The main result of the paper is the following common fixed point theorem for compatible and weakly compatible self-mappings on Menger spaces under -contractive conditions, which improves the result of B. Singh and S. Jain [J. Math. Anal. Appl. 301, No. 2, 439–448 (2005; Zbl 1068.54044)]:
Let be the class of all functions , such that is non-decreasing, upper semi-continuous from the right and
for all Let be self-maps on a complete Menger space with a continuous t-norm of Hadžić-type. If
(ii) either or is continuous;
(iii) is compatible and is weakly compatible;
(iv) there exists such that
for all , and , where
then and have a unique common fixed point.