le \frac{1}{{\alpha }^{\frac{q}{q-1}}{\beta }^{\frac{1}{q-1}}}{\left(‖{u}^{\ast }-{\nu }^{\ast }‖\right)}^{\frac{1}{q-1}},$

$u\in \psi \left({\nu }^{\ast }\right)+\frac{2}{\frac{1}{{\alpha }^{\frac{q}{q-1}}{\beta }^{\frac{1}{q-1}}}}{\left(‖{u}^{\ast }-{\nu }^{\ast }‖\right)}^{\frac{1}{q-1}}{B}_{X}.$

$\psi \left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\frac{\gamma \left(\beta \right)}{16}\right)\subset \psi \left({v}^{\ast }\right)+L{\left(‖{u}^{\ast }-{v}^{\ast }‖\right)}^{\frac{1}{q-1}}{B}_{X},$ (3.22)

${\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)=〈{u}^{\ast },u〉-f\left(u\right).$ (3.23)

$\psi \left({u}^{\ast }\right)\subset \partial {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right),\forall {u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,{r}_{3}\left(\beta \right)\right).$ (3.24)

$\stackrel{¯}{x}\in \psi \left({u}^{\ast }\right)+L{\left(‖{u}^{\ast }‖\right)}^{\frac{1}{q-1}}{B}_{X}\subset \psi \left({u}^{\ast }\right)+L{r}_{3}^{\frac{1}{q-1}}\left(\beta \right){B}_{X},\forall {u}^{\ast }\in {B}_{X}\left(0,{r}_{3}\left(\beta \right)\right).$

$r\left(\beta \right)$ 的定义，可知 $r\left(\beta \right)\le {r}_{3}\left(\beta \right)\le \frac{\beta {\alpha }^{q}{r}^{q-1}\left(\beta \right)}{{16}^{q}}$ ，结合 $r\left(\beta \right)$ 以及L的定义可得 $L{r}^{\frac{1}{q-1}}\left(\beta \right)=\frac{2}{\frac{1}{{\alpha }^{\frac{q}{q-1}}{\beta }^{\frac{1}{q-1}}}}{r}^{\frac{1}{q-1}}\left(\beta \right)\le \frac{r\left(\beta \right)}{16}$ ，从而

$\psi \left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\frac{r\left(\beta \right)}{16}\right)\supset \psi \left({u}^{\ast }\right)+{B}_{X}\left(\stackrel{¯}{x},L{\gamma }^{\frac{1}{q-1}}\left(\beta \right)\right)\ne \varnothing ,\forall {u}^{\ast }\in {B}_{X}\left(0,r\left(\beta \right)\right).$

${u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,r\left(\beta \right)\right)$${x}_{{u}^{\ast }}\in \psi \left({u}^{\ast }\right)\cap {B}_{X}\left(\stackrel{¯}{x},\frac{r\left(\beta \right)}{16}\right)$ ，则由(3.22)可知对于任意的 ${v}^{\ast }\in {B}_{{X}^{\ast }}\left(0,r\left(\beta \right)\right)$${z}_{{v}^{\ast }}\in \psi \left({v}^{\ast }\right)$ 使得

$‖{x}_{{u}^{\ast }}-{z}_{{v}^{\ast }}‖\le L{‖{u}^{\ast }-{v}^{\ast }‖}^{\frac{1}{q-1}},$ (3.25)

${x}_{{u}^{\ast }}\in \partial {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)$${z}_{{v}^{\ast }}\in \partial {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left(v\ast \right)$

$〈{z}_{{v}^{\ast }},{u}^{\ast }-{v}^{\ast }〉\le {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)-{\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left(v\ast \right)$

$〈{x}_{{u}^{\ast }},{v}^{\ast }-{u}^{\ast }〉\le {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({v}^{\ast }\right)-{\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right).$

$\begin{array}{l}0\le {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({v}^{\ast }\right)-{\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)-〈{x}_{{u}^{\ast }},{v}^{\ast }-{u}^{\ast }〉\\ \le 〈{z}_{{v}^{\ast }}-{x}_{{u}^{\ast }},{v}^{\ast }-{u}^{\ast }〉\le L{‖{v}^{\ast }-{u}^{\ast }‖}^{1+\frac{1}{q+1}},\end{array}$

$\partial {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)=\left\{\nabla {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)\right\}=\left\{{x}_{{u}^{\ast }}\right\},\forall {u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,r\left(\beta \right)\right).$

$\psi \left({u}^{\ast }\right)=\left\{\nabla {\left(f+{\delta }_{{B}_{X}\left[\stackrel{¯}{x},\gamma \left(\beta \right)\right]}\right)}^{\ast }\left({u}^{\ast }\right)\right\}=\left\{{x}_{{u}^{\ast }}\right\}\subset {B}_{X}\left(\stackrel{¯}{x},\frac{r\left(\beta \right)}{16}\right),\forall {u}^{\ast }\in {B}_{{X}^{\ast }}\left(0,r\left(\beta \right)\right),$

$\psi \left({B}_{{X}^{\ast }}\left(0,{r}_{3}\left(\beta \right)\right)\right)\subset {B}_{X}\left(\stackrel{¯}{x},\frac{r\left(\beta \right)}{16}\right).$ (3.26)

$\delta \left(\beta \right)$ 的定义可知 $\delta \left(\beta \right)=\frac{r\left(\beta \right)}{16}$ ，结合(3.17)，(3.26)及 $\psi$ 的定义，可得

$\beta {\alpha }^{q}{‖x-\vartheta \left({u}^{\ast }\right)‖}^{q}\le f\left(x\right)-f\left({x}_{{u}^{\ast }}\right)-〈{u}^{\ast },x-{x}_{{u}^{\ast }}〉,\forall x\in {B}_{X}\left(\stackrel{¯}{x},\delta \left(\beta \right)\right),$

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