李陶，方欣(中国科学技术大学地球和空间科学学院，安徽合肥 230052)



钠激光雷达对中间层顶大气温度和风场的探测

李陶，方欣
(中国科学技术大学地球和空间科学学院，安徽合肥 230052)

摘 要:中高层大气温度和风场是研究中高层大气波动的重要参数．钠高光谱分辨率激光雷达能够对中间层顶(80－105km)大气温度和风场进行高精度观测．2011年中国科学技术大学成功研制了我国首台高光谱分辨率钠测温测风激光雷达系统．文中对该激光雷达系统进行了详细介绍，其中包括探测的基本原理，发射机，接收机和采集控制部分的设计．给出了钠测温测风激光雷达于2011年12月9日晚同时探测的大气温度、纬向风、经向风和钠原子密度的结果．结果发现中间层顶区域大气温度、纬向风、经向风变化范围较大，分别是175K～235K，－70～60m/s，－100～110m/s，有明显的半日或周日潮汐振荡的成分．探测结果表明中国科学技术大学钠测温测风激光雷达可对中间层顶区域温度和大气风场进行高时空分辨率的探测，其探测数据对于研究中高层大气动力学具有重要的意义．

关键词:大气温度;纬向风;经向风;中高层大气

引用格式:李陶，方欣．钠激光雷达对中间层顶大气温度和风场的探测［J］．安徽师范大学学报:自然科学版，2015，38(2) :103－109．

引言

中间层顶区域是大气波动(重力波、潮汐波、行星波等)的活跃区域．这些波动携带的能量和动量会随着波的破碎和耗散过程直接影响背景大气，从而驱动中间层大气环流以及各大气层间的祸合．因此对该区域的大气温度和风场的探测对于研究这些中高层大气物理过程具有非常重要的意义．中频雷达和流星雷达可以实现对该区域风场的垂直分辨探测，但很难获得温度信息．由于对钠荧光光谱认识的提升，推动了激光雷达对中间层顶区域温度和风场探测的快速发展．Gibson和Thomas等于1979年首次实现了对中间层顶区域温度的测量，在钠层峰值附近温度误差±15K［1］．Fricke和von Zahn(1985)利用准分子泵浦染料激光器实现了在10分钟内1km垂直分辨率和5K精度的温度廓线测量．1990年美国科罗拉多州立大学(CSU) She带领的研究组和伊利洛伊大学Gardner研究组合作，研制了利用双频技术的高光谱分辨率窄带系统，并于1991年开始了对中间层顶(80－105 km)温度的常规观测(She et al．，1992)［2］，于1997年开始同时对温度和水平风场的常规观测，并于2002年开始了对温度和风场的24hr连续常规观测(She et al．，2003)．美国科罗拉多州立大学于2000年在ALOMAR建了另外一台Na测温测风激光雷达(She et al．，2002)［3］．美国伊利洛伊大学也于1994年与科罗拉多州立大学合作在Urban建了一台类似的Na温测风激光雷达，其技术指标和性能与CSU激光雷达近似．在亚洲，日本的信州大学的N．激光雷达系统可以探测中间层顶区域的温度(Kawahara et al．，2003)［4］，但目前未有风场数据的报道．近几十年来，国际上钠测温测风激光雷达为中高层大气研究做出了突出贡献．

针对国内对中间层顶区域探测资料医乏的现状，中国科学技术大学(USTC)成功研制了一台钠测温测风激光雷达系统，是国内首次通过了国内外专家鉴定的钠测温测风激光雷达系统(CLi et al．［5］)．本文将对该钠测温测风激光雷达系统设计进行详细介绍，并给出其钠原子密度、温度和风场的探测结果，及其对重力波动量通量的探测结果．

1　钠高光谱分辨率激光雷达系统

钠高光谱分辨率激光雷达系统由发射机、接收机、采集部分和控制部分组成．图1给出了该激光雷达系统的原理示意图．其探测的基本原理是:通过发射钠原子荧光谱D2a线的三个频率窄带激光(υ0，υ0+ 630MHz，υ0－630MHz)，其中υ0对应的波长为589．158nm．系统分别接收这三个频率的回波信号．由于温度和风速的变化，会引起钠荧光光谱的多普勒增宽和多普勒频移，这会体现在三个频率回波信号强度的变化．根据实际探测的三个频率回波信号光子数，可反演出大气温度和风场．利用任一频率回波信号光子数可反演出钠原子密度．

下面将分别对系统各个部分进行阐述．

1．1发射机

钠高光谱分辨率激光雷达系统发射机主要由:半导体连续激光器(CW YAG)、环形染料激光器(Ring dye laser)、脉冲泵浦激光器(30Hz Pulsed YAG)和脉冲染料放大激光器(PDA)、激光频率绝对锁定跟踪系统(Doppler free system)、三频激光发生器(AOM)和自动准直系统(Beam steering)组成．

半导体连续激光器泵浦环形激光器用于产生整个激光雷达系统所需的窄带激光光源．半导体连续激光器输出波长532nm的激光，其最大输出率为6W左右．半导体连续激光器泵浦环形染料激光器Matisse DS，产生单模种子光源．环形染料激光器通过外部参考腔(reference cavity)采用侧边缘锁定技术(side offringe)可使得激光器输出的激光线宽均方根值在500KHz以内．通过扫描外部参考腔的方法实现精细扫描激光频率，其扫描范围约为6GHz．最后输出的波长为589．158nm．

由于环形染料激光器本身的参考腔只能相对锁定激光器的频率，随着环境的变化(温度、震动等)和自身压电陶瓷原件的松弛，其输出的激光频率会产生漂移，可能在几秒或几分钟内产生几百MHz的频率漂移．而对于测量大气风场来说，1MHz的频移将会引起风速测量误差约为0．6m/s(Li，2005)［6］．因此，激光器输出激光的频移将引起很大的测量误差．我们需绝对锁定环形染料激光器输出激光的频率．激光频率绝对锁定跟踪系统用于保证环形染料激光器的输出波长精确稳定在589．158nm．我们选用充满钠原子的蒸汽池(钠原子池，Sodium cell)中的钠原子在受激辐射情况下输出的荧光信号谱线作为频率的绝对标准．激光频率绝对锁定和自动跟踪系统结合环形染料激光器外部参考腔的侧边缘(side offringe)锁定技术(Fritschel，1989)［7］，利用相位锁定技术(Arie et al．，1992; Gianlucaet al．，2003; Smith et al．，2008)，通过基于La bview的锁定控制软件，在外部参考腔的带有压电陶瓷的镜片上加上频率为的参考抖动信号，将光电倍增管探测的荧光信号和参考抖动信号进行相位比较，得到偏差信号，通过PID(比例积分微分)控制算法将该偏差信号转换为NI(美国国家仪器公司)的多功能数据采集卡输出电压控制压电陶瓷执行动作．从而使得环形染料激光器始终锁定在钠无多普勒荧光光谱的D2a峰值凹陷处．图2是激光频率绝对锁定跟踪系统扫描得到的钠原子饱和荧光光谱．锁定系统精度为±2MHz．

三频激光发生器用于产生钠高光谱分辨率激光雷达系统所需的三个频率(υ0，υ0+630MHz，υ0－630MHz)的种子光源．我们主要利用声光相互作用的基本原理，实现激光频率频移的目的．环形激光器的输出激光以布拉格入射角入射到声光晶体器件(AOM)．其工作原理图3如图所示．通过外部施加丁TL电平信号给声光晶体的控制信号源，使得声光晶体是否处在调制频率的状态．在第一个脉冲周期(0～1/ 30s)内，施加给AOM1和AOM2的控制信号源TTL电平为低电平，两块声光器件都不调制激光频率，声光调制器系统输出的激光频率为D2apeak，图3上图所示．在第二个脉冲周期(1/30～2/30)，施加给AOM1控制信号源低电平，施加给AOM2控制信号源高电平AOM1不调制激光频率，AOM2向上频移激光频率315MHz，由于两次通过AOM2，声光调制系统输出的激光频率为υ0+ 630MHz，图3中间图所示．在第三个脉冲周期(2/30～3/30)内，施加给AOM1控制信号源高电平，施加给AOM2控制信号源低电平，AOM1向下频移激光频率315MHz，AOM2不调制激光频率，同样两次通过AOM1，声光调制系统输出的激光频率为υ0－630MHz，图3下图所示．整个声光调制器系统最终输出光的频率是30Hz，交替输出，其顺序为υ0，υ0+630MHz和υ0－630MHz．

脉染料冲放大器用于放大三个频率种子激光的功率，并将其转换成脉冲激光．其由声光调制器输出的三种频率激光在染料池DC1，DC2和DC3中被脉冲ND: YAG激光器输出的532nm激光泵浦，从而实现激光的三级放大．图4是我们自主研制的脉冲染料放大器的原理图．从声光调制器输出的589nm激光由1．5～3mm小孔光阑入射，经介质反射镜M1反射后由平凸透镜L1汇聚通过第一级染料流动池DC1，被染料介质吸收后的激光由50微米小孔Pinhole 2改善光斑形状，经由双凹透镜L2和双凸透镜L3汇聚通过第二级染料流动池DC2内染料介质后，由小孔Pinhole 3改善光斑形状，经双凹透镜L4和双凸透镜L5改善光束发散角后通过第三级染料流动池DC3．脉冲ND: YAG激光器输出的532nm从PDA左上部左侧入射，经第一片光束分束器BS1分束，部分反射光由棱柱形透镜CL1汇聚到染料流动池DC1内染料上．透射光再由第二片分束器分束，部分反射光由棱柱形透镜CL2汇聚到染料流动池DC2内染料上，透射光由反射晶体P1和P2反射进第三个染料流动池DC3和589nm光束成小角度通过染料介质．激光染料介质由若丹明640(Rodamine 640，也叫Rhodamine 101)和奇通红620(Kiton Red 620，也叫Sulforhodamine B)溶解在1000m1高纯度甲醇混合而成．第一级和第二级使用同一种浓度染料介质，Rodamine 640含量为9．16mg，Kiton Red含量为44．3mg．第三级使用一种浓度染料介质，Rodamine 640含量为1．9mg，Kiton Red含量为14．3mg．染料介质通过染料循环器抽运，以7L/min的流速循环．在17W激光泵浦情况下，当输入的589nm频率为υ0激光为350mW时，PDA输出功率约为1．65W，当输入的种子激光υ0+630MHz的功率为300mW时，PDA输出功率约为1．5W，当输入的种子激光υ0－630MHz的功率为250mW时，PDA输出功率约为1．45W．PDA输出激光的脉冲宽度和上升沿时间通过快速光电二极管和示波器测量，测得其输出脉冲宽度约10ns，上升沿时间约4ns．

自动准直系统用于准直钠测温测风激光雷达系统收发光轴，保证系统工作在最佳效率状态．我们采用的回波信号法进行准直，通过驱动电机控制发射天线在倾斜方向和旋转方向进行扫描，电机每动作一次后记录回波信号强度．准直过程可以分为三步:1)回波信号追踪; 2)单轴扫描;3)矩阵扫描．图5和图6给出了自动准直系统其中一个调整镜架在倾斜和旋转方向(对应于地理东西和南北方向)扫描的回波信号曲线．脉冲累加次数为24，积分距离为4km，激光发射平均功率约1．4W．图中显示了三个不同高度(22km，30km和90km)的回波信号，从图中可以看出近似的梯形函数关系，根据梯形的半高宽算出倾斜方向视场角约1．05mrad，旋转方向视场角为0．975mrad．图7给出自动准直系统进行矩阵扫描的二维强度图结果．脉冲累加次数为48，图中显示的信号是21km－23km的2km范围的积分，由中心计算公式(1)计算出倾斜方向位置为27μrad，旋转方向中心位置为70μrad．(中心位置的弧度值表示的含义是最后定准的中心位置偏离自动准直镜架初始位置的角度)．最终发射光束位置定位在矩阵扫描的等值线图顶部的中心位置．由误差传递公式，在能量波动为5%的情况下，根据实际扫描信号计算出自动准直系统的准直误差约为10μrad．

1．2接收机

钠测温测风激光雷达接收系统主要用于接收钠原子共振荧光后向散射回波信号．其主要由望远镜、斩光盘系统、后继接收光路组成．图8是钠测温测风激光雷达接收系统和采集系统结构图．钠测温测风激光雷达系统选用牛顿反射式望远镜．望远镜直径为76cm，F/#为F/2．4．在观测经纬向风场模式下，两台望远镜其中一台向正东倾斜和天顶的角30°，另一台向正北倾斜和天顶角倾斜30°．望远镜接收的回波信号由光纤传输到后续接收通道，包括斩光盘系统和后继祸合光路．光纤芯径为2000μm，数值孔径NA =0．37，长度15m，内部透过率＞90%．斩光盘系统由一高速旋转的盘片、驱动电机和电机控制器组成．其工作转速为5400转/分钟，盘片直径为200mm．光纤出射光经过斩光盘片后由焦距f =30mm，直径为25．4mm的透镜准直后通过滤光片．滤光片的直径为25．4mm，中心波长为589．1nm，带宽为1nm．经滤光片后的光信号再经过f =40mm、直径为25．4mm的透镜汇聚到光电倍增管阴极接收面上．光谱响应范围为300～720nm，在589nm处其量子效率可达40%．该倍增管集成了内部放大转换电路，直接输出0～5V的光子电脉冲信号，脉冲宽度为8ns，单光子脉冲分辨率为20ns．

1．3采集和控制部分

光电倍增管输出的脉冲信号由PCI插槽的MCA－3系列光子计数卡P7882进行采集，采集卡时间间隔设置为1μs(对应距离分辨率为150m)，采样长度为2048．采集软件基于LabVIEW开发环境开发．利用光子计数卡提供的动态链接库接口函数，可方便对光子计数卡进行控制．程序界面左侧用于常规参数的输入和关键参数的显示，右侧是回波信号显示．程序内部集成有环形染料激光器报警功能模块，用于提醒观测人员在跳模情况下对系统进行适当调整操作．

时序控制部分是保证钠测温测风激光雷达系统按照一定秩序正常运行的关键，只有在时序正确的情况下，才能获取到正确有效的回波信号．需要协同工作的设备有:接收系统斩光盘，声光晶体，AOM系统斩光盘，控制计算机，光子计数卡，脉冲泵浦Nd: YAG激光器．图9给出了该激光雷达系统时序控制框图．接收斩光盘输出脉冲外触发第一台数字延迟脉冲发生器DG645－1．DG645－1一个通道输出同步脉冲激光器(30Hz pulsed YAG)，一个通道输出同步第二台数字延迟脉冲发生器DG645－2．DG645－2的三个通道输出分别同步三频激光发生器中声光晶体1，声光晶体2和AOM斩光盘．脉冲激光器的Q开关输出、控制计算机和采集计算机的同步信号及DG645－2的二个通道输出信号输入给自制的转换电路，转换后的信号作为光子计数卡的触发信号和光谱标记信号．

2　探测结果

系统在进行探测时，其存储原始回波信号文件是150m距离分辨率，1分钟时间分辨率的数据文件．在数据反演过程中，我们先进行坏数据的判断，将无效的回波信号剔除．接着对回波信号进行预处理，为了提高系统的信噪比，积分15分钟范围内回波信号，扣除背景噪声，并以2km汉宁窗进行平滑处理．最后，预处理后的回波信号，计算得到大气温度、纬向风、经向风和钠原子密度信息．图10给出2011年12月09日晚探测的大气温度、纬向风、经向风和钠原子密度结果．从图中可以看出，整晚温度在175K～235K范围内．随着时间的变化，温度变化非常剧烈，14:00UT在93km处为整晚最低值175K，温度最大值出现81Km左右．在87～105km高度内，可以看到较为明显的相位向下移动的波动传播结构，有较为明显的12小时半日潮汐波动，100km高度处幅度约为15K．纬向风结果显示纬向风整晚变化范围是－70m/s～60m/s．也可看到较为明显的12小时半日潮汐波动，幅度值约在101km高度最大，约55m/s振幅．经向风整晚在－100～110m/s范围内变化，也可看到较为明显的相位向下传播波动结构．可以看到明显的周期12小时半日潮汐，其振幅在105km高度达到最大值，约100m/s．90km处可以看到较为明显的周期24小时的周日潮汐，其幅度值约为30m/s．钠原子密度随高度和时间变化也比较剧烈，其最大值出现在18:30UT左右93km高度处，约4400原子数/cm3．

3　小结

本文详细地叙述了钠高光谱分辨率测温测风激光雷达的基本原理，并对激光雷达系统的发射子系统，接收子系统和控制子系统进行了详细的叙述．最后给出了测量结果，结果显示在中间层顶区域(80－105km)，大气温度和风场的变化与大气重力波和潮汐波有直接的关系．目前我们己经积累了大量的高精度探测数据，这对于研究中高层大气动力学，大气重力波和潮汐波的相互关系具有重要的科学意义．同时该激光雷达对于提高我国空间环境的保障能力具有非常重要的应用价值．

参考文献:

［1］GIBSON A，THOMAS L，BHATTACHACHARYYA S．Laser observation of ground-state hyperfine structure of sodium and of temperatures in the upper atmosphere［J］．Nature (London) 1979，281:131－132．

［2］FRICKE K H，VONZAHN U．Mesopause temperature derived from probing the hyperfine structure of the D2 resonance line of sodium by lidar［J］．J Atmos Terr Phys，1985，47:499－512．

［3］SHE C Y，YU J R，LATIFI H，BILLS R E．High-spectral-resolution fluorescence lidar for mesospheric sodium temperature measurements［J］．Appl Opt，1992，31:2095－2106．

［4］She C Y，J Sherman，Yuan T，B P Williams，K Arnold，T D Kawahara，T Li，L Xu，J D Vance and D A Krueger，The first 80-hour continuous lidarcampaign for simultaneous observation of mesopause region temperature and wind［J］．Geophys Res Lett，2003，30(6) :1319．

［5］SHE C Y，VANCE J D，WILLIAMS B P，KRUEGER D A，MOOSMULLER H，D Gibson-Wilde and D Fritts．Lidar studies of atmospheric dynamics near the polar mesopause［J］．EOS，2002，83:289－293．

［6］LI Tao，FANG Xin，LIU Wei，GU Sheng-Yang and DOU Xiankang．A narrow band sodium lidar for the measurements of mesopause region temperature and wind［J］．Applied Optics，2012，51(21) :5401－5411．

［7］LI T，Sodium Lidar Observed Variability In Mesopause Region Temperature And Horizontal Wind: Planetary Wave Influence And Tidal-Gravity Wave Interactions: PhD Dissertation［D］．Fort Collins: Colorado State University．2005．

High-Spectra Resolution Sodium Lidar for Mesopause Region Temperature and Wind Measurements

LI Tao，FANG Xin
(School of Earth and Space Sciences，University of Science and Technology of China，Hefei 230052，China)

Classification No: O175．8

Document code: A

Paper No:1001－2443(2015) 02－0110－07

Received date: 2014－04－10

Foundation item: Supported by the National Natural Science Foundation of China(11026213)．

Corresponding author: YAN Jie(1990－)，Female，born in Jiangsu，master，major in differential equation．

引用格式:严洁，肖建中．带积分边界条件的奇异多点边值问题的正解［J］．安徽师范大学学报:自然科学版，2015，38(2) :110－116．

In this paper，we consider the existence of positive solutions for the following singular integral boundary value problem (BVP for short)

where m≥3，0＜η1＜…＜ηm－2＜1，αi≥0(i = 1，2，…，m－2) withf∈C(［0，∞)，［0，+∞) ) and g∈L1［0，1］is nonnegative; the function a∈C((0，1)，［0，+∞) ) may be singular at t = 0，1．

The multi-point boundary value problems for ordinary differential equations arise in areas of applied mathematics and physics．For example，many problems in the theory of elastic stability，non Newtonian fluid and the turbulence theory of gases in porous media can be handled by the method of multi-point problems (see［1，2］)．In 1980s，Bitsadze and Samarkii firstly researched the nonlocal elliptic boundary value problems (see［3］)．Motivated by their work，the study of multi-point boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev (see［4］)．Since then，multi-point boundary value problems for nonlinear ordinary differential equations has received much attention for many authors．To identify a few，we refer the reader to［5－7］and references therein．

In［6］，the following second order three-point boundary value problem was considered，where 0＜α＜1，0＜η＜1，f∈C(R，R)．The authors obtained the existence results of signchanging solutions for the above three-point boundary value problem by using the fixed-point index method．

A class of boundary value problems with integral boundary conditions for ordinary differential equations arise in the study of various physical，biological and chemical processes，such as heat conduction，undergroundwater flow，thermoelasticity and plasma physics．The existence of positive solutions for such class of problems has attracted much attention (see［8，9］)．

In［8］，Feng et al．investigated the existence and nonexistence of positive solutions of the following second order boundary value problem with integral boundary conditions in Banach space．

In［9］，Liu et al．investigated the existence of positive solutions for the singular second order integral boundary value problem where a，b∈C［0，1］，c∈C((0，1)，(0，+∞) )，f∈C((0，+∞)，［0，+∞) ) and g，h∈L1［0，1］are nonnegative; c may be singular at t = 0，1 and f may be singular at u = 0．The authors established the existence of positive solutions for the above problem by applying the fixed point index theorems．

Motivated by the works mentioned above，in this paper，we consider the BVP(1)．To the best of our knowledge，there are very few results for multi-point boundary value problems with integral boundary conditions．By applying Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type，we establish the existence of positive solutions．

1 Preliminaries

Let E = C［0，1］be a Banach space with the norm，and letbe a cone in E，where σ is a positive constant．For ΩP，the closure of Ω．The function u∈C［0，1］∩C2(0，1) is said to be a positive solution of the BVP (1)，if it satisfies (1) and u(t)≥0 for t∈(0，1)．The function e will be defined by e(s) = s(1－s)，s∈［0，1］．

We will frequently use the following constants:

For convenience in presentation，we here list two assumptions to be used throughout the paper．

(H1) f∈C(［0，+∞)，［0，+∞) )，g∈L1［0，1］is nonnegative such that k＞0．

(H2) a: (0，1)→［0，+∞) is continuous，and，and a(t)0 on any subset of (0，1)．

Our approach is based on the following Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type．

Lemma 1．1(see［10］) Let E a Banach space，and PE be a cone in E．Assume Ω1，Ω2are open subset ofE withand A:be a completely continuous operator such that，either

(1)‖Au‖≤‖u‖，u∈P∩Ω1，‖Au‖≥‖u‖，u∈P∩Ω2; or

(2)‖Au‖≥‖u‖，u∈P∩Ω1，‖Au‖≤‖u‖，u∈P∩Ω2，then A has a fixed point in

2 Lemmas

Lemma 2．1 Assume that (H1) (H2) hold，y(t)∈C［0，1］．Then the BVPhas a unique solution can be expressed in the form

Proof．Integrating both sides of (5) on［0，t］twice，we have By (5)，we get

Combining with (10)，(11) and (12)，we get

Thus，by (9) and (8)，we obtain

Multiplying g(t) on the both sides of (13)，and integrating from 0 to 1，and then solving∫10g(t) u(t) dt，we have

Combining with (13)，(14) and (7) we get

which shows the lemma．

Remark 2．1 For any t，s∈［0，1］，it is easy to check that

Lemma 2．2 If (H1) hold，then for any t，s∈［0，1］，it holds where

Proof．Firstly，from (8)，(9) and (15)，we have

Combining with (7) and (17)，we get ?

From (7) and (19)，we get

Thus，combining with (18) and (20)，Lemma 2．2 is proved．It follows from Lemma 2．1 that u∈E is a solution of the BVP (1) if and only if u is a fixed point of T．Lemma 2．3 For any t1，t2，s∈［0，1］，it holdsProof．For any t1，t2，s∈［0，1］，if t1，t2≤s，then

| l(t1，s)－l(t2，s) | = (1－s) | t1－t2|≤| t1－t2|．If t1，t2≥s，then | l(t1，s)－l(t2，s) | = s | t1－t2|≤| t1－t2|．Without loss of generality，we suppose t1＜t2and t1≤s≤t2．Since

－(t2－t1)≤(s－1) (t2－t1)≤l(t1，s)－l(t2，s)≤s(t2－t1)≤(t2－t1)，

which shows that (22) is also true in this case．

Lemma 2．4 If (H1)，(H2) are satisfied，then T: P→P is a completely continuous operator．

Proof．For each u∈P，by the definition of T and the assumptions (H1) and (H2)，we have (Tu) (t)≥0，t∈［0，1］．It follows from Lemma 2．2 that

Combining with (23) and (24)，we have

i．e．，Tu∈P．This shows that T(P)P．

Step 1．T is bounded．If D = { u∈P:‖u‖≤r} is a bounded subset of P，then for each u∈D，we have u≤r，let M = max{ f(u) : u≤r} be a constant．By (21) and (16)，

thus T(D) is bounded in P．

Step 2．T(D) is equicontinuous．For any ε＞0，there exists δ=ε＞0．where the constant

for any | t1－t2|＜δ，(Tu) (t1)，(Tu) (t2)∈T(D)，from Lemma 2．2 and Lemma 2．3，

So T(D) is equicontinuous．

Step 3．T is continuous．Take { um}∞m =0with‖um－u0‖→0(m→∞)．There exists R0such that { um}∈［0，

R］．Let M= max{ f(u) :0≤u≤R}，L=1mk．Since | H(t，s) a(s) f(u(t) ) |≤mke(s) a(s)≤

0000402m02

L0a(s)，then from (H2) and Lebesgue dominated convergence theorem，we have

Thus T is continuous．

Above knowable，T is a completely continuous operator．

3 Main results

In the following，we shall give the main results of this paper．

Let A，B be two positive constants defined by A =

Theorem 3．1 Suppose that (H1)，(H2) are satisfied，and there exist two positive constants R1and R2with R1≠R2such that

then the BVP (1) has at least one positive solution u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}．

Proof．Without loss of generality，we suppose R1＜R2．Let Ω1= { u∈C［0，1］:‖u‖＜R1}，Ω2= { u∈C［0，1］:‖u‖＜R2}．Then from (H3) and Lemma 2．2，for any u∈P∩Ω1，we have‖u‖= R1，

thus‖Tu‖≤‖u‖，u∈P∩Ω1．(25)

On the other hand，from (H4) and Lemma 2．2，for any u∈P∩Ω2，we have‖u‖= R2，and σR2≤u≤R2，

thus‖Tu‖≥‖u‖，u∈P∩Ω2．(26)

Thus，from Lemma 1．1 and Lemma 2．4，T has a fixed point u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}，which is one positive solution of the BVP (1)．

Theorem 3．2 Suppose that (H1)，(H2) are satisfied，and there exist two positive constants R1and R2with R1≠R2such that then the BVP (1) has at least one positive solution u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}．

Proof．Without loss of generality，we suppose R1＜R2．Let Ω1= { u∈C［0，1］:‖u‖＜R1}，Ω2= { u∈

C［0，1］:‖u‖＜R2}．Then from (H5) and Lemma 2．2，for any u∈P∩Ω1，we have‖u‖= R1， thus‖Tu‖≤‖u‖，u∈P∩Ω1，(27)

On the other hand，from (H6) and Lemma 2．2，for any u∈P∩Ω2，wehave‖u‖= R2，and σR2≤u≤ R2， thus‖Tu‖≥‖u‖，u∈P∩Ω2．

Thus，from Lemma 1．1 and Lemma 2．4，T has a fixed point u*∈P，with min { R1，R2}≤‖u*‖≤max{ R1，R2}，which is one positive solution of the BVP (1)．

Corollary 3．3 Suppose that (H1)，(H2)，(H7) and (H8) are satisfied，

Then the BVP (1) has at least one positive solution u∈P．

To illustrate how our main results can be used in practice we present an example．

Example 3．4 Consider the singular integral boundary value problem

Proof．From the BVP (29)，we can get m = 3，．By calculating，．Thus，applying Corollary 3．3，we know that the BVP (29) has a positive solution．

References:

［1］KAPER H G，KNAPP M，KWONG M K．Existence theorems for second order boundary value problem［J］．Differ Integral Equ，1991，4:543－554．

［2］TIMOSHENKO S．Theory of elastic stability［M］．New York: McGraw-Hill，1961．

［3］BITSADZE A V．On the theory of nonlocal boundary value problems［J］．Sov Math Dokl，1984，30:8－10．

［4］IL’IN V A，MOISEEV E I．Nonlocal boundary value of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects ［J］．Differ Equ，1987，23(7) :803－810．

［5］FENG W，WEBB J R L．Solvability of a three-point nonlinear boundary value problem at resonance［J］．Nonlinear Anal，1997，30:3227－3238．

［6］XIAN X，SUN J X．On sign-changing solution for some three-point boundary value problems［J］．Nonlinear Anal，2004，59:491－505．

［7］REN J L，GE W G．Positive solution for three-point boundary value problems with sign changing nonlinearities［J］．Appl Math Lett，2004，17:451 －458．

［8］FENG M Q，JI D H，GE W G．Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces［J］．J Comput Appl Math，2008，222:351－363．

［9］LIU L S，HAO X A，WU Y H．Positive solutions for singular second order differential equations with integral boundary conditions［J］．Math Compt Model，2013，57:836－847．

［10］GUO D，LAKSHMIKANTHAM V．Nonlinear Problems in Abstract Cones［M］．San Diego: Academic Press，1988．

Abstract:Middle and upper atmosphere temperature and wind are important parameters to study atmospheric waves．A high-spectra resolution sodium lidar was developed at the University of Science and Technology of China (USTC) in 2011．It can be used for temperature and wind measurements in mesopause region(80－105km)．First，the lidar system is introduced in details，including principle of measurements，the transmitter，receiver，and acquisition and control subsystems．The results of mesopause region temperature，zonal wind，meridional wind and sodium density，observed simultaneously by this lidar on the night of December 9，2011，are then present．The results show that mesopause temperature，zonal wind，meridional wind vary in significantly with 175～235K，－70m～60m/s and－100～100m/s respectively，likely induced by the solar diurnal and/or semidiurnal tides．These results further indicate that the USTC high-spectra resolution sodium lidar is an effective tool for mesopause region temperature and wind measurements with high temporal and spatial resolutions，and the observational data enable us to study the middle and upper atmosphere dynamics in great high resolutions． In this paper，we are concerned with the existence of positive solutions for singular second order multi-point boundary value problems with integral boundary conditions．The arguments are based upon a specially constructed cone and the fixed point theorem of cone expansion-compression type．Meanwhile，an example is given to demonstrate the main results．

Key words:atmospheric temperature; zonal wind; meridional wind; middle and upper atmosphere boundary value problems; positive solutions; fixed point theorem; existence of positive solutions

Positive Solutions for Singular Multi-Point Boundary Value Problems with Integral Boundary Conditions

YAN Jie，XIAO Jian-zhong
(School of Mathematics and Statistics，Nanjing University of Information Science and Technology，Nanjing 210044，China)

作者简介:李陶(1974－)，男，安徽巢湖人，安徽师范大学物理系1992级校友．中国科学技术大学地球和空间科学学院教授，博士生导师，中国科学院“百人计划”入选者(2009年)，国家杰出青年基金获得者(2012年)．

基金项目:国家自然科学基金(41225017)．

收稿日期:2015－03－03

DOI:10．14182/J．cnki．1001－2443．2015．02．001 10．14182/J．cnki．1001－2443．2015．02．002

文章编号:1001－2443(2015) 02－0103－07

文献标志码:A

中图分类号:TN249