The sequence is bounded, so there exists a convergent subsequence.
On the other hand, this sequence is bounded below; thus there exists μ ≥ 0 such that lim n → ∞ ψ ( max { d ( x n + 1, x n ), d ( y n + 1, y n ) } ) = μ.
end{aligned} (3.1) By Lemma 2.8, ({u_{n}}) is bounded in E. Thus, there exists (C>0) such that (Vert u_{n} Vert _{2}leq C).
Then (f u)) is bounded in Ω, and thus, there exists a positive constant N such that biglvert f u bigrvert leq N, quadforall t in I, u inOmega.
Hence ({u_{n}}) is bounded in (H_{T}^{1}), thus there exists an (uin H^{1}_{T}) such that (u_{n}rightharpoonup u) in (H^{1}_{T}) and (u_{n}rightarrow u) in (C [0,T], {mathbb{R}}^{N})), where a subsequence is considered when necessary.
The functions (f t,x,Hx)) and (h t,s,x)) are bounded for (tin[0,b]) and (xinmathbb{X}), thus there exists a constant (N>0) such that int_{0}^{b}bigl| f bigl s,x_{lambda}(s),(Hx_{lambda}) (s bigr bigr| ^{2},dsleq bN^{2}.
In fact, if u is a solution of the problem (4.2), then Lemma 4.6 implies that u is bounded in (L^{infty}_{omega} (Q_{omega})), thus there exists a constant (R> min{ { r,C(L_{0}) } }), where r is defined as (3.6), R is independent of L, u, such that | u|_{L_{omega}^{infty} (Q_{omega} )} leq C(L_{0}) < R, quad L in [0, L_{0}], (4.83) where (L_{0} ) is defined as (4.1).
Thus, there exists (hat{C}>0) such that (| Y t,x |leqhat{C}), so that (Y t,x)) is bounded, and the proof is complete.
