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* → 2^{E*} (*i* = 1, 2,..., *N*) *be maximal monotone operators such that*. *Let**be such that* lim_{n→ ∞}*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* ⊂ *C*_{k+1}. By induction, we can conclude that *F* ⊂ *C*_{
n
}for all *n* ≥ 1.

**Step 2**. lim_{n→∞}*D*_{
f
}(*x*_{
n
}, *x*_{0}) 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 lim_{n→ ∞}*D*_{
f
}(*x*_{
n
}, *x*_{1}) exists.

**Step 3**. lim_{n→ ∞}||∇*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* lim_{n→ ∞}*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 *λ*^{i} *>* 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.