The Domínguez–Lorenzo condition and multivalued nonexpansive mappings

A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued

“…Dhompongsa et al [3] Recall that a bounded closed and convex subset E of a Banach space X has the fixed point property for nonexpansive mappings (FPP) (respectively, for multivalued nonexpansive mappings (MFPP)) if every nonexpansive mapping of E into E has a fixed point (respectively, every nonexpansive mapping of E into 2 E with compact convex values has a fixed point). The following concepts and result were introduced and proved by Bruck [14,15].…”

“…The class of spaces having property (D) contains several well-known ones including k-uniformly rotund, nearly uniformly convex, uniformly convex in every direction spaces, and spaces satisfying Opial condition (see [3,[19][20][21][22][23] and references therein).…”

The fixed point property for multivalued nonexpansive mappings

“…This relationship gives an iterative method which reduces at each step the value of the Chebyshev radius for a chain of asymptotic centers. Consequently, in [6] they defined the Domínguez-Lorenzo condition ((DL)-condition, in short) which implies weak normal structure and the w-MFPP. Definition 1.…”

The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings

“…Consequently, they can give an affirmative answer of a problem in [21] proving that every nonexpansive self-mapping T : E → KC(E) has a fixed point where E is a nonempty bounded closed convex subset of a nearly uniformly convex Banach space. Recently, Dhompongsa et al [5], gave an existence of a fixed point for a multivalued nonexpansive and (1 − χ)-contractive mapping T : E → KC(X) such that T (E) is a bounded set and which satisfies the inwardness condition, where E is a nonempty bounded closed convex separable subset of a reflexive Banach space which satisfies the Domínguez-Lorenzo condition, i.e., an inequality concerning the asymptotic radius and the Chebyshev radius of the asymptotic center for some types of sequences. Consequently, they could show that if X is a uniformly nonsquare Banach space satisfying property WORTH and T : E → KC(X) is a nonexpansive mapping such that T (E) is a bounded set and which satisfies the inwardness condition, where E is a nonempty bounded closed convex separable subset of X, then T has a fixed point.…”

“…Finally, using this fact, we prove that if C NJ (X) is less than an appropriate positive number, then every multivalued nonexpansive mapping T : E → KC(E) has a fixed point. In particular, we give a partial answer to the question which has been asked in [5].…”