Large contractions and surjectivity in Banach spaces. (English) Zbl 1539.47091
Som, Tanmoy (ed.) et al., Applied analysis, optimization and soft computing. ICNAAO-2021, Varanasi, India, December 21–23, 2021. Singapore: Springer. Springer Proc. Math. Stat. 419, 3-12 (2023).
Summary: In [Proc. Am. Math. Soc. 124, No. 8, 2383–2390 (1996; Zbl 0873.45003)], T. A. Burton proposed a new concept of contraction-type mapping by introducing the notion of large contraction. In his paper, a fixed-point result for a single-valued large contraction in a complete metric space is given and some applications to integral equations are deduced. In this paper, we will continue the study of the above-mentioned mappings in the context of a Banach space X. More precisely, we will show that any large contraction \(t:X\rightarrow X\) is a norm-contraction in the sense of A. Granas [Bull. Acad. Pol. Sci., Cl. III 5, 867–871 (1957; Zbl 0078.11701)]. Then, as an application, a surjectivity theorem for the field \(1_X-t\) generated by \(t\) is proved. In the second part of this work, we extend the concept of large contraction to the multivalued case and we prove fixed-point theorems for multivalued large contractions \(T:X\rightarrow P(X)\) in a Banach space \(X\). Additionally, some surjectivity results for the field \(1_X-T\) generated by the multivalued operator \(T\) are given. The results of this paper extend and complement several fixed-point theorems and surjectivity results in the recent literature.
For the entire collection see [Zbl 1531.00060].
For the entire collection see [Zbl 1531.00060].
MSC:
| 47H10 | Fixed-point theorems |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H04 | Set-valued operators |
Keywords:
Banach space; large contraction; multivalued large contraction; fixed point; surjectivity theoremReferences:
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