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Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. (English) Zbl 1172.54027

The main result of the paper is the following common fixed point theorem for compatible and weakly compatible self-mappings on Menger spaces under ϕ-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 Φ be the class of all functions ϕ: + + , such that ϕ is non-decreasing, upper semi-continuous from the right and

n=0 ϕ n (t)<

for all t>0· Let P,Q,L,M be self-maps on a complete Menger space (X,,Δ) with Δ a continuous t-norm of Hadžić-type. If

(i) L(X)Q(X),M(X)P(X);

(ii) either P or L is continuous;

(iii) (L,P) is compatible and (M,Q) is weakly compatible;

(iv) there exists φΦ such that

F Lx,My (φ(t))min{F Px,Lx (t),F Qy,My (t),F Px,qy (t),F Qy,Lx (βt),[F Px,ξ *F ξ,My ]((2-β)t)}

for all x,y,ξX, β(0,2) and t>0, where

F 1 *F 2 (t):=sup t 1 +t 2 =t min{F 1 (t 1 ),F 2 (t 2 )}(t>0),

then L,M,P and Q have a unique common fixed point.

54H25Fixed-point and coincidence theorems in topological spaces
54E70Probabilistic metric spaces