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\mu^{*} \in W , \lim_{n\rightarrow\infty}\|\mu _{n}-\mu^{*}\| .

- \{\mu_{n}\} .

([ 28 ])

\|x-y\|^{2}=\|x\|^{2}-\|y\|^{2}-2\langle x-y,y \rangle .
(2.4)
\left \{ \begin{array}{@{}l} {F(u_{n},u)+\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall u\in C};\\ {G(v_{n},v)+\varphi(v)-\varphi(v_{n})+\frac{1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0,\quad \forall v\in Q};\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}))};\\ {y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} B^{*}(Au_{n}-Bv_{n})), \quad\forall n\geq1}; \end{array} \right .
(3.1)

0<\alpha\leq\alpha_{n}\leq\beta<1 ( \alpha, \beta\in (0,1) );

\liminf_{n\rightarrow\infty}r_{n}>0 \lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0 .

\{(x_{n}, y_{n})\} ( 1.12 ).

, , -, \{(x_{n},y_{n})\} ( 1.12 ).

Now we prove conclusion (I).

\begin{aligned} \|u_{n}-x\|=\bigl\| T_{r_{n}}^{F}x_{n}-T_{r_{n}}^{F}x \bigr\| \leq\|x_{n}-x\|, \end{aligned}
(3.2)
\begin{aligned} \|v_{n}-y\|=\bigl\| T_{r_{n}}^{G}y_{n}-T_{r_{n}}^{G}y \bigr\| \leq\|y_{n}-y\|. \end{aligned}
(3.3)
\begin{aligned}[b] \|x_{n+1}-x\|^{2}={}\bigl\| \alpha_{n}u_{n}+(1- \alpha_{n})T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\| ^{2} \\ ={}\alpha_{n}^{2}\|u_{n}-x\|^{2}+(1- \alpha_{n})^{2}\bigl\| T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\| ^{2} \\ {} +2\alpha_{n}(1-\alpha_{n})\bigl\langle u_{n}-x,T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\rangle \\ \leq{}\alpha_{n}^{2}\|u_{n}-x \|^{2}+(1-\alpha_{n})^{2}\bigl\| u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n})-x\bigr\| ^{2} \\ {} +2\alpha_{n}(1-\alpha_{n})\bigl\langle u_{n}-x,T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\rangle \\ \leq{}\alpha_{n}^{2}\|u_{n}-x \|^{2}+(1-\alpha_{n})^{2}\bigl\| u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n})-x\bigr\| ^{2} \\ {} +\alpha_{n}(1-\alpha_{n}) \bigl(\|u_{n}-x \|^{2}+\bigl\| u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})-x\bigr\| ^{2}\bigr) \\ ={} \alpha_{n}\|u_{n}-x\|^{2}+(1- \alpha_{n})\bigl\| u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})-x \bigr\| ^{2} \\ \leq{}\alpha_{n}\|u_{n}-x\|^{2}+(1- \alpha_{n}) \bigl(\|u_{n}-x\|^{2}+\bigl\| \rho _{n}A^{*}(Au_{n}-Bv_{n})\bigr\| ^{2} \\ {} -2\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle \bigr) \\ \leq{}\|x_{n}-x\|^{2}+(1-\alpha_{n})\bigl\| \rho_{n}A^{*}(Au_{n}-Bv_{n})\bigr\| ^{2} \\ {} -2(1-\alpha_{n})\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle . \end{aligned}
(3.4)
\begin{aligned} \bigl\| \rho_{n}A^{*}(Au_{n}-Bv_{n}) \bigr\| ^{2}=\rho_{n}^{2}\bigl\langle A^{*}(Au_{n}-Bv_{n}),A^{*}(Au_{n}-Bv_{n}) \bigr\rangle \\ =\rho_{n}^{2}\bigl\langle Au_{n}-Bv_{n},AA^{*}(Au_{n}-Bv_{n}) \bigr\rangle \\ \leq\lambda_{A}\rho_{n}^{2}\langle Au_{n}-Bv_{n},Au_{n}-Bv_{n}\rangle \\ =\lambda_{A}\rho_{n}^{2}\|Au_{n}-Bv_{n} \|^{2}. \end{aligned}
(3.5)
\begin{aligned} \|x_{n+1}-x\|^{2} \leq{}\|x_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{A} \rho_{n}^{2}\| Au_{n}-Bv_{n} \|^{2} \\ {}-2(1-\alpha_{n})\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle . \end{aligned}
(3.6)
\begin{aligned} \|y_{n+1}-y\|^{2} ={}\|y_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{B} \rho_{n}^{2}\|Au_{n}-Bv_{n} \|^{2} \\ {} +2(1-\alpha_{n})\rho_{n} \langle Bv_{n}-By,Au_{n}-Bv_{n} \rangle . \end{aligned}
(3.7)
\begin{aligned} \|x_{n+1}-x\|^{2}+\|y_{n+1}-y \|^{2} \\ \quad\leq\|x_{n}-x\|^{2}+\|y_{n}-y \|^{2}-\rho_{n}(1-\alpha_{n}) \bigl(2- \rho_{n}(\lambda _{A}+\lambda_{B})\bigr) \|Au_{n}-Bv_{n}\|^{2}. \end{aligned}
(3.8)
\Gamma_{n+1}(x,y)\leq\Gamma_{n}(x,y)-\rho_{n}(1- \alpha_{n}) \bigl(2-\rho_{n}(\lambda _{A}+ \lambda_{B})\bigr)\|Au_{n}-Bv_{n} \|^{2}.
(3.9)
\lim_{n\rightarrow\infty}\|Au_{n}-Bv_{n} \|=0.
(3.10)
\begin{aligned} \|x_{n+1}-x_{n}\|^{2}=\|x_{n+1}-x-x_{n}+x \|^{2} \\ =\|x_{n+1}-x\|^{2}-\|x_{n}-x\|^{2}-2 \langle x_{n+1}-x_{n},x_{n}-x\rangle \\ =\|x_{n+1}-x\|^{2}-\|x_{n}-x\|^{2}-2 \bigl\langle x_{n+1}-x^{*},x_{n}-x\bigr\rangle +2\bigl\langle x_{n}-x^{*},x_{n}-x\bigr\rangle . \end{aligned}
\limsup_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0.
(3.11)
\limsup_{n\rightarrow\infty}\|y_{n+1}-y_{n} \|=0.
(3.12)
\lim_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0,
(3.13)
\lim_{n\rightarrow\infty}\|y_{n+1}-y_{n} \|=0.
(3.14)
F(u_{n+1},u)+\phi(u)-\phi(u_{n+1})+\frac {1}{r_{n+1}}\langle u-u_{n+1},u_{n+1}-x_{n+1}\rangle \geq0, \quad \forall u\in C,
F(u_{n},u)+\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall u \in C.
F(u_{n+1},u_{n})+\phi(u_{n})- \phi(u_{n+1})+\frac {1}{r_{n+1}}\langle u_{n}-u_{n+1},u_{n+1}-x_{n+1} \rangle \geq0
(3.15)
F(u_{n},u_{n+1})+\phi(u_{n+1})- \phi(u_{n})+\frac {1}{r_{n}}\langle u_{n+1}-u_{n},u_{n}-x_{n} \rangle \geq0.
(3.16)
\frac{1}{r_{n+1}}\langle u_{n}-u_{n+1},u_{n+1}-x_{n+1} \rangle +\frac{1}{r_{n}}\langle u_{n+1}-u_{n},u_{n}-x_{n} \rangle \geq0,
\biggl\langle u_{n+1}-u_{n},\frac{u_{n}-x_{n}}{r_{n}}- \frac{u_{n}-x_{n+1}}{r_{n+1}}\biggr\rangle \geq0,
\begin{aligned} 0\leq\biggl\langle u_{n+1}-u_{n},u_{n}-x_{n}- \frac {r_{n}}{r_{n+1}}(u_{n+1}-x_{n+1})\biggr\rangle \\ =\biggl\langle u_{n+1}-u_{n},u_{n}-u_{n+1}+u_{n+1}-x_{n}- \frac {r_{n}}{r_{n+1}}(u_{n+1}-x_{n+1})\biggr\rangle . \end{aligned}
\begin{aligned} \|u_{n+1}-u_{n}\|^{2}\leq\biggl\langle u_{n+1}-u_{n},x_{n+1}-x_{n}+\biggl(1- \frac {r_{n}}{r_{n+1}}\biggr) (u_{n+1}-x_{n+1})\biggr\rangle \\ \leq\|u_{n+1}-u_{n}\|\cdot\biggl[\|x_{n+1}-x_{n} \|+\biggl|1-\frac{r_{n}}{r_{n+1}}\biggr|\cdot\| u_{n+1}-x_{n+1}\|\biggr]. \end{aligned}
\|u_{n+1}-u_{n}\|\leq\|x_{n+1}-x_{n} \|+\biggl|1-\frac{r_{n}}{r_{n+1}}\biggr|\cdot\| u_{n+1}-x_{n+1} \|.
(3.17)
\lim_{n\rightarrow\infty}\|u_{n+1}-u_{n} \|=0.
(3.18)
\lim_{n\rightarrow\infty}\|v_{n+1}-v_{n} \|=0.
(3.19)
\begin{aligned} \|x_{n+1}-x\|^{2} \leq{}\|u_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{A} \rho_{n}^{2}\| Au_{n}-Bv_{n} \|^{2} \\ {} -2(1-\alpha_{n})\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle \end{aligned}
(3.20)
\begin{aligned}[b] \|y_{n+1}-y\|^{2} ={}\|v_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{B} \rho_{n}^{2}\|Au_{n}-Bv_{n} \|^{2}\\ {} +2(1-\alpha_{n})\rho_{n} \langle Bv_{n}-By,Au_{n}-Bv_{n} \rangle . \end{aligned}
(3.21)
\begin{aligned} \|x_{n+1}-x\|^{2}+\|y_{n+1}-y \|^{2} \\ \quad\leq\|u_{n}-x\|^{2}+\|v_{n}-y \|^{2}-\rho_{n}(1-\alpha_{n}) \bigl(2- \rho_{n}(\lambda _{A}+\lambda_{B})\bigr) \|Au_{n}-Bv_{n}\|^{2}, \end{aligned}
(3.22)
\begin{aligned} \|u_{n}-x\|^{2}=\bigl\| T_{r_{n}}^{F}x_{n}-T_{r_{n}}^{F}x \bigr\| ^{2} \\ \leq\langle x_{n}-x,u_{n}-x\rangle \\ =\frac{1}{2}\bigl(\|x_{n}-x\|^{2}+ \|u_{n}-x\|^{2}-\|x_{n}-u_{n} \|^{2}\bigr), \end{aligned}
(3.23)
\begin{aligned} \|v_{n}-y\|^{2}=\bigl\| T_{r_{n}}^{G}y_{n}-T_{r_{n}}^{G}y \bigr\| ^{2} \\ \leq\langle y_{n}-y,v_{n}-y\rangle \\ =\frac{1}{2}\bigl(\|y_{n}-y\|^{2}+ \|v_{n}-x\|^{2}-\|y_{n}-v_{n} \|^{2}\bigr). \end{aligned}
(3.24)
\begin{aligned} \|x_{n}-u_{n}\|^{2}+ \|y_{n}-v_{n}\|^{2} \\ \quad\leq\|x_{n}-x\|^{2}-\|x_{n+1}-x \|^{2}+\|y_{n}-y\|^{2}-\|y_{n+1}-y \|^{2} \\ \qquad{} -\rho_{n}(1-\alpha_{n}) \bigl(2- \rho_{n}(\lambda_{A}+\lambda_{B})\bigr) \|Au_{n}-Bv_{n}\|^{2}. \end{aligned}
(3.25)
\begin{aligned} \lim_{n\rightarrow\infty}\|x_{n}-u_{n} \|=0, \end{aligned}
(3.26)
\begin{aligned} \lim_{n\rightarrow\infty}\|y_{n}-v_{n} \|=0. \end{aligned}
(3.27)
\begin{aligned}[b] \|u_{n}-Tu_{n}\|={}\|u_{n}-x_{n+1}+x_{n+1}-Tu_{n} \| \\ \leq{}\|u_{n}-x_{n+1}\|+\|x_{n+1}-Tu_{n} \| \\ ={}\|u_{n}-u_{n+1}-u_{n+1}-x_{n+1}\| \\ {} +\bigl\| \alpha_{n} u_{n}+ (1- \alpha_{n})T \bigl(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}) \bigr)-Tu_{n}\bigr\| \\ \leq{}\|u_{n}-u_{n+1}\|+\|u_{n+1}-x_{n+1} \|+\alpha_{n} \|u_{n}-Tu_{n}\| \\ {} + (1- \alpha_{n})\bigl\| T\bigl(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-Tu_{n}\bigr\| \\ \leq{}\|u_{n}-u_{n+1}\|+\|u_{n+1}-x_{n+1} \|+\alpha_{n} \|u_{n}-Tu_{n}\| \\ {} + (1- \alpha_{n})\bigl\| -\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr\| . \end{aligned}
(1-\alpha_{n})\|u_{n}-Tu_{n}\| \leq \|u_{n}-u_{n+1}\|+\|u_{n+1}-x_{n+1}\|+ (1- \alpha_{n})\bigl\| -\rho_{n} A^{*}(Au_{n}-Bv_{n}) \bigr\| .
(3.28)
\lim_{n\rightarrow\infty}\|Tu_{n}-u_{n} \|=0.
(3.29)
\lim_{n\rightarrow\infty}\|Sv_{n}-v_{n} \|=0.
(3.30)
\begin{aligned} \|x_{n}-Tx_{n}\|\leq\|x_{n}-u_{n}+u_{n}-Tu_{n}+Tu_{n}-Tx_{n} \| \\ \leq\|x_{n}-u_{n}\|+\|u_{n}-Tu_{n} \|+\|Tu_{n}-Tx_{n}\| \\ \leq2\|x_{n}-u_{n}\|+\|u_{n}-Tu_{n} \|. \end{aligned}
(3.31)
\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n} \|=0.
(3.32)
\begin{aligned} \|y_{n}-Sy_{n}\|\leq\|y_{n}-v_{n} \|+\|v_{n}-Sv_{n}\|+\|Sv_{n}-Sy_{n}\| \leq2\|y_{n}-v_{n}\|+\|v_{n}-Sv_{n} \|, \end{aligned}
(3.33)
\lim_{n\rightarrow\infty}\|y_{n}-Sy_{n} \|=0.
(3.34)

Since \{x_{n}\} and \{y_{n}\} converge weakly to x^{*} and y^{*} , respectively, then it follows from ( 3.32 ), ( 3.34 ) and Lemma 2.3 that x^{*}\in F(T) and y^{*}\in F(S) . Since every Hilbert space satisfies Opial’s condition, Opial’s condition guarantees that the weakly subsequential limit of \{(x_{n}, y_{n})\} is unique.

Samples from different tissues were processed for single-cell suspensions. See Additional file Durango Mens DB021 Western Boot Dark Brown/Texas Flag 13 M US 2pAQyW
: Supplemental Experimental Procedures for details.

Cell staining procedures were detailed in the Additional file SenNi Mens and Womens Lightweight Casual Athletic Breathable Shoes Flashing Sneakers Blue mkoRNzpdrX
: Supplemental experimental procedures.

Single-cell homogenate from the LNs, spleen, lungs, and bone marrow were resuspended at a concentration of 1 × 10 cells per ml and incubated with CFSE (1μM) at 37°C for 10min. The reaction was then quenched by addition of ice-cold FBS. After CFSE labeling, 1 × 10 γδ T cells were plated per well of a 96-well plate. 1 × 10 DCs were added. After 5days, the cells were harvested. In some experiments, neutralizing IL-1β or IL-23 mAbs were added in this culture system, or transwell plates were used. Cells were analyzed by flow cytometry.

Co-housing protocols were described in Additional file 1 : Supplemental experimental procedures.

See Additional file 1 : Supplemental experimental procedures for details.

The detailed protocols for oral and fecal sample microbiota sequencing were described in Additional file 1 : Supplemental experimental procedures.

Results were exhibited as means ± SEM. Statistical analysis was performed using GraphPad Prism software. The statistical significance of differences between groups was determined by the Student’s test. All data were analyzed using two-tailed tests unless otherwise specified, and a value < 0.05 was considered statistically significant.

Antigen-presenting cells

Cervical lymph nodes

Dendritic cells

Germ free

IL-17 receptor

Inguinal lymph nodes

Interferon regulatory factor 8

Mesenteric lymph nodes

Pathogen-associated molecular pattern

Specific pathogen free

IL-17-producing γδ T cells

The authors thank Dr. Yang-Xin Fu for providing critical reagents for this study.

Funding

This work was partly supported by the Kentucky Research Challenge Trust Fund. Huang-ge Zhang is supported by a Research Career Scientist (RCS) Award.

Availability of data and materials

All data generated or analyzed during this study are included in this published article and its Additional files. The oral and fecal microbiota datasets were deposited in the Biosample database ( www.ncbi.nlm.nih.gov ) with submission ID: SUB2183894 and SUB2183406, respectively.

Authors’ contributions

CF participated in the design and coordination of the research project; collected, performed, and analyzed data; and contributed to the manuscript writing. YHC participated in the design and coordination of the research project and collected, performed, and analyzed data. XS, VRJ, FX, and SM participated in some of the experiments and analyzed the data. YLW performed a single-cell sequence experiment, and YHC supervised this experiment, analyzed the data, and revised the manuscript. HGZ, BH, and JH participated in the experimental design and provided critical reagents for the study. JY participated in the design and coordination of the research project, supervised the whole project, analyzed the data, revised the manuscript, and approved the final version of the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Ethics approval and consent to participate

This study only used animals and the protocol was approved by the IACUC of University of Louisville (IACUC#15241).

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver ( http://creativecommons.org/publicdomain/zero/1.0/ ) applies to the data made available in this article, unless otherwise stated.

Additional file 1: Includes supplemental experimental procedures and five figures. (DOCX 2846kb)
Department of Microbiology and Immunology, University of Louisville School of Medicine, Louisville, USA
Department of Medicine, James Graham Brown Cancer Center, University of Louisville School of Medicine, Louisville, USA
Department of Microbiology and Immunology, Stanford University, Stanford, USA
Department of Oncology, Zhejiang University the Second Affiliated Hospital, Hangzhou, China
Tumor Immunobiology Program, James Graham Brown Cancer Center, University of Louisville School of Medicine, Louisville, USA
©The Author(s).2017

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