The 2.0 API provides support for /search:

http://api.stackexchange.com/2.1/search?site=[site]&intitle=[search string]

If you want to constrain this to your posts, 2.0's /search/advanced path allows you to filter by user, or by basically anything else:

http://api.stackexchange.com/2.1/search/advanced?site=[site]&title=[search string]&user=[your user ID]

Beware - /search calls the title parameter intitle, while /search/advanced calls it title.

The API v2 provides support for /search:

http://api.stackexchange.com/2.2/search?site=[site]&intitle=[search string]

If you want to constrain this to your posts, v2's /search/advanced path allows you to filter by user, or by basically anything else:

http://api.stackexchange.com/2.2/search/advanced?site=[site]&title=[search string]&user=[your user ID]

Beware - /search calls the title parameter intitle, while /search/advanced calls it title.