Hybrid Fixed-Set Results for Multivalued Mappings in Banach Algebras under a Weak Topology Setting

Abstract

In this paper, using the technique of measure of weak noncompactness and by removing the convexity hypothesis, we prove some fixed set results for the sum and the product of three multivalued mappings AB + C, acting on Banach algebras satisfying a certain sequential condition $(\mathcal{P})$ in the weak topology setting. In addition, by using a new definition of the multivalued mapping $\left(\frac{I-C}{A}\right)$, we get new fixed set theorems for the mappings of the form ${\left(\frac{I-C}{A}\right)}^{-1}B$ under some suitable conditions on the operators A, B, and C in Banach algebras. These hybrid theorems improve some recently obtained results by Ben Amar et al. We also apply these results to the theory of self-similarity.

Keywords:

• Fixed set
• measure of weak noncompactness
• multivalued mappings
• self similarity
• weak topology

AMS (2010) subject classification:

• 47H10
• 47J05
• 47J10