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V V V , V V V M V u M M(u) M u M(u) {u(k)}+ M u u M k=0 M M(u) {(k)}+ (k) M(u(k)) k=0 lim (k) =.

k+ u M V Au, - u f, - u M(u).

f V A : V V Au - Av (R)( u - v ) u, v V, V V R = max{ u, v } [0, +) V V [0, +) (0) = 0 () + + A A + {u(k)}k=0 V u V lim sup Au(k), u(k) - u 0 k lim inf Au(k), u(k) - v Au, u - v v V.

k A 1 ( A(t(u + v)), u + v - A(tu), u ) dt = 0 1 = A(u + tv), v dt u, v V.

0 Av, v ( v ) v v V, lim () = +.

V V + F : V R1 1 F (u) = FA(u) - f, u, FA(u) = A(tu), u dt, f V.

0 u, v V 1 F (u) - F (v) = A(v + t(u - v)), u - v dt - f, u - v.

0 F v V lim F (v) lim (() - f )d = +.

V v + v + V V 0 u(0) M k = 0, 1, 2,...

u(k+1) M(u(k)) J(u(k+1) - u(k)), v - u(k+1) f - Au(k), v - u(k+1) v M(u(k)), > 0 J : V V Jv, v = Jv v = ( v ) v v V, V V V V A F M 0 < < min{1, 1/0}, 0 = (d0 + -1(d1)), d0 = sup u, d1 = sup Au - f, V V uS0 uS0 S0 = {u M : F (u) F (u(0))}.

{u(k)}+ k=0 V F d0 < + A d1 < + 0 < 0 < + {u(k)}+ S0, u(k) d0, k = 0, 1, 2,...

V k=0 u(0) S0 u(N) S0 u(N+1) S0 v = u(k) 1 ( u(k+1) - u(k) ) u(k+1) - u(k) = J(u(k+1) - u(k)), u(k+1) - u(k) V V f - Au(k), u(k+1) - u(k) Au(k) - f u(k+1) - u(k), V V u(k+1) - u(k) -1 Au(k) - f.

V V t [0, 1] | A(u(k) + t(u(k+1) - u(k))) - Au(k), u(k+1) - u(k) | (d)( t(u(k+1) - u(k)) ) u(k+1) - u(k) V V (d)( u(k+1) - u(k) ) u(k+1) - u(k), V V d = max{ u(k) + t(u(k+1) - u(k)), u(k) }.

V V u(k) + t(u(k+1) - u(k)) - u(k) t(u(k+1) - u(k)) V V V u(k+1) - u(k) -1 Au(k) - f.

V V d u(k) + -1 Au(k) - f d0 + -1 (d1).

V V (d) 0 | A(u(k) + t(u(k+1) - u(k))) - Au(k), u(k+1) - u(k) | 0( u(k+1) - u(k) ) u(k+1) - u(k).

V V F (u(k+1)) - F (u(k)) = A(u(k) + t(u(k+1) - u(k))), u(k+1) - u(k) dt- f, u(k+1) - u(k) = = A(u(k+1) + t(u(k) - u(k+1))) - Au(k), u(k+1) - u(k) dt+ + f - Au(k), u(k) - u(k+1).

v = u(k) F (u(k+1)) - F (u(k)) | A(u(k+1) + t(u(k) - u(k+1))) - Au(k), u(k+1) - u(k) |dt- J(u(k+1) - u(k)), u(k+1) - u(k) 0( u(k+1) - u(k) ) u(k+1) - u(k) V V - ( u(k+1) - u(k) ) u(k+1) - u(k) = V V = -( u(k+1) - u(k) ) u(k+1) - u(k), V V = 1/ - 0 > F (u(k+1)) + ( u(k+1) - u(k) ) u(k+1) - u(k) F (u(k)).

V V k = 0, 1,..., N N F (u(k+1)) + ( u(k+1) - u(k) ) u(k+1) - u(k) F (u(0)), V V k= F (u(N+1)) F (u(0)) u(N+1) S V m {u(k )}+ u V m + m= M u M u F F (u) -( u ) ( u ) u - A(0) u - f u.

V V V V V V V {F (u(k))}+ k= {F (u(k))}+ k=lim ( u(k+1) - u(k) ) u(k+1) - u(k) = 0.

V V k+ lim u(k+1) - u(k) = 0.

V k+ v M(u(k)) Au(k), u(k) - v = Au(k), u(k) - u(k+1) + Au(k), u(k+1) - v Au(k), u(k) - u(k+1) + J(u(k+1) - u(k)), v - u(k+1) + + f, u(k+1) - u(k) + f, u(k) - v ( Au(k) + f ) u(k+1) - u(k) + V V V + ( u(k+1) - u(k) ) v - u(k+1) + f, u(k) - v = V V = C(k)(v) + f, u(k) - v, C(k)(v) = ( Au(k) + f ) u(k+1) - u(k) + V V V + ( u(k+1) - u(k) ) v - u(k+1).

V V u M(u) m {u(m)}+ u(m) M(u(k )) m=lim u(m) = u m+ v = u(m) m m m lim sup Au(k ), u(k ) - u(m) lim sup C(k )(u(m))+ m+ m+ m + lim sup f, u(k ) - u(m) = 0.

m+ m m m m lim sup Au(k ), u(k ) - u lim sup Au(k ), u(k ) - u(m) + m+ m+ m + lim sup Au(k ), u(m) - u 0.

m+ m m lim inf Au(k ), u(k ) - v 0 v V.

m+ M(u) m {(m)}+ (m) M(u(k )) m=lim (m) =.

m+ v = (m) m m m m m C(k )((m)) Au(k ), u(k ) - + Au(k ), - (m) + f, (m) - u(k ), m m m 0 = lim inf C(k )((m)) lim inf Au(k ), u(k ) - + m+ m+ m m + lim inf Au(k ), - (m) + lim inf f, (m) - u(k ) m+ m+ Au, u - + f, - u M(u), u V (, )V M(u) M u M V (Au, - u)V (f, - u)V M(u), f V A : V V L 1 (A(t(u + v)), u + v)V dt - (A(tu), u)V dt = 0 = (A(u + tv), v)V dt u, v V, Au - Av L u - v V u, v V, V 0 M (Au, u - 0)V ( u ) u u V, V V lim () = +.

+ u0 M n = 0, 1, 2,...

u(n+1) M(u(n)) u(n+1) - u(n), v - u(n+1) V f - Au(n), v - u(n+1) v M(u(n)), V > 0 M(u(n)) M(u(n)) M u(n+1) M {u(n)}+ M n= 0 < <.

L { u(n) }+ n= V F : V RF (u) = FA(u) - (f, u)V, FA(u) = (A(tu), u)V dt.

u, v V F (u) - F (v) = (A(v + t(u - v)), u - v)V dt - (f, u - v)V.

F M {u(n)}+ S0, n = 0, 1, 2,..., n= S0 = {u M : F (u) F (u0)} F u0 S0 u(n) Su(n+1) S v = u(n) - u(n) - u(n+1) f - Au(n), u(n) - u(n+1).

V V t [0, 1] | A(u(n) + t(u(n+1) - u(n))) - Au(n), u(n+1) - u(n) | V A(u(n) + t(u(n+1) - u(n))) - Au(n) u(n) - u(n+1) V V L t u(n) - u(n+1).

V F (u(n+1)) - F (u(n)) = A(u(n) + t(u(n+1) - u(n))), u(n+1) - u(n) dt - f, u(n+1) - u(n) = V V = A(u(n) + t(u(n+1) - u(n))) - Au(n), u(n+1) - u(n) dt+ V + f - Au(n), u(n) - u(n+1).

V F (u(n+1)) - F (u(n)) A(u(n) + t(u(n+1) - u(n))) - Au(n), u(n+1) - u(n) dtV 2 - u(n) - u(n+1) u(n) - u(n+1) tdtV V 2 - u(n) - u(n+1) - u(n) - u(n+1), V V = 1/ - L/2 > F (u(n+1)) F (u(n)) - u(n) - u(n+1) F (u(n)) n = 0, 1, 2,...

V n = 0, 1, 2,..., N - N F (u(n+1)) + u(n) - u(n+1) F (u0), V n= u(n+1) S (x1, x2, x3) () = (1(), 2(), 3()), = (1, 2) R C1(), | [1(), 2()] | c > 0.

w() = (w1(), w2(), w3()) G() = |[1w(), 2w()]| k = /k, k = 1, 2, [, ] (, ) | | R km m ( G T kw) + G P + G Q = 0, k,m= P Q km T km T = T RkRm, k,m=Rk() = kw() k = 1, w(1, 2) = (1, 2), (1, 2), P D(w) + GP0 = 0, km D(w) = m( G T k w) + G P + G Q.

k,m= P w P P x3 = (x1, x2) C1(R ), 3() (1(), 2()),.

F : R R F (x1, x2, x3) = (x1, x2), F F F F (x) =,, = (1(x1, x2), 2(x1, x2), 0).

x1 x2 x3 N : R R (x1, x2) N(x1, x2, x3) =, x R, |(x1, x2)| = [1, 2], 1 = (1, 0, 1(x1, x2)), 2 = (0, 1, 2(x1, x2)) | (x1, x2) | = = (1(x1, x2))2 + (2(x1, x2))2 + 1 > 0 (x1, x2) R.



= v : R3, v3() F (v()),, v | = |.

M M P0() = ()N(w()), : R () 0, I(w);

() = 0, I-(w).

I(w) = { : w3() = F (w())} I-(w) = \ I(w) I-(w) = { : w3() > F (w())} w M D(w()) + G() () N(w()) = 0,.

u M u M (u) = v : R3 : tv > 0, u + t v t [ 0, tv ].

M M (w) M (D(w), ) d + G N(w), d = 0.

I-(w) G N(w), = 0.

I(w) (w) w + t M M t [0, t] F (w) + t 3 = w3 + t 3 F (w + t ), t 3 F (w + t ) - F (w) = t (, F (w)) + o(t).

t > I(w) t +0 ([1, 2], ) = 3 - (, F (w)) G N(w), 0, I(w).

w (D(w()), ()) d 0 (w).

M w M w w M |D(w())| () =,.

| [1w(), 2w()] | w D(w()) = -| D(w()) | N(w()), I(w).

I(w) g = (g1, g2, g3) D(w)/| D(w) | + N(w) g =, w = w().

| D(w)/| D(w) | + N(w) | (g, D(w)/|D(w)|) > 0, (g, N(w)) > 0.

a b a = -b (a, b) > -1 c = (a + b)/|a + b| (a, c) > 0 (b, c) > (a, a + b) 1 + (a, b) (a, c) = = = (b, c) > 0.

|a + b| |a + b|, 0 > (g, D(w())) >, B (), (g, N(w())) | (w()) | >, B (), B() = { R2 : | - | } > 0 C(R2) 0 () 1, () = 1, B/3(0), () = 0, B(0), () = ( - )g (w) > M t [ 0, 1 ] F w3() + t3() - F (w() + t()) = w3() + t ( - ) g3- F (w()) + t ( - ) (g, F (w()) + c() (t ( - ))2 = = w3() - F (w())+ +t ( - ) (N(w()), g)|(w())| - c() t ( - ), c() = (2F (x())g, g) x() = w() + ()(), () [0, 1] |c()| c0 B() min{0, /(2 c0)} (N(w()), g)|(w())| - c() t ( - ) - c0/(2c0) = /2 > 0.

B() ( - ) = t ( - ) ((N(w()), g)|(w())| - c() t ( - )) = 0.

w w3() F (w()) M t [0, 1] w3() + t3() - F (w() + t()) 0,, (w) M (D(w()), ()) d = ( - ) (D(w()), g) d B() ( - ) d = mes(B/3() ) > 0, B/3() D(w()) = 0, I-(w).

I-(w) g = D(w)/|D(w)|, 0 > 0 I-(w) w3() > F (w()) w F w3() - F (w()) 1 > 0.

= g (w) M w w u w = + u M = v : R3, v = 0, 3 + v3 F ( + v),, M(u) = {v M : u + s(v - u) M, s [0, 1]}.

u M (D(() + u()), v() - u()) d 0 v M(u).

u w = + u (w) M v = u + t M(u) w w u = w - M M(u) = M u M M (1, 2) k = 1, 2 k = 3 - k gk = | k | Gk = | k w | 1 R R m (Rk, R ) = km.

Fk k k = const k R k = 1, km F km Fk = F Rm/Gm.

m= km km km F G T = F Gk /Gm.

12 F = F = kk F = tk(k) k gk /Gk k = Gk/gk t1, t2 : R+ R+ k : R+ k k tk C(R+), tk() = 0, tk() > tk(), > 1, c, c0, c1, c2 > 0, p1 > 1, p2 > 1, k = 1, k k c0p - c1 tk() c2p, k C(), k() c > 0.

Rk km T T k = m Q : R :

G = |[1(), 2()]| R Q Q C(), C(), () c > 0,.

P = D(w) tk(|kw|/gk) D(w) = k kgk kw + | [1, 2] | Q.

|kw| k= u() = w() - () w (1) W () p1,p C0 () 1 + 2.

p1,p2 Lp1 Lp (1) V = W () pk > 1, k = 1, p1,ppk k v 1v + 2v, kv = |kv|p d, V Lp1 LpLpk , V V V V V k = 1, k( + v) k(v) =, v V, gk tk(|x|) Tk(x) = x, x R, |x| D( + u) = k (kgk Tk(k(u))) + | [1, 2] | Q.

k= V 2 k (kgk Tk(k(u))) d = - kgk (Tk(k(u)), k) d, k=1 k= u kgk (Tk(k(u)), k(v - u)) d k= (| [1, 2] | Q, v - u) d.

u M M = {v V : 3() + v3() F (() + v()) }, v M(u) M(u) = {v M : u + s(v - u) M, s [0, 1]}, Ak : V V f V Aku, v = kgk (Tk(k(u)), kv) d, k = 1, 2, f, v = (|[1, 2]| Q, v)d.

0 < c1 gk() c2,.

k |kgk Tk(k(u))| c|k( + u)|p -1,.

Ak k | Aku, v | c |k(u + )|p -1|kv|d pk-k c k(u + ) kv c( u + k )p -1 v.

Lpk Lpk V Lpk V A = A1 + A u M A(u), v - u f, v - u v M(u).

A (t) = A(u + tv), w [0, 1] u, v, w V pk pk- Au, u - v0 (c1 ku - c2 ku - c3) u V, Lpk Lpk k= v0 V A A tk Ak x, y R3 k = 1, 2 (x, y) |x| |y| (Tk(x) - Tk(y), x - y) = (x, y) (x, y) = tk(|x|)|x| - tk(|x|) - tk(|y|) + tk(|y|)|y| |x| |y| tk(|x|)|x| - tk(|x|)|y| - tk(|y|)|x| + tk(|y|)|y| = = (tk(|x|) - tk(|y|)) (|x| - |y|).

tk (tk() - tk()) ( - ) 0,, 0.

(Tk(x) - Tk(y), x - y) 0 x, y R3, k = 1, 2.

v, u V k = 1, Akv - Aku, v - u = = kgk gk (Tk(k(v)) - Tk(k(u)), k(v) - k(u)) d 0.

Ak A k = 1, Aku, u - v0 = kgk (Tk(k(u)), ku - kv0) d = tk(|k(u)|) = kgk gk (k(u), k(u) - k(v0)) d |k(u)| kgk gktk(|k(u)|) (|k(u)| - |k(v0)|) d k k kgk gk c0|k(u)|p - c1 - c2|k(u)|p -1|k(v0)| d pk pk- c k(u + ) - c k(u + ) k(v0 + ) - c 0 2 Lpk Lpk Lpk pk pk- C1 ku - C2 ku - C3, Lpk Lpk M + v(i) i=0 M v V Wp () p > 1 L1() + v(i) i=0 v [L1()]v v M k = 1, 2 k : V R k(v) = Ik(|k( + v)|/gk) k gk gk d, Ik() = tk(s)ds.

k k Ik() = tk(s)ds c2 sp -1ds = cp.

0 k(v + s(u - v)) lim = Akv, u, k(v) = Ak v.

ss : V R (v) = 1(v) + 2(v) - f, v, v V (v) = Av - f A M V M u (u) = min (v).

vM u v M(u) M(u) u+s(v-u) M s [0, 1] u M (u + s(v - u)) (u), s [0, 1].

(u + s(v - u)) - (u) = s (u + ss(v - u)), v - u, 0 s 1, A(u + ss(v - u)) - f, v - u 0.

s A (x1, x2) R [0, 1] ((x1, x2) + (1 - )(x1, x2)) (x1, x2) + (1 - )(x1, x2), = x R3 : x3 (x1, x2) M(v) P : R :

P (x) = min x - z, z |P (x) - P (z)| |x - z| x, z R.

v M M v v v v P = (P1, P2, P3 ) : v P () = P (() + v()),, v M v v M(v) = { V : (() + () - P (), N(P ())) 0, }, N : R3 R (u) M v v3() (v1(), v2()), P () M(v) M e R + te 0 t te, te > 0 : + te t = te, + te t > te.

/, te > e R.

> 0 : R 0 t te -, () = ( + te) = -t + te te - t te.

(1) W () p1,p lim (x) = 0.





(1) W () p1,p3() > (1(), 2()), = {x : x = (), } = d > d = inf x - z.

x, z p = min{p1, p2} > 2.

R (2) Wp () C() p (2, +) (2) [Wp ()]V [C()]3 p (1, +) {vn}+ M v V n + n= w M(v) w > n (0, w] n w M(vn), w = w + e3, e3 = (0, 0, 1), v M V {vn}+ v n= V [C()] v v M w M vn vn (() + w() - P (), N(P ())) = vn v vn = (( + w)() - P () P (), N(P ())) = v vn v = (( + w)() - P (), N(P ()) N(P ()))+ v vn vn +(P () - P (), N(P ())) = v v = (( + w)() - P (), N(P ()))+ v vn v +(( + w)() - P (), N(P ()) - N(P ()))+ v vn vn +(P () - P (), N(P ())).

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

r = d/4 d r > |( + v)( + te) - ( + v)( + tee)| < r {t, e}, 0 r, |( +w)( +te)-( +w)( +tee)| < r {t, e}, 0 r, 1 = {t, e} R R : te - t te, e R3, |e| = = { R : = + te {t, e} }.

v w r r > 0 r r R r = x R : | - x| r, w( +tee) = v( +tee) = 0 e ( + w)( + tee) r, ( + v)( + tee) r.

[0, r ] v v (( + w)() - P (), N(P ())) = v v v = (w() - v(), N(P ())) + (( + v)() - P (), N(P ())) v -|w() - v()| + |( + v)() - P ()| -2 r + d - r > 0.

4d ([1(x1, x2), 2(x1, x2)], e3) (N(x1, x2, x3), e3) = = |[1(x1, x2), 2(x1, x2)]| = > 0 x R, |[1(x1, x2), 2(x1, x2)]| v (e3, N(P ()) v min(e3, N(P ()) = m > 0, v ()(e3, N(P ())) ()m m \.

v v (( + w)() - P (), N(P ())) m [0, w], w = min r,.

4 d V {vn}+ v n= V [C()] > 0 n : | vn() - v() | < n > n, R > 0 : vn(), v(), w() BR(0), BR(0) = {x : |x| < R, x R } v vn v |(( + w)() - P (), N(P ()) - N(P ()))| v vn v max |( + w)() - P ()| |N(P ()) - N(P ())| vn v c |N(P ()) - N(P ())|.

vn {P ()}+ N n= > 0 n :

v vn v |(( + w)() - P (), N(P ()) - N(P ()))| < n > n,.

v vn vn |(P () - P (), N(P ()))| v vn |P () - P ()| |v() - vn()|, > 0 n :

v vn vn |(P () - P (), N(P ()))| < n > n,.

[ 0, w ] n = n = m/vn vn v v (( + w)() - P (), N(P ())) (( + w)() - P (), N(P ()))+ v vn v -|(( + w)() - P (), N(P ()) - N(P ()))|v vn vn -|(P () - P (), N(P ()))| > > - /3 - /3 > 0 n > n,, w M(vn) {vn}+ M v V n= w M(v) {wn}+ n=wn M(vn), n = 1, 2, 3,..., lim wn - w = 0.

V n+ {k}+ k=[0, ] w w k nk > nk-1 w M(vn) k n nk wn = vn 1 n < n1 wn = w nk n < nk+k k = 1, 2, 3,...

w k w - w = e3 V V k k k 0 k + (x1, x2) R [0, 1] ((x1, x2) + (1 - )(x1, x2)) (x1, x2) + (1 - )(x1, x2).

M M() = M M.

P q GP GP (1, 2) = q[1w(), 2w()].

u M v M tk(|k( + u)|/gk) kgk (k( + u), k(v - u))d|k( + u)| k=-q ([1( + u), 2( + u)], v - u)d (|[1, 2]| Q, v - u)d.

p2 > p1 2.

R W2 () Lr() r [1, +) [W2 ()]3 V Lr = [Lr()]3 r [1, +) B : V V Bu, v = ([1(u + ), 2(u + )], v) d u, v V.

r > 1 1 + + = 1.

p1 p2 r B p1, p2, r | Bu, v | |1(u + )| |2(u + )| |v|d 1(u + ) 2(u + ) v Lp1 Lp2 Lr ( u + c)2 v.

V V u M A(u), v - u - q B(u), v - u f, v - u v M.

B u v V ([1(u + ), 2(u + )], v)d = 1 = - ([1(u + ), (u + )], 2v)d - ([(u + ), 2(u + )], 1v)d.

2 () 2[1, ] = [21, ] + [1, 2], 1[, 2] = [1, 2] + [, 12].

[1, 2] = (2[1, ] + 1[, 2]).

w = u + wi = ui + {ui}+ [C2()]i= ui u V i v V ([1wi, 2wi], v)d = = - ([1wi, wi], 2v)d + ([wi, 2wi], 1v)d.

p1, p2, r ([1wi, 2wi], v)d - ([1w, 2w], v)d = = ([1wi, 2(wi - w)], v)d + ([1(wi - w), 2w], v)d 1wi 2(wi - w) v + Lp1 Lp2 Lr + 1(wi - w) 2w v c ui - u v.

Lp1 Lp2 Lr V V V {ui} i=([1wi, wi], 2v)d - ([1w, w], 2v)d = = ([1(wi - w), wi)], 2v)d + ([1w, wi - w], 2v)d C 1(wi - w) 2v wi + 1w 2v wi - w Lp1 Lp2 Lr Lp1 Lp2 Lr c ui - u v, V V ([wi, 2wi], 1v)d - ([w, 2w], 1v)d c ui - u v.

V V i B {ui}+ V u V v V i=lim Bui, v = Bu, v.

i wi = ui+ w = u+ | Bui, v - Bu, v | = = ([1wi, 2wi], v)d - ([1w, 2w], v)d = = ([1wi, wi], 2v)d - ([1w, w], 2v)d + + ([wi, 2wi], 1v)d - ([w, 2w], 1v)d ([1wi, wi], 2v)d - ([1w, w], 2v)d + + ([wi, 2wi], 1v)d - ([w, 2w], 1v)d.

Di = ([1wi, wi], 2v)d - ([1w, w], 2v)d = = ([1wi, ui - u], 2v)d - g, u - ui.

g : V R g, = ([1, w], 2v)d.

([1, w], 2v)d |1| |w| |2v|d 1 w 2v c.

Lp1 Lr Lp2 V ui u lim g, u - ui = 0, i V {ui}+ ui c i = 1, 2,...

i=1 V V Lr lim ui - u = 0.

Lr i ([1wi, ui - u], 2v)d 1(ui + ) ui - u 2v c ui - u Lr Lp1 Lr Lplim Di lim ([1wi, ui - u], 2v)d + | g, u - ui | = 0.

i+ i+ B (i) B (ii) {ui} V u V i=lim sup Bui, ui - u i lim inf Bui, ui - v Bu, u - v v V.

i B (i) (ii) {ui}+ i=V u V v V | Bu, u - v - Bui, ui - v | | Bu - Bui, u - v | + | Bui, u - ui |.

i + B | Bui, u - ui | ( ui + c)2 u - ui, V Lr i + lim Bui, ui - v = Bu, u - v.

i (ii) B (ii) {ui}+ u V i=lim Bui, ui - v = Bu, u - v v V.

i M X T : X X f X u M T u, v - u f, v - u v M T M v0 M T v, v - v0 ( v - v0 ) v v M, lim () = +.

V V + M 2pp2 > p1 >, p2 - f V q R 2pp2 > p1 =, p2 - q0 > f V q |q| < q0;

c0 > q0 > f c0, |q| q0.

V p2 > 2p2/(p2 - 1) p2 > 3 p1 > 2/(1 - p-1) > | Bu, v | |1(u + )| |2(u + )| |v|d 1(u + ) 2(u + ) v, Lp1 Lp2 Lr r p1pr =.

p1p2 - p1 - pp1(p2 - 1) > 2p2, p1(p2 - 1) - p2 > p2, p1 >, p1(p2 - 1) - ppp1 > p1 = r.

p1(p2 - 1) - p r p1 p > 0 r > r 1 1 + + = 1, (p1 - ) (p2 - ) r p1 - > r.

ap bq cr 1 1 abc + +, + + = 1, p, q, r > 1, a, b, c p q r p q r r | Bu, v | c 1(u + ) (p1-) + 2(u + ) (p2-) + v.

Lp1 Lp2 Lr r r r v c 1(v + ) + 2(v + ) Lr Lp1 Lp c 1(v + ) (p1-) + 2(v + ) (p2-) + 1.

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