1. 普通的二分查找,利用中间数比较，将上界下标或者下界下标减1或者加1，针对查找某个数是否存在的情况，或者是某个有序数组单一存在的某个数：

private static int binary_search(int[] nums, int key) {
        int low = 0, high = nums.length-1;
        while(low<=high) {
            int mid = low + ((high-low)>>1);
            if (nums[mid] > key) high = mid-1;
            else if (nums[mid] < key) low = mid + 1;
            else return mid;
        }
        return -1;
    }

2. 查找下界，当有数组中存在多个重复的数据时，二分查找原型就不是那么好用了，这个时候就需要确定这个数对应的上界和下界。下界的求解过程就是我们在二分查找基础上改动，因为求解下界也就是左侧的下标，那么我们就是采用右侧往左压缩的过程，直到一个合适的索引为止,如下代码中所示。将等号取在第一个等式上，往左侧压缩：

private static int get_lower(int[] nums, int key) {
        int low=0, high = nums.length-1;
        System.out.println("test");
        while(low<=high) {
            int mid = low + (high-low)/2;
            if (nums[mid] >= key) high = mid-1;
            else low = mid+1;
        }
        if (low<nums.length && nums[low]== key) return low;
        return -1;
    }

3. 查找上界，原理和下界相同，不过是往右侧压缩：

private static int get_upper(int[] nums, int key) {
        int low = 0, high = nums.length-1;
        while(low<=high) {
            int mid = low + (high-low)/2;
            if (nums[mid] <= key) low = mid+1;
            else high = mid-1;
        }
        if(high>=0 && nums[high]==key) return high;
        return -1;
    }

部分参考链接

http://zhangxiaoya.github.io/2015/06/26/upper-bound-and-lower-bound-of-binary-search/