trong â Mβ = β 0 0 0 v N :R2×R2 →R3 l ma trên 2ì2ì3 N = 0 0 1 0 0 0 0 0 0 0 0 0 , tùc l ,N(u, v) = (0, u1v1,0), vỵi u, v ∈R2. Vẳ thá, clBf(0,0) ={Mβ|β∈ {−1} ∪[1,∞)}, clB(f,g)(0,0) ={(Mβ, N)|β ∈ {−1} ∪[1,∞)}, Bf(0,0)∞={Mβ|β ≥0}, B(f,g)(0,0)∞ ={(Mβ,02×2×3)|β ≥0}. Chånu= (1,0)∈S2. Ta câ, vỵi måi (c∗, k∗)∈C∗×K(g(0,0))∗,
hc∗, f0(0,0)ui+hk∗, g0(0,0)ui= 0. Do â, i·u ki»n (i) trong nh lỵ 4.1 khổng thọa.
Cho u= (u1, u2)∈S2 sao cho (f, g)0(0,0)u∈ −[C×clK(g(0,0))]. Khi â, u= (u1,0)
vỵi u1 =±1. Ta câ
T2(−K, g(0,0), g0(0,0)u) =A2(−K, g(0,0), g0(0,0)u),
v do â, vỵik∗ = (0,0,−1)∈N(−K, g(0,0)),
supk∈T2(−K,g(0,0),g0(0,0)u)hk∗, ki=−4
(quan sĂt rơng hiằn tữủng envelope-like xÊy ra). BƠy giớ, vỵi måi(Mβ, N)∈clB(f,g)(0,0),
tỗn tÔi(c∗, k∗) = (1,0,0,−1)∈Λ1(0,0)thäa
hc∗,2Mβ(u, u)i+hk∗,2N(u, u)i= 2β >supk∈T2(−K,g(0,0),g0(0,0)u)hk∗, ki
v , vỵi måi(Mβ, N)∈ B(f,g)(0,0)∞\ {0}, tỗn tÔi c∗ = 1 ∈C∗\ {0}vỵi hc∗, f0(0,0)ui= 0
thäa
hc∗, Mβ(u, u)i=β >0.
Vẳ thá, (a0
) cõa Nhªn x²t 4.2 v do õ (ii) (a) trong nh lỵ 4.1 thäa. Hìn núa, cho
w = (w1, w2) ∈ v⊥ \ {(0,0)}, tùc l , w1 = 0 v w2 6= 0, n¸u g0(0,0)w = (0,0, w2) ∈
clcone[cone(−K−g(0,0))−g0(0,0)u] ={(k1, k2, k3) ∈R3|k3 ≥0}, th¼ w2 >0. Vẳ thá,
vợi måiMβ ∈Bf(0,0)∞, tỗn tƠi c∗ = 1 ∈C∗ \ {0} vỵi hc∗, f0(0,0)ui= 0 thäa
hc∗, f0(0,0)w+Mβ(u, u)i=w2+β >0,
v , vỵi måiMβ ∈Bf(0,0)∞\ {0}, tỗn tÔi c∗ = 1∈ C∗ \ {0} vỵi hc∗, f0(0,0)ui= 0 thäa
hc∗, Mβ(u, u)i =β > 0. Vẳ thá, bi Nhên xt 4.2 (ii), iÃu kiằn (ii) (b) cừa nh lỵ 4.1
thäa. H» qu£ l ,(0,0)∈ LFE(2, f, S).
V¼ f 6∈ C1 tÔi (0,0), c¡c H» qu£ 7, 8 cõa [7], ành lỵ 4.5 cừa [25] v cĂc hằ quÊ 4.4
v 4.5 trản khng Ăp dng ữc. Hỡn nỳa, vẳd2(f, g)((0,0), u) =∅, nh lỵ 3 cừa [7]
Kát luên v hữợng nghiản cựu m rởng · t i
Trong · t i nghi¶n cựu ny, Ưu tiản, chúng tổi giợi thiằu khĂi niằm và cĂc têp tiáp xúc cĐp mởt v cĐp hai v khÊo sĂt mởt số tẵnh chĐt cừa chúng. Tiáp theo, chúng tổi à xuĐt khĂi niằm Ôo hm suy rởng kiu xĐp x cĐp mởt v cĐp hai v ữa ra cĂc tẵnh chĐt cừa chúng. Cuối cũng, dũng cĂc Ôo hm suy rởng kiu xĐp x ny dữợi giÊ thiát khÊ vi cht (trong cĂc iÃu kiằn tối ữu cƯn) hay khÊ vi (trong c¡c i·u ki»n tèi ÷u õ), chúng tổi thiát lêp cĂc iÃu kiằn tối ữu cĐp hai mợi cho cĂc nghiằm yáu a phữỡng v c¡c nghi»m chc chn àa phữỡng, vợi tẵnh chĐt envelope-like ữủc lm ró hỡn, cừa b i to¡n tèi ÷u vectì khỉng trỡn trong cĂc khổng gian vổ hÔn chiÃu (P).
Trong ká hoÔch nghiản cựu tữỡng lai, chúng tổi s m rởng hữợng nghiản cựu cừa à t i b¬ng c¡ch x²t b i to¡n tối ữu vectỡ khổng trỡn vợi rng buởc bao hm thùc kh¡ têng qu¡t sau ¥y:
(P1) minCf(x), sao chox∈S, 0∈F(x),
trong â f :X → Y l Ănh xÔ ỡn tr v F : X → 2Z l Ănh xƠ a tr, X v Z l c¡c
khỉng gian Banach,Y l khỉng gian ành chu©n, S ⊂X, v C ⊂Y l nõn lỗi õng.
Chúng tổi s thiát lêp cĂc iÃu kiằn tối ữu cƯn v ừ c§p mët v cĐp hai cho cĂc nghiằm yáu v nghiằm chưc chưn cừa b i to¡n (P1) bơng cĂc quy tưc nhƠn tỷ Fritz-
John-Lagrange. Chúng tổi dũng cĂc Ơo hm suy rëng kiºu x§p x¿ chof, Ôo hm theo
hữợng a tr choF, v c¡c nân ti¸p xúc v têp tiáp xúc cĐp mởt v cĐp hai dữợi cĂc giÊ
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