Fixed point theorems for generalized Lipschitzian semigroups (Q1599613)

Let \(K\) be a nonempty subset of a \(p\)-uniformly convex Banach space \(E\), \(G\) a left reversible semitopological semigroup, and \({\mathcal S}=\{T_t: t\in G\}\) a generalized Lipschitzian semigroup of \(K\) into itself, that is, for \(s\in G\), \(\|T_sx-T_sy\|\leq a_s\|x-y\|+ b_s(\|x-T_sx\|+ \|y-T_s y\|)+c_s (\|x-T_s y\|+\|y-T_s x\|)\), for \(x,y\in K\) where \(a_s,b_s, c_s>0\) such that there exists a \(t_1\in G\) such that \(b_s+c_s<1\) for all \(s\succeq t_1\). It is proved that if there exists a closed subset \(C\) of \(K\) such that \(\cap_s\overline {\text{co}} \{T_tx:t\succeq s\}\subset C\) for all \(x\in K\), then \({\mathcal S}\) with \([(\alpha+ \beta)^p(\alpha^p \cdot 2^{p-1}-1)/ (c_p-2^{p-1}\beta^p) \cdot N^p]^{1/p}<1\) has a common fixed point, where \(\alpha= \lim \sup_s(a_s+ b_s+c_s)/(1-b_s-c_s)\) and \(\beta= \lim\sup_s (2b_s+2 c_s)/(1-b_s-c_s)\).