Suppose f:C→X where C is a closed subset of X. Necessary and sufficient conditions are given for f to have a fixed point. All results hold when X is complete metric space. Several results hold in a much more general setting.
Singh, D., Sharma, M., & Sharma, R. (2012). Fixed Point Theorems via Absorbing Maps. Research Journal of Applied Sciences, Engineering and Technology, 4(14), 2134-2139. https://www.airitilibrary.com/Article/Detail?DocID=20407467-201207-201510260012-201510260012-2134-2139
JUNGCK, G., & RHOADES, B. (1993). SOME FIXED POINT THEOREMS FOR COMPATIBLE MAPS. International Journal of Mathematics and Mathematical Sciences, 1993(), 417-428. https://doi.org/10.1155/S0161171293000535