Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces

Let be a Hilbert space and a nonempty closed convex subset of . Let be a maximal monotone mapping and a bounded demicontinuous strong pseudocontraction. Let be the unique solution to the equation . Then is bounded if and only if converges strongly to a zero point of A as which is the unique solution in , where denotes the zero set of , to the following variational inequality , for all .

Throughout this work, we always assume that is a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of and a nonlinear mapping. We use and to denote the domain and the range of the mapping . and denote strong and weak convergence, respectively.

Recall the following well-known definitions.

(1)A mapping is said to be monotone if

(2)The single-valued mapping is maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies .

(3) is said to be pseudomonotone if for any sequence in which converges weakly to an element in with we have

(4) is said to be bounded if it carries bounded sets into bounded sets; it is coercive if as .

(5)Let be linear normed spaces. is said to be demicontinuous if, for any we have as .

(6)Let be a mapping of a linear normed space into its dual space . is said to be hemicontinuous if it is continuous from each line segment in to the weak topology in .

(7)The mapping with the domain and the range in is said to be pseudocontractive if

(8)The mapping with the domain and the range in is said to be strongly pseudocontractive if there exists a constant such that

Remark 1.1.

For the maximal monotone operator , we can defined the resolvent of by . It is well know that is nonexpansive.

Remark 1.2.

It is well-known that if is demicontinuous, then is hemicontinuous, however, the converse, in general, may not be true. In reflexive Banach spaces, for monotone mappings defined on the whole Banach space, demicontinuity is equivalent to hemicontinuity.

To find zeroes of maximal monotone operators is the central and important topics in nonlinear functional analysis. We observe that is a zero of the monotone mapping if and only if it is a fixed point of the pseudocontractive mapping . Consequently, considerable research works, especially, for the past 40 years or more, have been devoted to the existence and convergence of zero points for monotone mappings or fixed points of pseudocontractions, see, for instance, [1–23].

In 1965, Browder [1] proved the existence result of fixed point for demicontinuous pseudocontractions in Hilbert spaces. To be more precise, he proved the following theorem.

Theorem Bo

Let be a Hilbert space, a nonempty bounded and closed convex subset of and a demicontinuous pseduo-contraction. Then has a fixed point in .

In 1968, Browder [4] proved the existence results of zero points for maximal monotone mappings in reflexive Banach spaces. To be more precise, he proved the following theorem.

Theorem Bt

Let be a reflexive Banach space, a maximal monotone mapping and a bounded, pseudomonotone and coercive mapping. Then, for any , there exists such that , or is all of .

For the existence of continuous paths for continuous pseudocontractions in Banach spaces, Morales and Jung [15] proved the following theorem.

Theorem MJ.

Let be a Banach space. Suppose that is a nonempty closed convex subset of and is a continuous pseudocontraction satisfying the weakly inward condition. Then for each , there exists a unique continuous path , , which satisfies the following equation .

In 2002, Lan and Wu [14] partially improved the result of Morales and Jung [15] from continuous pseudocontractions to demicontinuous pseudocontractions in the framework of Hilbert spaces. To be more precise, they proved the following theorem.

Theorem LW.

Let be a bounded closed convex set in . Assume that is a demicontinuous weakly inward pseudocontractive map. Then has a fixed point in . Moreover; for every , defined by converges to a fixed point of .

In this work, motivated by Browder [3], Lan and Wu [14], Morales and Jung [15], Song and Chen [19], and Zhou [22, 23], we consider the existence of convergence of paths for maximal monotone mappings in the framework of real Hilbert spaces.

Lemma 2.1.

Let be a nonempty closed convex subset of a Hilbert space and a demicontinuous monotone mapping. Then is pseudomonotone.

Proof.

For any sequence which converges weakly to an element in such that

we see from the monotonicity of that

Combining (2.1) with (2.2), we obtain that

By taking , we arrive at

which yields that

Noticing that

we have

Let , for all and . By taking and in (2.7), we see that

Noting that , , , and is demicontinuous, we have as , and hence

This completes the proof.

Lemma 2.2.

Let be a nonempty closed convex subset of a Hilbert space , a maximal monotone mapping, and a bounded, demicontinuous, and strongly monotone mapping. Then has a unique zero in .

Proof.

By using Lemma 2.1 and Theorem B2, we can obtain the desired conclusion easily.

Lemma 2.3.

Let be a nonempty closed convex subset of a Hilbert space , a maximal monotone mapping, and a bounded, demicontinuous strong pseudocontraction with the coefficient . For , consider the equation

where . Then, One has the following.

(i)Equation (2.10) has a unique solution for every .

(ii)If is bounded, then as .

(iii)If , then is bounded and satisfies

where denotes the zero set of .

Proof.

(i) From Lemma 2.2, one can obtain the desired conclusion easily.

(ii) We use to denote the unique solution of (2.10). That is, . It follows that . Notice that

From the boundedness of and , one has .

(iii) For , one obtains that

It follows that

That is, , for all . This shows that is bounded. Noticing that , one arrives at

This completes the proof.

Lemma 2.4.

Let be a nonempty closed convex subset of a Hilbert space and a maximal monotone mapping. Then . If one defines by , for all , then is a nonexpansive mapping with and , where denotes the set of fixed points of .

Proof.

Noticing that is maximal monotone, one has . It follows that . For any , one sees that

which yields that is nonexpansive mapping. Notice that

That is, . On the other hand, for any , we have

This completes the proof.

Set . Let denote the Banach space of all bounded real value functions on with the supremum norm, a subspace of , and an element in , where denotes the dual space of . Denote by the value of at . If , for all , sometimes will be denoted by . When contains constants, a linear functional on is called a mean on if . We also know that if contains constants, then the following are equivalent.

(1).

(2), for all .

To prove our main results, we also need the following lemma.

Lemma 2.5 (see [20, Lemma 4.5.4]).

Let be a nonempty and closed convex subset of a Banach space . Suppose that norm of is uniformly Gâteaux differentiable. Let be a bounded set in and . Let be a mean on . Then

if and only if

Now, we are in a position to prove the main results of this work.

Theorem 2.6.

Let be a Hilbert space and a nonempty closed convex subset of . Let be a maximal monotone mapping and a bounded demicontinuous strong pseudocontraction. Let be as in Lemma 2.3. Then is bounded if and only if converges strongly to a zero point of as which is the unique solution in to the following variational inequality:

Proof.

The part is obvious and we only prove . From Lemma 2.3, one sees that as . It follows from Lemma 2.4 that as . Define , , where is a Banach limit. Then is a convex and continuous function with as . Put

From the convexity and continuity of , we can get the convexity and continuity of the set . Since is continuous and is a Hilbert space, we see that attains its infimum over ; see [20] for more details. Then is nonempty bounded and closed convex subset of . Indeed, contains one point only. Set , where . Notice that is nonexpansive. Since every nonempty bounded and closed convex subset has the fixed point property for nonexpansive self-mapping in the framework of Hilbert spaces, then has a fixed point in , that is, . It follows from Lemma 2.4 that . On the other hand, one has . In view of Lemma 2.5, we obtain that

By taking in (2.23), we arrive at

Combining (2.14) with (2.23) yields that . Hence, there exists a subnet of such that . From (iii) of Lemma 2.3, one has

Taking limit in (2.25), one gets that

If there exists another subset of such that , then is also a zero of . It follows from (2.26) that

By using (iii) of Lemma 2.3 again, one arrives at

Taking limit in (2.28), we obtain that

Adding (2.27) and (2.29), we have

which yields that

It follows that . That is, converges strongly to , which is the unique solution to the following variational inequality:

Remark 2.7.

From Theorem 2.6, we can obtain the following interesting fixed point theorem. The composition of bounded, demicontinuous, and strong pseudocontractions with the metric projection has a unique fixed point. That is, .

The third author was supported by the National Natural Science Foundation of China (Grant no. 10771050).

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