Let
X
be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space
X
⁎
. Let
T
:
X
⊇
D
(
T
)
→
2
X
⁎
and
A
:
X
⊇
D
(
A
)
→
2
X
⁎
be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for
T
+
A
under weaker sufficient conditions. These theorems improved the wellknown maximality results of Rockafellar who used condition
D
(
T
)
∘
∩
D
(
A
)
≠
∅
and Browder and Hess who used the quasiboundedness of
T
and condition
0
∈
D
(
T
)
∩
D
(
A
)
. In particular, the maximality of
T
+
∂
ϕ
is proved provided that
D
(
T
)
∘
∩
D
(
ϕ
)
≠
∅
, where
ϕ
:
X
→
(

∞
,
∞
]
is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.