Báo cáo hóa học: "RICCATI INEQUALITY AND OSCILLATION CRITERIA FOR PDE WITH p-LAPLACIAN" pptx
Báo cáo hóa học: "RICCATI INEQUALITY AND OSCILLATION CRITERIA FOR PDE WITH p-LAPLACIAN" pptx
... INEQUALITY AND OSCILLATION CRITERIA FOR
PDE WITH p-LAPLACIAN
ZHITING XU
Received 1 November 2003; Revised 25 D ecember 2004; Accepted 25 December 2004
Oscillation criteria for PDE with p-Laplacian div(A(x)
Du
p−2
Du)+p(x)|u|
p−2
u ... C
2
r
l
ρ(s)g(s)
Z(s)
q
ds
1/q
,
(3.6)
10 Riccati inequality and oscillation criteria
Note that for α>1anda...
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... conditions for oscillation of delay differential equations,” Journal of
Mathematical Analysis and Applications, vol. 219, no. 1, pp. 165–178, 1998.
4 J. Shen and X. Tang, “New oscillation criteria for ... H. Tang and J. S. Yu, Oscillation of delay difference equation,” Computers & Mathematics with
Applications, vol. 37, no. 7, pp. 11–20, 1999.
7 M. Bohner, “Some osci...
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... (0,s
] for some positive real number
s
≥ max{α
X
,β
X
},andinequality(3.9)istrueforallx, y in X. Then {Sx : x ∈ X and Sx ∈
Fx} and {Tx : x ∈ X and Tx ∈ Gx} are nonempty, closed sets and are ... TX
and Gx ⊆ SX for all x in X, ϕ ∈ Γ,
ˆ
ϕ(t) <tfor all t in (0,s
] for some positive real number
s
≥ max{α
X
,β
X
}, and that inequality (3.12)istrueforallx, y in X. Then...
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... function; for example, this leads to the distortion and growth estimates for
univalent functions as well as the rotation theorem. In the next theorem, we derive the
distortion and growth estimates for ... =
ϕ(r)
1 − r
2
.
The other inequality for |f’ (z)| is similar. Since the function f is univalent, the
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... l
2
δ(a) for all
a ∈
A
and all l Î ℂ,
(3) δ(ab)=δ(a)b
2
+ a
2
δ(b) for all
a
,
b ∈
A
,
(4) δ(a*) = δ(a)* for all
a ∈
A
.
Theorem 4.2 Suppose that
f
: A →
A
is a mapping with f(0) = 0 for which ... ∈
A
and therefore
δ(a)+δ(b)=2δ
a+b
2
= δ( a + b
)
for all
a
,
b ∈ A
. Moreover, letting m = 0 and passing the limit n ® ∞ in (3.6), we get (3.5).
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