liitroduction
In this paper we give common fixed point theorems for mappings fulfilling some nonlinear contraction type conditions on a 2-metric space.
For the notions see 0 1. The concept of a %metric space (XI d ) has been investigated by S. GAHLER in [7]. For other papers on 2-metric spaces and related topics see for example [24]. Our definitions of f-orbitally completeness of (X, d ) and f-orbitally continuity of a mapping y are slight modifications of the wellknown corresponding definitions considered in papers of L. C I R I ~ [ 2 ] and [3] for selfmappings on a metric space. For definitions of diameters of orbits and their applications to fixed point theorems in a metric space see for example F. BROWDER [l], L. ~I R I I ~ [4], L. RHOADES [18], I. Rus [22] and W. WALTER [26]. I n particular Lemma i. 1 is dual to the result of D. T. NHAN (see Lemmas 1.1 and I .2 of [ 171).
In 5 2 we prove some common fixed point theorems for contractive type mappings in a %metric space. We generalize the results of L. RHOADES [IS], K. ISEKI, 1'. L. SHARM-4 and B. K. SHARMA [Y] and A. K. SHARMA [ 2 3 ] . For a wider review of the generalized contraction mappings on a metric space see 1181.
In 5 3 we give a t first coincidence theorems for mappings in a 2-metric space.
For the reference to coincidence theorems for contraction mappings in a metric space see for example R. MACHUCA [14] and K. GOEBEL [XI.
Theorems 3.1 and 3.2 provide us with a possibility to formulate some commoii fixed point theorems for commuting mappings in a 2-metric space without an assumption of a continuity of these mappings and with no continuity property of the 2-metric d. In Theorems 3.5 and 3.6 the results of [13] have been generalized.