Let (X, d) a complete metric space endowed with a partial order (le) such that ((X,d,le )) is regular.
Let (X, d) be a complete metric space endowed with a partial order (le) such that ((X,d,le )) is regular.
If (q in Q) and (t in M_n) there exists an essential right ideal (E) of (R) such that (qE le R).
Assume that there exist constants (0 le kappa < frac{4Gamma (q+1)}{(6+q)}) and (M>0) such that (vert f t,u vert le frac{ kappa }{T^{q}}vert uvert +M) for all (t in [0,T]), (u in C[0,T]).
) If the following conditions satisfied (i) for ( 0 < beta <span class="lh lhl">le 1,) ( 0 < tau <span class="lh lhl">le 1 ) such that ( 0 le beta -tau < 1).
We decompose the domain Ω into K rectangles (Omega_{k}), (1 le k le K), such that overline{Omega}= bigcup_{k=1}^{K} overline{Omega}_{k} quadtext{and}quad Omega_{k}cup Omega_{l}=emptyset,quad 1 le k neq l le K.
Thus, there must exist (1 le t le k) such that (#(Scap Y_{i_t}) = l ne m = #(S' cap Y_{i_t})).
end{aligned}We suppose also that (E) admits a Hermitian metric such that (|s|le kappa |sigma |), where (kappa ) is real and strictly positive, and we equip (Lotimes det E^*) with the associated Hermitian metric.
(6) Moreover, assume that there exist (m,M>0) such that m le A_{x}(t) le M quadtextit{for every } x in C^{1}(I), t in I. (7) Finally, suppose that there exists (phiin L^{1}(I)) such that biglvert F_{x}(t) bigrvert lephi(t) quadtextit{for every } xin C^{1}(I), textit{ a.e.
L-Lipschitz if there exists (L>0) such that |Tx-Ty|le L|x-y| for all (x,yin C).
