On approximation of fixed points of multivalued pseudocontractive mappings in Hilbert spaces

Felicia Obiageli Isiogugu

doi:10.1186/s13663-016-0548-x

Abstract

It is proved that if a multivalued k-strictly pseudocontractive mapping S of Chidume et al. (Abstr. Appl. Anal. 2013:629468, 2013) is of type one, then $I-S$ is demiclosed at zero. Also, under this condition, the Mann (respectively, Ishikawa) sequence weakly (respectively, strongly) converges to a fixed point of a multivalued k-strictly pseudocontractive (respectively, pseudocontractive) mapping S without the condition that the fixed point set of S is strict, where S is of type one if for any pair $r,g \in D(S)$ , $$\|u-v\|\leq\Phi(Sr,Sg) \quad\mbox{for all } u\in P_{S}r, v\in P_{S}g, $$ and Φ denotes the Hausdorff metric. The results obtained give a partial answer to the problem of the removal of the strict fixed point set condition, which is usually imposed on multivalued mappings. Thus, the results extend, complement, and improve the results on multivalued and single-valued mappings in the contemporary literature.

Highlights

1 Introduction The approximation of fixed points of multivalued mappings with respect to the Hausdorff metric, using Mann [ ] or Ishikawa [ ] iteration scheme, has never been successful without imposing the condition that either the fixed point set of S is strict or that S is a multivalued mapping for which PS satisfies some contractive conditions
Theorem . ([ ]) Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let S : C → P(C) be a multivalued k-strictly pseudocontractive mapping with F(S) = ∅ such that for every r ∈ C, Sr is weakly closed and Sz = {z} for all z ∈ F(S)
They suggested to approximate the fixed points of multivalued mappings S directly instead of PS and without imposing the strict fixed point set condition on S. This suggestion was due to the fact that it has not been established that if a multivalued map S belongs to a class of maps, PS necessarily belongs to the same class of maps and that the fixed point set of S need not be strict. They introduced the ‘type-one’ condition, which guarantees the weak convergence of the Mann sequence {rn} without imposing the condition that the fixed point set of S is strict to a fixed point of a multivalued quasinonexpansive mapping S in a real Hilbert space

Summary

Introduction

The approximation of fixed points of multivalued mappings with respect to the Hausdorff metric, using Mann [ ] or Ishikawa [ ] iteration scheme, has never been successful without imposing the condition that either the fixed point set of S is strict or that S is a multivalued mapping for which PS satisfies some contractive conditions (see, e.g., [ , – ], and references therein). A multivalued mapping S : D(S) ⊆ H → CB(H) from the domain of S into the family of all closed and bounded subsets of H is said to be k-strictly pseudocontractive in the sense of Chidume et al [ ] if there exists k ∈ ( , )

Results

Conclusion