REGULARIZATION AND HOLDER TYPE ERROR ESTIMATES FOR ...

H.T. Nguyen, P.V. Tri, L.D. Thang, N.V. Hieu - Regularization and Hölder type...
This gives e kMx 0
≥ M k
kɛ
Hence, we obtain
or kMx 0 ≥ ln( M k
kɛ
). Therefore
x 0 ≥ 1
kM ln(Mk kɛ ).
f(x) ≤ 1
ɛx k 0
≤
(kM)k
ɛ ln k ( M k
kɛ ).
Lemma 3.2
For 0 ≤ m ≤ M, we have the following inequality
e −mx
(
ɛx k + e −Mx ≤ (kM)k ɛ m M −1 ln( M k ) k(
m
kɛ ) M −1) .
Proof.
Since the inequality
1
ɛx k +e −Mx ≤ (kM) k ɛ −1 (
ln( M k
kɛ ) ) −k,
we obtain
e −mx
ɛx k + e −Mx =
≤
≤
e −mx
(ɛx k + e −Mx ) m M (ɛx k + e −Mx ) 1− m M
1
(ɛx k + e −Mx ) 1− m M
(
[(kM) k ɛ −1 ln( M k ) −k
] 1−
m
M
kɛ )
(
≤ (kM) k(1− m M ) ɛ m M −1 ln( M k ) k(
m
kɛ ) M −1)
≤ (kM) k ɛ m M −1 (
ln( M k
kɛ ) ) k(
m
M −1) .
In next Theorem, we shall study the existence, the uniqueness and the stability of
a (weak) solution of Problem (6)-(8). In fact, one has
Theorem 3.1 The problem (11) has uniquely a weak solution u ɛ ∈ satisfying (10).
The solution depends continuously on g in L 2 (0, π)).
Proof
The proof is divided into two steps. In Step 1, we prove the existence and the
uniqueness of a solution of (6)-(8). In Step 2, the stability of the solution is given.
Denote W = ([0, T ]; L 2 (0, π) ∩ L 2 (0, T ; H0 1(0, π)) ∩ C1 (0, T ; H0 1 (0, π)).
Step 1. The existence and the uniqueness of a solution of (6)-(8)
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