Let be a sequence that as.
Let be a sequence that converges to an element in.
Let ((X, sigma)) be a b-metric-like space and ({x_{n}}) be a sequence that converges to u with (sigma u,u =0).
Let now (x,y in X), and let ({x_{n}}) be a sequence that converges to x in ((X,D_{T})).
Let us present some auxiliary result: Lemma 5 Let Γ : R d → P c c ( R d ) be an usc multifunction and ( x n ) n be a sequence that converges to x ∈ R d. Suppose that there exists a constant M > 0 such that d ( 0, Γ ( x n ) ) ≤ M for every n ∈ N. Then d ( 0, Γ ( x ) ) ≤ M. Proof By the upper semicontinuity of Γ it follows that d ( 0, Γ is lower semicontinuous (see [[39], Lemma 9.3.1]).
Let t and τ be fixed, and for any (T>0), (tgeqtau), (t, tauin[0,T]), let ({ u^{n}_{tau},sigma^{n})}subset Htimes{mathcal{H}}(sigma _{0})) be a sequence that converges to some ({ u_{tau},sigma)}subset Htimes {mathcal{H}}(sigma_{0})), and let (u^{n}(t)) and (u(t)) be the solutions of problem (2.5 - 2.8 2.5 - 2.8bols (sigma^{n}), σ and initial data (u^{n}_{tau}), (u_{tau}), respectively.
