K¸t luªn v h÷îng nghi¶n cùu mð rëng

trong â Mβ = β 0 0 0 v N :R2×R2 →R3 l ma trªn 2×2×3 N = 0 0 1 0 0 0 0 0 0 0 0 0 , tùc l ,N(u, v) = (0, u1v1,0), vîi u, v ∈R2. V¼ th¸, 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, vîi 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 ành lþ 4.1 khæng thäa.

Cho u= (u1, u2)∈S2 sao cho (f, g)0(0,0)u∈ −[C×clK(g(0,0))]. Khi â, u= (u1,0)

vîi 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, vîik∗ = (0,0,−1)∈N(−K, g(0,0)),

supk∈T2(−K,g(0,0),g0(0,0)u)hk∗, ki=−4

(quan s¡t r¬ng hi»n t÷ñng envelope-like x£y ra). B¥y gií, vîi måi(Mβ, N)∈clB(f,g)(0,0),

tçn t¤i(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 , vîi måi(Mβ, N)∈ B(f,g)(0,0)∞\ {0}, tçn t¤i c∗ = 1 ∈C∗\ {0}vîi hc∗, f0(0,0)ui= 0

thäa

hc∗, Mβ(u, u)i=β >0.

V¼ th¸, (a0

) cõa Nhªn x²t 4.2 v do â (ii) (a) trong ành lþ 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. V¼ th¸,

vîi måiMβ ∈Bf(0,0)∞, tçn t¤i c∗ = 1 ∈C∗ \ {0} vîi hc∗, f0(0,0)ui= 0 thäa

hc∗, f0(0,0)w+Mβ(u, u)i=w2+β >0,

v , vîi måiMβ ∈Bf(0,0)∞\ {0}, tçn t¤i c∗ = 1∈ C∗ \ {0} vîi hc∗, f0(0,0)ui= 0 thäa

hc∗, Mβ(u, u)i =β > 0. V¼ th¸, bði Nhªn x²t 4.2 (ii), i·u ki»n (ii) (b) cõa ành lþ 4.1

thäa. H» qu£ l ,(0,0)∈ LFE(2, f, S).

V¼ f 6∈ C1 t¤i (0,0), c¡c H» qu£ 7, 8 cõa [7], ành lþ 4.5 cõa [25] v c¡c h» qu£ 4.4

v 4.5 ð tr¶n khæng ¡p döng ÷ñc. Hìn núa, v¼d2(f, g)((0,0), u) =∅, ành lþ 3 cõa [7]

K¸t luªn v h÷îng nghi¶n cùu mð rëng · t i

Trong · t i nghi¶n cùu n y, ¦u ti¶n, chóng tæi giîi thi»u kh¡i ni»m v· c¡c tªp ti¸p xóc c§p mët v c§p hai v kh£o s¡t mët sè t½nh ch§t cõa chóng. Ti¸p theo, chóng tæi · xu§t kh¡i ni»m ¤o h m suy rëng kiºu x§p x¿ c§p mët v c§p hai v ÷a ra c¡c t½nh ch§t cõa chóng. Cuèi còng, dòng c¡c ¤o h m suy rëng kiºu x§p x¿ n y d÷îi gi£ thi¸t kh£ vi ch°t (trong c¡c i·u ki»n tèi ÷u c¦n) hay kh£ vi (trong c¡c i·u ki»n tèi ÷u õ), chóng tæi thi¸t lªp c¡c i·u ki»n tèi ÷u c§p hai mîi cho c¡c nghi»m y¸u àa ph÷ìng v c¡c nghi»m chc chn àa ph÷ìng, vîi t½nh ch§t envelope-like ÷ñc l m rã hìn, cõa b i to¡n tèi ÷u vectì khæng trìn trong c¡c khæng gian væ h¤n chi·u (P).

Trong k¸ ho¤ch nghi¶n cùu t÷ìng lai, chóng tæi s³ mð rëng h÷îng nghi¶n cùu cõa · t i b¬ng c¡ch x²t b i to¡n tèi ÷u vectì khæng trìn vîi r ng buëc bao h m thùc kh¡ têng qu¡t sau ¥y:

(P1) minCf(x), sao chox∈S, 0∈F(x),

trong â f :X → Y l ¡nh x¤ ìn trà v F : X → 2Z l ¡nh x¤ a trà, X v Z l c¡c

khæng gian Banach,Y l khæng gian ành chu©n, S ⊂X, v C ⊂Y l nân lçi âng.

Chóng tæi s³ thi¸t lªp c¡c i·u ki»n tèi ÷u c¦n v õ c§p mët v c§p hai cho c¡c nghi»m y¸u v nghi»m chc chn cõa b i to¡n (P1) b¬ng c¡c quy tc nh¥n tû Fritz-

John-Lagrange. Chóng tæi dòng c¡c ¤o h m suy rëng kiºu x§p x¿ chof, ¤o h m theo

h÷îng a trà choF, v c¡c nân ti¸p xóc v tªp ti¸p xóc c§p mët v c§p hai d÷îi c¡c gi£

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