Abstract
We propose a new discrete version of nonlinear oscillator with damping dynamical system governed by a general maximal monotone operator. We show the weak convergence of solutions and their weighted averages to a zero of a maximal monotone operator 
. We also prove some strong convergence theorems with additional assumptions on 
. This iterative scheme gives also an extension of the proximal point algorithm for the approximation of a zero of a maximal monotone operator. These results extend previous results by Brézis and Lions (1978), Lions (1978) as well as Djafari Rouhani and H. Khatibzadeh (2008).
be a real Hilbert space with inner product 
and norm 
. We denote weak convergence in 
by 
and strong convergence by 
. Let 
be a nonempty subset of 
which we will refer to as a (nonlinear) possibly multivalued operator in 
. 
is called monotone (resp. strongly monotone) if 
(resp. 
for some 
) for all 
, 
. 
is maximal monotone if 
is monotone and 
is surjective, where 
is the identity operator on 
.
is a maximal monotone operator and 
, has been investigated by many authors specially for asymptotic behavior. We refer the reader to [
, 
and 
, we get
(resp. 
) is nonnegative (resp. positive) sequence and 
. This discrete version gives also an algorithm for approximation of a zero of maximal monotone operator 
. This algorithm extends proximal point algorithm which was introduced by Martinet in [
and 
and then generalized by Rockafellar [
. Under suitable assumptions, we investigate weak and strong convergence of 
and 
to an element of 
if and only if 
is bounded. Therefore, 
if and only if 
is bounded provided 
. Our results extend previous results in [
, and we assume the following assumptions on the sequence 
: 
or its weighted average to an element of 
. First we recall the following elementary lemma without proof.
is a nonnegative sequence and 
is a positive sequence such that 
. If 
as 
, then 
as 
.
is a solution to (1.5) and 
satisfies (1.6). If 
and 
, then 
as 
if and only if 
is bounded.
by (1.5); we get
is bounded and this proves necessity. Now, we prove sufficiency. By monotonicity of 
, we have
. Multiplying both sides of the above inequality by 
and using (1.5), we deduce
to 
, we get
and suppose that 
and 
as 
. By assumptions on 
, 
and Lemma 2.1, we have
, we get 
exists. If 
, we obtain again 
exists. Therefore, 
, and hence 
exists. This follows that 
exists. It implies that 
and hence 
and 
as 
. Now we prove 
. Suppose that 
. By monotonicity of 
and Assumption (1.6), we get
, we get: 
. By maximality of 
, we get 
.
is 
(the domain of 
), as a trivial consequence of Theorem 2.2, we have that If 
is bounded then 
.
be a solution to (1.5) and 
. If 
satisfies (1.6), then 
as 
if and only if 
is bounded.
implies that 
, from (1.5) and (2.7), we get
to 
and letting 
, since 
satisfies (1.6), we have
, we have 
as 
. Assume 
as 
, by the monotonicity of 
, we have 
. Letting 
, we get 
. Similar to the proof of Theorem 2.2, 
exists. This implies that 
as 
.
under suitable assumptions on operator 
and the sequence 
.
is compact and 
. If 
satisfies (1.6), then 
as 
if and only if 
is bounded.
, we get 
and 
as 
. Therefore, there exists a subsequence 
of 
such that 
as 
and 
is bounded. The compacity of 
implies that 
has a strongly convergent subsequence (we denote again by 
) to 
. By the monotonicity of 
, we have 
. Letting 
, we obtain 
. Now, the proof of Theorem 2.2 shows that 
exists. This implies that 
as 
.
is strongly monotone operator and 
. If 
satisfies (1.6), then 
as 
if and only if 
is bounded.
as 
, and 
exists. Since 
is strongly monotone, we have
and summing from 
to 
, we have
, we get:
. This implies that 
as 
.
, where 
is a proper, lower semicontinuous and convex function and 
.
, where 
is a proper, lower semicontinuous, and convex function. Assume that 
is nonempty (i.e., 
has at least one minimum point) and 
. If 
satisfies (1.6), then 
as 
.
is subdifferential of 
and 
, by Assumption (1.6), we have
and summing from 
to 
and letting 
, we get
, we deduce
, we have
. By (2.20) and (2.21), we have that
, then 
. This implies that 
. On the other hand, for each 
by (1.5), we get (2.7). The proof of Theorem 2.2 implies that there exists 
. Then the theorem is concluded by Opial's Lemma (see [