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Let (X, d) be a metric space, Y be a nonempty subset of
X, and let T : Y ï¿½?? P(X) be a non-self multivalued mapping. In this
paper, by a new technique we study the fixed point theory of multivalued
mappings under the assumption of the existence of a bounded sequence
(xn)n in Y such that T nxn ï¿½?? Y, for each n ï¿½?? N. Our main result
generalizes fixed point theorems due to Matkowski (Diss. Math. 127,
1975), WÂ¸egrzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and
Zaslavski (Fixed Point Theory 8:303ï¿½??307, 2007), PetruÂ¸sel et al. (SetValued
Var. Anal. 23:223ï¿½??237, 2015) and provides a solution to the
problems posed in PetruÂ¸sel et al. (Set-Valued Var. Anal. 23:223ï¿½??237,
2015) and Rus and SÂ¸erban (Miskolc Math. Notes 17:1021ï¿½??1031, 2016).
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