Coincidence Theorems for Hybrid Contractions

Abstract

A coincidence theorem of GOEBEL [3] (cf. Theorem 1 below) proved in 1968 has recently received some attention. In particular, OKADA [9], SINGH-VIILENDRA [ 131, and SINOH-KULSHRESTHA [ 121 extended Theorem 1 to L-spaces, 2-metric spaces and metric spaces, respectively. Further, PARK [lo] gave a generalization of it. The purpose of this note is to generalize GOEBEL'S result to a hybrid of multivalued and singlevalued maps satisfying a contractive condition. Consequently, fixed point theorems for multivalued maps of COVITZ-NADLER [ 11, NADLER [7] and SMITHSON [14] are extended. Also coincidence theorems for a pair of multivalued maps are presented. Consistent with [8, p. 6201, X will denote an arbitrary nonempty set, ( M , d ) a metric space, id the identity on M and C L ( M ) (resp. C B ( M ) ) the nonempty closed (resp. closed and bounded) subsets of M . For A , BCCL(M) and E =-0, N(E, A ) = {ZC M : d(x, a) -=E for some aEA) , E A , B = { E > O : A c N ( E , B ) , B c N ( E , A ) } , and, Theorem 1. (GOEBEL [3]). Let S, T : X -, M such that S ( X ) c T ( X ) and T ( X ) is a complete subspace of M. Further, assume there exists k , O e k e 1 such that (1.1) d(Sz, By) s k d ( T x , T y )for all x, yCX. Then S and T have a coincidence, that is, there is zEX such that SZ = Tz.Note that if X = M and T =id above, we have the BANACH contraction principle.The following result generalizes Theorem 1 and the method of proof of this and other results is a variant of the technique employed in [3].

On general hybrid contractions

“…Our results (theorems and corollaries) extend, generalize, improve and unify a number of results in the existing single-valued, multi-valued and hybrid fixed point theory (see for instance, [3,5,7,8,[10][11][12][13][14][15]17] and references thereof). We mention a few here.…”

Coincidence and fixed points of nonlinear Hybrid contractions

“…HA AND Y.J. CHO slngle-valued mappings have recently been studied by Mukherjee [II], Naimpally et al [12], Rhoades et al [I] and Singh et al [2]. In this paper, we consider a very general type of condition involving two multi-valued mappings and a slngle-valued mapping and establish coincidence and fixed point theorems (cf.…”

Common fixed points of hybrid maps and an application

“…I(q, u) ∈ S(q, u)) for all u ∈ Y . for some t, t 1 ∈ Y and for every u ∈ Y , (ii) there exists a function F ∈ Ψ such that F (H(S(x, u), T (y, u )), d(I(x, u), J(y, u )), d(I(x, u), S(x, u)), d(J(y, u ), T (y, u )), d(I(x, u), T (y, u )), d(J(y, u ), S(x, u))) ≤ 0, (5) for all x, y, u, u ∈ Y . Then (a) If I(Y × Y ) is a closed subset of X , then there exists b ∈ Y such that I(b, u) ∈ S(b, u) for all u ∈ Y ; (b) If J(Y × Y ) is a closed subset of X , then there exists c ∈ Y such that J(c, u ) ∈ T (c, u ) for all u ∈ Y .…”