Abstract
AbstractLet (X, d) be a metric space, G be a graph associated with X and f : X → X be an operator which satisfies two main assumptions: (1) f is generalized G-monotone; (2) f is a G-contraction with respect to d. In the above framework, we will present sufficient conditions under which: (i) f is a Picard operator; (ii) the fixed point problem x = f(x), x ∈ X is well-posed in the sense of Reich and Zaslavski; (iii) the fixed point problem x = f(x), x ∈ X has the Ulam-Hyers stability property; (iv) f has the Ostrowski stability property; (v) f satisfies to some Gronwall type inequalities. Some open questions are presented.KeywordsMetric spaceDirected graph G-contractionGeneralized contractionIncreasing operatorDecreasing operatorGeneralized G-monotone operatorFixed point(Weakly) Picard operatorStabilityGronwall lemmaOpen problem2010 Mathematics Subject Classification47H1034G2045N0506A0647H0947H0754E3554H25
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