V Q^ min /(x,M).
vdi moi x,y > 0.
L^y € > 0 Ik sd nhd tily <j, ta chiing minh tdn tai (x,y) G [0,OO) X [0,OO) sao c h o / ( x , y ) > 7 - e , tiic la lA + 7gx - g A + Bx + y hay ey + (Bx + ^ ) ( £ - 7 ) + a A + Bx + y ^
N^u e > 7 thi b^t dang thiic tren hi^n nhien dung vdi mgi (x, y) G [0, oo) x [0, oo).
Cdn n^u e < 7 thi ta chgn y = J(Bx + >l)(7 - e) va x tily y thudc [0, oo), khi đ
bat dang thiic tren thda man. Vay
sup /(x,y) = 7 = max{7, - } .
x,v>o A * Trudng hgp i <\'. Chumg minh tuong tu la nhan dugc
sup /(x,y) = - = raax{7, - } .
z,!/>0 -^ -^
Mat khac, ta cung cd
d _ Bjx + i-yA - Q)
d^^^""'^' ~ iy + Bx + A)^ '
do đ / la ham sd ludn dcm dieu giam theo bién x. Dd / don dieu tang theo bién
y, ta cd ngay didu kien la 7 ^ j .
Ta kiém tra dieu kien nghiem duy nhát ciia he phucmg U-inh
4> = f{^,<P), ^ = fi<P,i^)- ^ = fi<P,i^)-
He phuong trinh nay tuong ducmg vdi
(p^ + B(l)-ip + Ăp = ^4) + a, Tp'^ + B4np + Axp = ')'i> + ct.
Trilr \6 theo v^ cdc phuong trinh nky ta nhan dugc
{<!> - V)(0 + xP + A)=j{(P- ij;).
Hay
i<f> - tp)i(f> + ip + A --f) = 0.
N€u 7 ^ A thi he phuong tiinh tren cd nghiem duy nhat (f> = Tp = ẹ
Nhu vay, néu 7 ^ ^ va 7 ^ yl thi theo h6 di 3.2, mgi nghiem cua (3.4) hdi tu va gidi han nay la sd £.
Ta xet trudng hgp cdn lai di he phuong trinh cd nghiem duy nhát. Gia sir
7 > > l v a 0 + V' = 7 - ^ - Cdng vé theo vé hai phuong trlnh cua he tren la dugc
(02 ^ ^2^ ^ 2B4nl; + yl(,^ + i/^) = 7(0 + V') + 2a,
hoac tuong duong,
i(j) + lA)^ + 2(B - 1)HJ = (7 - A)i(P + iP) + 2Q.
Trong phuong trinh nay, thay (/> + V' bdi 7 - >1 ta thu dugc
{B - l)<p^ = Q.
Vi vay, néu B ^ 1 thi dang thiic iB - 1)HJ = a khdng the xay ra nen khdng the
cd (/) + V = 7 - >1, tiic la phuong trinh (0 - i/')(<^ + V + -4 - ^) = 0 cd nghiem duy
nhát (l) = ip^ ị
Nhu vay, n6u B ^l, ^,'r > A thi mgi nghiem cua (3.4) la hgi tụ
Xet trudng hgp i3 > 1. Ta cd
Mat khac, do (/. ^ V nen ((^ + VÓ > ^^H'- Tir hai ket qua nay, ket hctp \ di
0 + ^ = 7 - yl, ta cd
( 7 - ^ ) ^ > ' "