Is there any existence of 'One Person Company' in the business in India?

Yes

One Person Company (OPC): Sec. 2 (62) of the companies Act, 2013, defines OPC as a “company which has only one person as a member”.
Rule 3 of the Companies (Incorporation) Rules, 2014 provides that:
(a) Only a natural person being an Indian citizen and resident in India can form one person company,
(b) It cannot carry out non-banking financial investment activities.
(c) Its paid up share capital is not more than Rs. 50 Lakhs
(d) Its average annual turnover of three years does not exceed Rs. 2 Crores.