Now, in this section, we prove our main results of this article.
Theorem 3.1. Let E be a real reflexive Banach space and f : E → ℝ a Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let A
i
: E → 2E* (i = 1, 2,..., N) be maximal monotone operators such that
. Let
be such that limn→ ∞e
n
= 0. Define a sequence
in E as follows:
If
for each i = 1, 2,..., N, then the sequence {x
n
} converges strongly to a point
Proof. We divide our proof into six steps as follows:
Step 1. F ⊂ C
n
for all n ≥ 1.
Since
is closed and convex for each i = 1, 2,..., N, we get that
is a nonempty, closed and convex subset of E. It is easy to see that C
n
is closed and convex for all n ≥ 1. Indeed, for each z ∈ C
n
, it follows that D
f
(z, y
n
) ≤ D
f
(z, x
n
+ e
n
) is equivalent to
This shows that C
n
is closed and convex for all n ≥ 1. It is obvious that F ⊂ C 1 = E.
Now, suppose that F ⊂ C
k
for some
. For any p ∈ F, by Lemma 2.3, we have
This implies that F ⊂ Ck+1. By induction, we can conclude that F ⊂ C
n
for all n ≥ 1.
Step 2. limn→∞D
f
(x
n
, x0) exists.
From
and
we have
By (2.1), for any p ∈ F ⊂ C
n
, we have
Combining (3.3) and (3.4), we know that limn→ ∞D
f
(x
n
, x1) exists.
Step 3. limn→ ∞||∇f(y
n
) - ∇f(x
n
+ e
n
)|| = 0
Since
for m > n ≥ 1, by (2.1), it follows that
Letting m, n → ∞, we have D
f
(x
m
, x
n
) → 0. Since f is totally convex on bounded subsets of E, f is sequentially consistent by Butnariu and Resmerita [20]. It follows that ||x
m
- x
n
|| → 0 as m, n → ∞. Therefore, {x
n
} is a Cauchy sequence. By the completeness of the space E, we can assume that x
n
→ q ∈ E as n → ∞. In particular, we obtain
Since e
n
→ 0, we also obtain
Since 
We know from [23] that, if f is bounded on bounded subsets of E, then ∇f is also bounded on bounded subsets of E. Moreover, if f is uniformly Fréchet differentiable on bounded subsets of E, then f is uniformly continuous on bounded subsets of E (see [24]). Using (3.5), we have
Also, we have
and hence,
and, since e
n
→ 0,
Since f is uniformly Fréchet differentiable on bounded subsets of E, ∇f is norm-to-norm uniformly continuous on bounded subsets of E by Lemma 2.1. Hence, we have
Step 4.
.
Denote
for each i ∈ {1, 2,..., N} and
for each n ≥ 1. We note that
for each n ≥ 1. For any p ∈ F, by (3.2), it follows that
Since
, by Lemma 2.3 and (3.8), it follows that
From (3.6) and (3.7), we get that
. Since f is sequentially consistent,
Thus, from (3.6) and (3.9), it follows that
and hence,
Again, since
, by Lemma 2.3 and (3.8), we know that
From (3.10) and (3.11), we have
Since f is sequentially consistent, it follows that
From (3.10) and (3.12), we have
and hence,
In a similar way, we can show that
,
and
Hence, we can conclude that
for each i = 1,2,..., N.
Step 5. 
For each i = 1, 2,..., N, we note that
and so
From (3.13) and
, we have
We note that
for each i = 1, 2,..., N. If (w, w*) ∈ G(Ai) for each i = 1, 2,..., N , then it follows from the monotonicity of A
i
that
Since x
n
→ q and e
n
→ 0, x
n
+ e
n
→ q. Therefore,
for each i = 1, 2,..., N. Thus, from (3.14), we have
By the maximality of A
i
, we have
for each i = 1, 2,..., N. Hence,
.
Step 6.
.
From
, we have
Since F ⊂ C
n
, we also have
Letting n → ∞ in (3.15), we obtain
Hence, we have
. This completes the proof.
As a direct consequence of Theorem 3.1, we also obtain the following result concerning a system of convex minimization problems in reflexive Banach spaces:
Theorem 3.2. Let E be a real reflexive Banach space and f : E → ℝ a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let g
i
: E → (- ∞, ∞] (i = 1, 2,..., N) be proper lower semi-continuous convex functions such that
. Let
be a sequence in E such that limn→ ∞e
n
= 0. Define a sequence
in E as follows:
If
for each i = 1, 2,..., N, then the sequence {x
n
} converges strongly to a point
.
Proof. By Rockafellar's theorem [25, 26], ∂g
i
are maximal monotone operators for each i = 1, 2,..., N. Let λi > 0 for each i = 1, 2,..., N. Then
if and only if
which is equivalent to
Using Theorem 3.1, we can complete the proof.
Remark 3.3. By means of the composite iterative scheme together with the shrinking projection method, we can construct the proximal point algorithms for finding a common element in the set
. Moreover, our algorithm is different from that of Reich and Sabach [11] which is based on a finite intersection of sets.
Remark 3.4. Theorems 3.1 and 3.2 also hold in a uniformly convex and uniformly smooth Banach space with the generalized duality mapping.