In fact, for (forall x, y in H), (gammain varrho^{-1} eta-eta ^{-1}),frac{eta}{varrho})), from Lemma 4.2, we have begammain varrho^{-1} eta-eta &leqbigammain varrho^{-1} eta-eta)ybigrVert + etagammabiglVert f(x)-f(y)bigrVert &leq Vert I-eta KVert Vert x-yVert +etagammavarrho Vert x-y Vert &leqbigl(1-eta^{2}+etagammavarrhobigr)Vert x-yVert &leq Vert x-yVert.
For every (x > 1) and (etainLambda_{n} ), (n rightarrowinfty), we have the following asymptotic formula: frac{-2ietavarphi x,eta)}{frac{-2ietavarphi xa)}- e^{-i eta x}-S_{o}(eta) e^{i eta x} = bigcirc biggl( frac{e^{x operatorname{Im} eta}}{eta} biggr),eta.4) where (S_{o}{fta)) is given by (1.12).
(2.16) From (1.12), for (etainLambda_{n} ), (S_{o} eta) ) has the asymptotic formula S_{o} eta) = bigcirc bigl( e^{2 operatorname{Im} eta} bigr), (2.17) by the aid of (2.17), (2.16) takes the form frac{-2ietavarphi x,eta)}{frac{-2ietavarphi xa)} = e^{-i eta x}+S_{o}(eta) e^{i eta x}+ bigcirc biggl( frac{e^{x operatorname{Im} eta}}{eta} biggr),etaich completes the proof of the lemma.
end{aligned} (4.2) So, if (0<etaleq2xi), then (I-eta T) is a nonexpansive mapping from C to H. From Theorem 4.1, Lemmas 2.2 and 4.1, we have the following result.
Therefore, from the second equation of system (1) and (30), we have begin{aligned} frac{partial I}{partial t}&=d_{2}Delta I + frac{beta SI t-tau )}{1+alpha I(t-tau)}-gamma I-eta I & leq d_{2}Delta I + frac{beta(frac{a-gamma}{b}+varepSI t-tau(t-tau)}{1+alpha I t-tau }-gamma I t-tau }-gammaI t-tau }-gamma (t > t_{1}+tau).
Let C be a nonempty closed convex subset of H and (T : C rightarrow H) be a ξ-inverse-strongly monotone mapping, then for all (x,yin C) and (eta>0), we have begin{aligned} bigl| (I-eta T x- I-eta T x- I-eta{2} =&bigl| (x-y)-eT x- I-etaigr| ^{2} =&| x-y |^{2}-2eT ybigre Tx-Ty,x-yrangle+eta^{2}| Tx-Ty|^{2} leq&| x-y|^{2}+eta(eta-2xi)| Tx-Ty|^{2}.
Then we apply the lower bound of S to the third equation of system (1), and we have begin{aligned} frac{partial I}{partial t}&=d_{2}Delta I + frac{beta SI t-tau )}{1+alpha I(t-tau)}-gamma I-eta I & geq d_{2}Delta I + frac{betaunderline{c}_{1}I(t-tau)}{1+alpha I(t-tau)}-gamma I-eta I, end{aligned} (39) for (t > t_{4}).
(operatorname{Fix}(T) = operatorname{Fix}(T((1 - eta)I + eta T)) = operatorname{Fix}(K)).
This shows that (operatorname{Fix}(T) = operatorname{Fix}(T((1 - eta)I + eta T))).
(1) If (x^ in operatorname{Fix}(T)), it is obvious that (x^ in operatorname{Fix}(T((1 - eta)I + eta T))).
