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Lp1 Lp v0 M Au, u - v0 - q Bu, u - vpk pk-1 pk (c1 ku - c2 ku - c|q| ku - c3) u M, Lpk Lpk Lpk k= A - q B Au, u - v0 - q Bu, u - vlim u u V V k k k (c1akp - c2akp -1 - c|q| akp - - c3) k= lim = +.

(a1+a2)+ (a1 + a2) p2 > 3 p1 > r = pp1 p2 r | Bu, u | c 1(u + ) + 2(u + ) + u Lp1 Lp2 Lr p1 p c 1u + 2u + 1, Lp1 Lp Au, u - v0 - q Bu, u - vpk pk- ((c1 - c|q|) ku - c2 ku - c3), u M.

Lpk Lpk k= q0 = c1/c A - q B Au, u - v0 - q Bu, u - vlim u u V V k k ((c1 - c|q|)akp - c2akp -1 - c3) k= lim = +.

(a1+a2)+ (a1 + a2) A B A-rB RR RMR = {u M : u R}.

V uR MR (A - qB)uR, v - uR f, v - uR v MR v0 MR c0 > A R > R0 v R f V : f cV V Av, v - v0 - f, v - v0 Av, v - v0 - c v - v0 > > 0.

V B q0 > 0 v R |q| qV |q Bv, v - v0 | |q| Bv v - v0 /2.

V V Av, v - v0 - q Bv, v - v0 > f, v - v0 v SR, SR = {v M : v = R} uR = R V (A - qB)uR, uR - v0 > f, uR - v0 v0 MR, uR < R V uR v M > v = (1 - )uR + v MR.

v v (A - qB)uR, v - uR f, v - uR, (A - qB)uR, v - uR f, v - uR v M.

V V V , V V V M V u M M(u) M h {h} Vh V Mh Vh uh Mh Mh(uh) Mh uh Mh(uh) (I) {hk}+ k= {vh }+ Mh v V k k k k=+ v M w M(v) {wh }+ k k=wh Mh (vh ), k = 1, 2, 3,..., k k k lim wh - w = 0.

V k k+ u M : Au, v - u f, v - u v M(u), f V uh Mh : Auh, vh - uh f, vh - uh vh Mh(uh).

A {uh }+ V u k k=hk 0 u {uh }+ k k= u V hk 0 uh Mh(uh ) Mh k k k (I) u M {u Mh (uh )}+ hk k k k=lim u - u = 0.

hk V hk vh = u k hk Auh, uh - u = k k = Auh, uh - u + Auh, u - u k k hk k hk f, uh - u + Auh u - u.

V k hk k hk V hk {uh }+ u k k=lim sup Auh, uh - u 0.

k k hk A lim inf Auh, uh - v Au, u - v v M.

k k hk (I) v M(u) {vh Mh (uh )}+ k k k k=lim vh - v = 0.

V k hk{uh }+ u k k=lim sup Auh, uh - v k k hk lim sup Auh, uh - vh + lim sup Auh, vh - v k k k k k hk0 hk lim sup f, uh - vh + lim sup Auh vh - v = V V k k k k hk0 hk= f, u - v.

Au, v - u f, v - u v M, u (I) (II) {v(k)}+ Mh k= v V k + v Mh w Mh(v) {w(k)}+ k=w(k) Mh(v(k)), k = 1, 2, 3,..., lim w(k) - w = 0;

V k+ (III) h {h} Mh(v) v Mh A : V V {hk}+ k + k= {u(0)}+ Mh V k hk k= c > 0 : u(0) c, k = 1, 2, 3,..., hk V c0 > 0 : F (u(0)) c0, k = 1, 2, 3,..., hk F : V R h = hk k = 1, 2, 3,...

n = 0, 1, 2,... u(n+1) Mh(u(n)) : J(u(n+1)-u(n)), v-u(n+1) h h h h h f - Au(n), v - u(n+1) v Mh(u(n)), h h h > 0 J : V V (III) (I) (III) A 0 < < min{1, 1/0}, 0 = (d0 + -1(d1)), d0 = sup u, d1 = sup Au - f, V V uS0 uSS0 = {u V : F (u) c0}, c {u(k)}+ S0 V h k= h uh {u(n)}+ k n=h h = hk k = 1, 2, 3,... {uh }+ k k= S0 V {uh }+ k k= M M(u) R Th K K = ;

KTh intK intK = K, K Th;

K, K Th K K K K Th (h) c < + h 0.

RK h = max RK, (h) = max, KTh KTh K RK K K K Wh Vh Th Wh = Xh Xh Xh, Vh =Xh Xh Xh, Xh Xh K Th 1 Xh Wp () XhWp () Wh W Vh V h = l, l = 1, 2,..., Nh K Th h = h : [C()]h Wh h(xl) = (xl) l = 1, 2,..., Nh.

M V Mh Vh Mh = {v Vh : 3() + v3() F (() + v()), h}, M(u) M Mh(uh) Mh uh uh Mh(uh) = {v Vh : (3() + v3() - P (), N(P ())) 0, h}.

Vh Mh Mh(v) tk : R+ R+ (I) (III) (III) (II) (I) {hk}+ {vh }+ Mh k k k=1 k= v V k + v M w M(v) w > 0 k (0, w] k (k) w Mh (vh ), k k (k) w = h (w) h (w + e3), e3 = (0, 0, 1), k = 1, 2, 3,..., k k v M V {vh } v k V [C()] > 0, i : | vh () - v() | < i > i,.

k {hk}+ k= h Th k = 1, 2, 3,...

k k v F (() + v()) = lim F ((h ) + v(h ) vh (h )) = k k k k k+ = lim F ((h ) + vh (h )) lim 3(h ) + (vh (h ))3 = k k k k k k k+ i+ = lim 3(h ) + (vh (h ))3 v3(h ) = 3() + v3(), k k k k k+ v M (k) w M w Mh k (k) vhk vhk (() + w () - P (), N(P ())) = vhk v vhk = (( + h w)() - P () P (), N(P ())) = k v vhk v = (( + h w)() - P (), N(P ()) N(P ()))+ k v vhk vhk +(P () - P (), N(P ())) = v v = (( + h w)() - P (), N(P ()))+ k v vhk v +(( + h w)() - P (), N(P ()) - N(P ()))+ k v vhk vhk +(P () - P (), N(P ())).

v v (( + h w)() - P (), N(P ())) = k v v v = (( +w)()-P (), N(P ()))+()(e3, N(P ())), h.

k k = 1, 2, 3,... h k h k (I) (I) (III) A p 2 t : R+ R+ t() - t() c0(1 + + )p-2( - ) > 0, c0 > 0 c1, c2 > x, y Rt(|x|) t(|y|) x - y |x| |y| t(|x|) t(|y|) c1 x - y, x - y (1 + |x|p + |y|p)1-2/p, |x| |y| t(|x|) t(|y|) x - y c2 | x - y | (1 + |x| + |y|)p-2.

|x| |y| k = 1, 2 tk : R+ R+ pk 2 Ak Tk(k(u)) - Tk(k(v)) Lqk k c Aku - Akv, u - v (c + ku + kv )p -2 u, v V, Lpk Lpk Aku - Akv Tk(k(u)) - Tk(k(v)) u, v V, V Lqk qk = pk/(pk - 1) c, c > k |Tk(k(u)) - Tk(k(v))|q d k k k k c0 (Tk(x) - Tk(y), x - y)q /2 (1 + |k(u)|p + |k(v)|p )(1-2/p )(qk/2)d q /k c0 (Tk(x) - Tk(y), x - y) d (2-q )/k k k c1 + |k(u)|p + |k(v)|p d.

2/qk 2/q k k |Tk(k(u)) - Tk(k(v))|q d c0 (Tk(x) - Tk(y), x - y) d (p -2)/pk k k k c1 + |k(u)|p + |k(v)|p d k c Aku - Akv, u - v (c + ku + kv )p -2, Lpk Lpk | Aku - Akv, w | |Tk(k(u)) - Tk(k(v))| |kw|d Tk(k(u)) - Tk(k(v)) kw.

Lqk Lpk A p-2 p- Au - Av (c0 + c1 u + c1 v ) u - v u, v V, V V V V p = max{p1, p2} c0, c1 > 0 A, (t) = t, (t) = c0 + c1tp- Aku - Akv V k c Aku - Akv, u - v (c + ku + kv )p -Lpk Lpk k c Aku - Akv u - v (c + ku + kv )p -2.

V Lpk Lpk V k Aku - Akv c u - v (c + ku + kv )p -V Lpk Lpk V p-2 p- u - v (c0 + c1 u + c1 v ).

V V V A = A1 + A A A A M Mh uh Mh : Auh, vh - uh f, vh - uh vh Mh.

L Lp p V Lp g L V p g, v g v v V.

L Lp p p = min{p1, p2} u f u (2) V2 = [Wp ()]3 (f - Au) L uh p Tk(k(u)) - Tk(k(uh)) ch Lqk k= c h Auh, vh - uh + Au, v - u f, vh - uh + f, v - u Auh-Au, uh-u.

Auh - Au, uh - u f - Au, uh - v + u - vh + Auh - Au, vh - u.

Mh Mh M v = uh Mh M vh = hu Auh - Au, uh - u f - Au u - hu + Auh - Au hu - u.

L Lp V V p D = Tk(k(u)) - Tk(k(uh)).

Lqk k= A h V {uh} D2 2 Tk(k(u)) - Tk(k(uh)) Lqk k= c1 Akuh - Aku, uh - u = c1 Auh - Au, uh - u k= Auh - Au Akuh - Aku c2D V V k= c1, c2 > 0 h u u - hu c u h2, u - hu c u h.

Lp V2 V VD2 c(h2 + 2Dh) D h(c + c2 + c).

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