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On a general class of multi-valued weakly Picard mappings. (English) Zbl 1117.47039

Let (X,d) be a metric space and T:X𝒫(X) be a multi-valued operator. T is said to be a generalized (α,L)-weak contraction if there exist a constant L0 and a function α:[0,)[0,1), with limsup rt + α(r)<1 for every t[0,), such that


for all x,yX. When α(t)=θ[0,1) for all t[0,), we say that T is a (θ,L)-weak contraction. The authors prove that a generalized (α,L)-weak contraction T has a fixed point whenever X is complete and T has nonempty bounded and closed values. Moreover, if T is a (θ,L)-weak contraction, then for any x 0 X there exists an orbit {x n } n0 converging to a fixed point u of T for which the following estimate holds:

d(x n ,u)minh n 1-hd(x 1 ,x 0 ),h 1-hd(x n-1 ,x n )

for a certain constant h<1.

47H04Set-valued operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54C60Set-valued maps (general topology)