© Hindawi Publishing Corp. FIXED POINT AND COINCIDENCE POINT THEOREMS FOR A PAIR OF SINGLE-VALUED AND MULTI-VALUED MAPS ON A METRIC SPACE

We obtain fixed point and coincidence point theorems for a pair of single-valued and multi-valued maps on a metric space satisfying a generalized nonexpansive type condition. 2000 Mathematics Subject Classification: 47H10, 54H25. Bogin [1] proved a fixed point theorem for a nonexpansive type self-map on a com-plete metric space. While Rhoades obtained a generalization of it (see [8, Theorem 1]) by replacing the constant coefficients in the governing inequality of the map with nonnegative real-valued functions of the independent variables, Ćiric ́ [3] obtained a generalization by further weakening the governing inequality without allowing the co-efficients to vary. Chandra et al. obtained a coincidence point theorem (see [2, Theo-rem 2.1]) for a pair of self-maps on a metric space unifying the results of Rhoades and Ćirić. They also obtained a corresponding version for multimaps (see [2, Theo-rem 2.2]). In this paper, we obtain proper generalizations of Theorems 2.1 and 2.2 of Chandra et al. [2]. Throughout, unless otherwise stated, (X,d) is a metric space, K(X) is the collec-tion of all nonempty compact subsets of X, CL(X) is the collection of all nonempty closed subsets of X, H is the extended Hausdorff metric on CL(X), F is a map-ping from X into CL(X), f, S are self-maps on X, I is the identity map on X, for any self-map h on X, (h) = {hx: x ∈ X}, R+ is the set of all nonnegative real numbers, N is the set of all positive integers, Ω: (R+)5 → R+ is monotonically in-creasing in each coordinate variable, for any t1, t2, t3, t4, t5 ∈ R+, Ω(t+1, t+2, t+3, t+4, t+5) = inf{Ω(s1,s2,s3,s4,s5) : sj ∈ (tj,+∞) for all j = 1,2,3,4,5}, Ω(t1, t+2, t+3, t+4, t+5) = inf{Ω(t1,s2,s3,s4,s5) : sj ∈ (tj,+∞) for all j = 2,3,4,5}, σj: R+ → R+ (j = 1,2) and ζ:R+ →R+ are defined as σ1(t)=Ω 0+,0+, t+, t+, t