#### Abstract

Based on a notion of relatively maximal (m)-relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar's theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this new model, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.

#### 1. Introduction

Let be a real Hilbert space with the inner product and with the norm on We consider the inclusion problem. Find a solution to

where is a set-valued mapping on

Rockafellar [1, Theorem ] discussed general convergence of the proximal point algorithm in the context of solving (1.1), by showing for maximal monotone, that the sequence generated for an initial point by the proximal point algorithm

converges strongly to a solution of (1.1) provided that the approximation is made sufficiently accurate as the iteration proceeds, where is the resolvent operator for a sequence of positive real numbers, that is bounded away from zero. We observe from (1.2) that is an approximate solution to inclusion problem

Next, we state the theorem of Rockafellar [1, Theorem ], where an approach of using the Lipschitz continuity of instead of the strong monotonicity of is considered, that turned out to be more application enhanced to convex programming. Moreover, it is well-known that the resolvent operator is nonexpansive, so it does not seem to be possible to achieve a linear convergence without having the Lipschitz continuity constant less than one in that setting. This could have been the motivation behind looking for the Lipschitz continuity of at zero which helped achieving the Lipschitz continuity of with Lipschitz constant that is less than one instead.

Theorem 1.1. Let be a real Hilbert space, and let be maximal monotone. For an arbitrarily chosen initial point let the sequence be generated by the proximal point algorithm (1.2) such that where , and the scalar sequences and respectively, satisfy and is bounded away from zero.

We further suppose that sequence is generated by the proximal point algorithm (1.2) such that

where scalar sequences and respectively, satisfy and

Also, assume that is bounded in the sense that the solution set to (1.1) is nonempty, and that is -Lipschitz continuous at 0 for Let

Then the sequence converges strongly to a unique solution to (1.1) with

where

As we observe that most of the variational problems, including minimization or maximization of functions, variational inequality problems, quasivariational inequality problems, minimax problems, decision and management sciences, and engineering sciences can be unified into form (1.1), the notion of the general maximal monotonicity has played a crucially significant role by providing a powerful framework to develop and use suitable proximal point algorithms in exploring and studying convex programming and variational inequalities. Algorithms of this type turned out to be of more interest because of their roles in certain computational methods based on duality, for instance the Hestenes-Powell method of multipliers in nonlinear programming. For more details, we refer the reader to .

In this communication, we examine the approximation solvability of inclusion problem (1.1) by introducing the notion of relatively maximal -relaxed monotone mappings, and derive some auxiliary results involving relatively maximal -relaxed monotone and cocoercive mappings. The notion of the relatively maximal -relaxed monotonicity is based on the notion of -maximal -relaxed monotonicity introduced and studied in [9, 10], but it seems more application-oriented. We note that our approach to the solvability of (1.1) differs significantly than that of  in the sense that is without the monotonicity assumption; there is no assumption of the Lipschitz continuity on and the proof turns out to be simple and compact. Note that there exists a huge amount of research on new developments and applications of proximal point algorithms in literature to approximating solutions of inclusion problems of the form (1.1) in different space settings, especially in Hilbert as well as in Banach space settings.

#### 2. Preliminaries

In this section, first we introduce the notion of the relatively maximal -relaxed monotonicity, and then we derive some basic properties along with some auxiliary results for the problem on hand.

Let be a real Hilbert space with the norm for , and with the inner product

Definition 2.1. Let be a real Hilbert space, and let be a multivalued mapping and a single-valued mapping on The map is said to be the following. (i)Monotone if (ii)Strictly monotone if is monotone and equality holds only if . (iii)-strongly monotone if there exists a positive constant such that (iv)-expanding if there exists a positive constant such that (v)Strongly monotone if (vi)Expanding if (vii)-relaxed monotone if there is a positive constant such that (viii)-cocoercive if there exists a positive constant such that (ix)Monotone with respect to if (x)Strictly monotone with respect to if is monotone with respect to and equality holds only if . (xi)-strongly monotone with respect to if there exists a positive constant such that (xii)-relaxed monotone with respect to if there exists a positive constant such that (xiii)-hybrid relaxed monotone with respect to if there exists a positive constant such that (xiv)-cocoercive with respect to if there exists a positive constant such that

Definition 2.2. Let be a real Hilbert space, and let be a mapping on Furthermore, let be a single-valued mapping on The map is said to be the following. (i)Nonexpansive if (ii)Cocoercive if (iii)Cocoercive with respect to if

Definition 2.3. Let be a real Hilbert space. Let be a single-valued mapping. The map is said to be relatively maximal -relaxed monotone (with respect to ) if (i) is -relaxed monotone with respect to for (ii) for

Definition 2.4. Let be a real Hilbert space. Let be a single-valued mapping. The map is said to be relatively maximal monotone (with respect to ) if (i) is monotone with respect to (ii) for

Definition 2.5. Let be a real Hilbert space, and let be -strongly monotone. Let be a relatively maximal -relaxed monotone mapping. Then the resolvent operator is defined by

Proposition 2.6. Let be a real Hilbert space. Let be an -strongly monotone mapping, and let be a relatively maximal -relaxed monotone mapping. Then the resolvent operator is single valued for

Proof. For any assume Then we have Since is relatively maximal -relaxed monotone, and is -strongly monotone, it follows that

Definition 2.7. Let be a real Hilbert space. A map is said to be maximal monotone if (i) is monotone, (ii) for

Note that all relatively monotone mappings are relatively -relaxed monotone for We include an example of the relative monotonicity and other of the relative -hybrid relaxed monotonicity, a new notion to the problem on hand.

Example 2.8. Let and for all Then is relatively monotone but not monotone, while is relatively -relaxed monotone for

Example 2.9. Let be a real Hilbert space, and let be maximal -relaxed monotone. Then we have the Yosida approximation where is the resolvent of that satisfies that is, is relatively -hybrid relaxed monotone (with respect to

#### 3. Generalization to Rockafellar's Theorem

This section deals with a generalization to Rockafellar's theorem [1, Theorem ] in light of the new framework of relative maximal -relaxed monotonicity, while solving (1.1).

Theorem 3.1. Let be a real Hilbert space, let be -strongly monotone, and let be relatively maximal -relaxed monotone. Then the following statements are mutually equivalent:(i)an element is a solution to (1.1), (ii)for an one has where

Proof. To show (i) (ii), if is a solution to (1.1), then for we have
Similarly, to show (ii) (i) we have

Theorem 3.2. Let be a real Hilbert space, let be -strongly monotone, and let be relatively maximal -relaxed monotone. Furthermore, suppose that is relatively -hybrid relaxed monotone and (o is -cocoercive with respect to

(i) For an arbitrarily chosen initial point suppose that the sequence is generated by the proximal point algorithm (1.2) such that

where , and the scalar sequence satisfies Suppose that the sequence is bounded in the sense that the solution set of (1.1) is nonempty.

(ii) In addition to assumptions in (i), we further suppose that, for an arbitrarily chosen initial point the sequence is generated by the proximal point algorithm (1.2) such that

where , and the scalar sequences and , respectively, satisfy and Then the following implications hold:

(iii) the sequence converges strongly to a solution of (1.1),

(iv) rate of convergences

where

Proof. Suppose that is a zero of We begin with the proof for where It follows from the definition of the generalized resolvent operator the relative -hybrid relaxed monotonicity of with respect to and the -cocoercivity of (o with respect to that or
Next, we move to estimate

For combining the previous inequality for all , we have
Hence, is bounded.
Next we turn our attention to convergence part of the proof. Since

we get
where
It follows that
It appears that (3.15) holds since (seems to hold) and
Hence, the sequence converges strongly to
To conclude the proof, we need to show the uniqueness of the solution to (1.1). Assume that is a zero of Then using we have that
is nonnegative and finite, and as a result, Consider and to be two limit points of then we have and both exist and are finite. If we express then it follows that Since is a limit point of the left hand side limit must tend to zero. Therefore, Similarly, we obtain This results in

Remark 3.3. When equals the subdifferential of , where is a functional on a Hilbert space can be applied for minimizing The function is proper if and is convex if where and Furthermore, the function is lower semicontinuous on if the set is closed in

The subdifferential of at is defined by

In an earlier work , Rockafellar has shown that if is a lower semicontinuous proper convex functional on then is maximal monotone on where is any real Banach space. Several other special cases can be derived.

Suppose that is strongly monotone and -cocoercive, and let be -locally Lipschitz (for ) such that is -relaxed monotone with respect to , that is,

where and Then is relatively maximal -relaxed monotone.

#### Acknowledgment

The author is greatly indebted to Professor Petru Jebelean and reviewers for their valuable comments and suggestions leading to the revised version.