Pergamon Nonlinear Analysis, Theory, Methods &Applications, Vol. 30, No. 1, pp. 471~76, 1997 Proc. 2nd World Congress of Nonlinear Analysts 8 1997 Elsevier Science Ltd PII: SO362-546X(97)00353-2 F’riated in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00 A COMPLETE COMPARISON OF 25 CONTRACTION CONDITIONS PAULA COLLACOt andJAIME CARVALHO E SILVAS t Department of Mathematics, University of Aveiro, 3800 Aveiro, Portugal $ Department of Mathematics, University of Coimbra, 3000 Coimbra, Portugal ABSTRACT: In thispaper we establish a comparison of the 25 contraction conditions considered in a previous paper of Rhoades [161, thuscompleting the comparison first made in a systematic way in that paper. 1. INTRODUCTION In [161, B.E. Rhoades gathered 25 contraction conditions most of them,alreadyknown in the literaturefor some time, in order to compare themandobtainfixed point theorems, Besides the Banach contractive condi- tion otherconditions proposed namely by Rakotch[ 131, Edelstein [6], Kannan[9], Bianchini[ 11, Reich [ 14 151, Sehgal [22], Chatterjea [2], Hardy andRogers [7], Zamftrescu [23] andCiric [3] werealsostudied. The comparison of the contractionconditions is importantbecause if condition (m) implies (n) and we have a fixed point theorem for condition(n), thenwe automatically obtain, asa corollary, a fixed point theorem for condition (m). Thecomparison made by B.E. Rhoades in [16] wasnot complete, but included alsogeneralizations of the 25 contractionconditions to one or two iterates of the function, uniformly on one, two or none of the vari- ables. But the comparison of the 25 basic contractionconditions is the important one because it implies similar comparisons for the other groups of 25 conditions over the iterates of the function f. In this paperwe present a complete comparison of all these contraction conditions. All these comparisons aresummed up in a tableandso,all the relations canbe easilyseen. Here we alsotry to deduce all the implications and all the non implications from the smallest number of results, as presented in Theorems 2.1 and3.1. Several authors generalized the paperof Rhoades [ 161, like Park [12] andRhoades [20], but these gener- alizations neverincluded the comparison of all the 25 basiccontraction conditions. In 1990, J: Kineses and V. Totik [lo] studied all these conditions from the fixed point theorems point of view, leaving behind the comparison between them. The proofsof the main theorems stated below will be given elsewhere. This work completes and includes previous work by the same authors [4-51. We will use the same numbering of Rhoades [ 161 to refer to the contractive conditions. 2. COMPARISON OF THE BASIC CONTRACTIVE CONDITIONS: IMPLICATIONS The fist theorem states all the implications needed to obtain all the implications existing amongst the 25 contraction conditions as shown in table 1. Most of them werealready provedby Rhoades [161. THEOREM 2.1: i. (1) + (2) j (3) j (10) * (22) = (25); ii. (1) 3 (7) 3 (8) * (23) * (20) + (22); iii. (2) j (8) * (10); iv. (2) a (15) * (17) j (25); v. (4)a (5)+(6)- (10); vi. (4)~(7)*(18)~(19)=,(21)~(22); 471