On random coincidence and fixed points for a pair of multivalued and single-valued mappings

• Main Authors: Ume Jeong S, Ješić Siniša N, Ćirić Ljubomir B
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2006/81045

<p/> <p>Let ( <inline-formula><graphic file="1029-242X-2006-81045-i1.gif"/></inline-formula>) be a Polish space, <inline-formula><graphic file="1029-242X-2006-81045-i2.gif"/></inline-formula> the family of all nonempty closed and bounded subsets of <inline-formula><graphic file="1029-242X-2006-81045-i3.gif"/></inline-formula>, and ( <inline-formula><graphic file="1029-242X-2006-81045-i4.gif"/></inline-formula>) a measurable space. A pair of a hybrid measurable mappings <inline-formula><graphic file="1029-242X-2006-81045-i5.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-81045-i6.gif"/></inline-formula>, satisfying the inequality (1.2), are introduced and investigated. It is proved that if <inline-formula><graphic file="1029-242X-2006-81045-i7.gif"/></inline-formula> is complete, <inline-formula><graphic file="1029-242X-2006-81045-i8.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-81045-i9.gif"/></inline-formula> are continuous for all <inline-formula><graphic file="1029-242X-2006-81045-i10.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-81045-i11.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-81045-i12.gif"/></inline-formula> are measurable for all <inline-formula><graphic file="1029-242X-2006-81045-i13.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-2006-81045-i14.gif"/></inline-formula> for each <inline-formula><graphic file="1029-242X-2006-81045-i15.gif"/></inline-formula>, then there is a measurable mapping <inline-formula><graphic file="1029-242X-2006-81045-i16.gif"/></inline-formula> such that <inline-formula><graphic file="1029-242X-2006-81045-i17.gif"/></inline-formula> for all <inline-formula><graphic file="1029-242X-2006-81045-i18.gif"/></inline-formula>. This result generalizes and extends the fixed point theorem of Papageorgiou (1984) and many classical fixed point theorems.</p>