Browder-Tikhonov正则化处理非强制型发展二阶半变分不等式
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本文主要探讨了一类涉及非强制算子的发展型二阶半变分不等式（Evolution Second Order Hemivariational Inequalities）的Browder-Tikhonov正则化方法。Browder-Tikhonov正则化是一种在数值计算和优化问题中常用的策略，当面对未知或部分未知的非光滑、非凸目标函数时，通过引入一个惩罚项来增强问题的可解性。作者肖义彬和黄南京分别来自中国电子科技大学应用数学学院和四川大学数学系，他们共同研究了这类问题。
在论文中，他们假设非强制算子是已知的但可能存在近似误差。他们采用Browder-Tikhonov正则化的思路，即在原始问题的基础上添加一个正则化项，以提高问题的结构清晰度和稳定性。通过这种方法，他们证明了正则化后的进化型半变分不等式问题是可解的，这在理论和实际应用中具有重要意义，因为它确保了求解过程的可行性。
接着，作者构造了一个基于正则化问题解序列的框架，并展示了这个序列的弱聚点就是原问题（即未正则化的进化二阶半变分不等式）的解。这意味着通过正则化，虽然可能引入了一定程度的近似，但我们能够得到一个收敛于原问题解的近似解序列。此外，文中还讨论了与该问题相关的关键词，如进化半变分不等式、演化包含、对D(L)的伪单调性、正则化、收敛性以及对偶性方法等。
这篇首发论文深入研究了在存在不确定性或近似性的背景下，如何通过Browder-Tikhonov正则化策略有效地处理一类进化型二阶半变分不等式。这对于理解此类复杂问题的求解策略，特别是在工程和科学计算中，提供了重要的理论支持和数值实践指导。
which is equivalent to the following evolution inclusion
Find u ∈ V such that u
0
∈ W and
f(t) ∈ u
00
(t) + Au
0
(t) + Bu(t) + ∂
2
G(t, u
0
(t)) for a.e. t ∈ [0, T ]
u(0) = θ
u
0
(0) = θ.
(2.2)
In the sequel, to formulate our main results, we present some important definitions, lemmas, theo-
rems and hypotheses.
Definition 2.1 (see [10]) Let V be a real reflexive Banach space with its dual V
∗
. A mapping T :
D(T ) ⊂ V → 2
V
∗
is said to be pseudomonotone if the following conditions hold:
(a)
The set T u is nonempty, bounded, closed and convex for all u ∈ D(T );
(b) T is upper semi-continuous from each finite dimensional subspace F of V into V
∗
endowed with
the weak topology;
(c)
if {u
i
} is a sequence in D(T ) converging weakly to u of D(T ), and if u
∗
i
∈ T u
i
is such that
lim suphu
∗
i
, u
i
− ui ≤ 0, then for each element v ∈ D(T ) there exists u
∗
(v) ∈ T u with the property
that
lim infhu
∗
i
, u
i
− ui ≥ hu
∗
(v), u − vi.
Definition 2.2
(see [10]) Let V be a real reflexive Banach space with its dual V
∗
, L : D(L) ⊂ V → V
∗
be a linear densely defined maximal monotone mapping. A mapping T : D(T ) ⊂ V → 2
V
∗
is said to be
pseudomonotone with respect to D(L) if and only if (a), (b) and the following condition holds:
(d)
if {u
i
} ∈ D(L)∩D(T ) is such that u
i
→ u weakly in V , Lu
i
→ Lu weakly in V
∗
, u
∗
i
∈ T (u
i
), u
∗
i
→
u
∗
weakly in V
∗
and lim suphu
∗
i
, u
i
− ui ≤ 0, then u
∗
∈ T (u) and hu
∗
i
, u
i
i → hu
∗
, ui.
Definition 2.3 (see [3, 16, 23]) A mapping T : D(T ) ⊂ V → 2
V
∗
is said to have (S
+
) property (be of
class (S
+
)) if and only if (a), (b) and the following condition holds:
(e)
for any sequence {u
i
} in D(T ) converging weakly to u ∈ D(T ), w
i
∈ T (u
i
), the condition
lim suphw
i
, u
i
− ui ≤ 0
implies the strong convergence of {u
i
} to u in V and there exists a subsequence {w
n
i
} of {w
i
}
such that {w
n
i
} converges weakly to w ∈ T u in V
∗
.
It is well known that the conditions
kJ(u)k
V
∗
= kuk
V
and hJ(u), ui = kuk
2
V
, ∀u ∈ V (2.3)
determine a unique mapping J from V to V
∗
, which is called the duality mapping. In our case it is
bijective bicontinuous, strictly monotone and of class (S
+
). For more details we can refer to [23].
Definition 2.4
(see [1]) Let L : D(L) ⊂ V → V
∗
be a linear densely defined maximal monotone
mapping. A multivalued mapping T : D(T ) ⊂ V → 2
V
∗
is said to have (S
+
) property with respect to
D(L) (be of class (S
+
) with respect to D(L)) if and only if (a), (b) and the following condition holds:
4
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