Finding Minimum Norm Fixed Point of Nonexpansive Mappings and Applications

Iterative algorithms for finding fixed point of nonexpansive mappings are very interesting topic due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonexpansive mappings. Related works can be found in 1–32 . On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. In an abstract way, we may formulate such problems as finding a point x† with the property


Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping T : 1.1 Iterative algorithms for finding fixed point of nonexpansive mappings are very interesting topic due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonexpansive mappings.Related works can be found in 1-32 .
On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution.In an abstract way, we may formulate such problems as finding a point x † with the property where C is a nonempty closed convex subset of a real Hilbert space H.In other words, x † is the nearest point or metric projection of the origin onto C, x † P C 0 , The so-called C-constrained pseudoinverse of A is then defined as the operator A † C with domain and values given by where x † ∈ S b is the unique solution to 1.6 .Note that the optimality condition for the minimization 1.5 is the variational inequality VI 5 is consistent and its solution set S b coincides with the solution set of VI 1.8 .On the other hand, VI 1.8 can be rewritten as where λ > 0 is any positive scalar.In the terminology of projections, 1.10 is equivalent to the fixed point equation It is not hard to find that for 0 < λ < 2/ A 2 , the mapping x → P C x − λA * Ax − b is nonexpansive.Therefore, finding the least-squares solution of the constrained linear inverse problem 1.6 is equivalent to finding the minimum-norm fixed point of the nonexpansive mapping x → P C x − λA * Ax − b .Motivated by the above least-squares solution to constrained linear inverse problems, we will study the general case of finding the minimum-norm fixed point of a nonexpansive mapping T : C → C: where Fix T {x ∈ C : Tx x} denotes the set of fixed points of T throughout we always assume that Fix T / ∅ .
We next briefly review two historic approaches which relate to the minimum-norm fixed point problem 1.11 .
Browder 1 introduced an implicit scheme as follows.Fix a u ∈ C, and for each t ∈ 0, 1 , let x t be the unique fixed point in C of the contraction T t which maps C into C: Browder proved that s − lim t↓0 x t P Fix T u.

1.13
That is, the strong limit of {x t } as t → 0 is the fixed point of T which is nearest from Fix T to u. Halpern 4 , on the other hand, introduced an explicit scheme.Again fix a u ∈ C. Then with a sequence {t n } in 0, 1 and an arbitrary initial guess x 0 ∈ C, we can define a sequence {x n } through the recursive formula 1.14 It is now known that this sequence {x n } converges in norm to the same limit P Fix T u as Browder's implicit scheme 1.12 if the sequence {t n } satisfies, assumptions A 1 , A 2 , and A 3 as follows: Some more progress on the investigation of the implicit and explicit schemes 1.12 and 1.14 can be found in 33-42 .We notice that the above two methods do find the minimum-norm fixed point x † of T if 0 ∈ C.However, if 0 / ∈ C, then neither Browder's nor Halpern's method works to find the minimum-norm element x † .The reason is simple: if 0 / ∈ C, we cannot take u 0 either in 1.12 or 1.14 since the contraction x → 1 − t Tx is no longer a self-mapping of C hence may fail to have a fixed point , or 1 − t n Tx n may not belong to C, and consequently, x n 1 may be undefined.In order to overcome the difficulties caused by possible exclusion of the origin from C, we introduce the following two remedies.
For Browder's method, we consider the contraction x → 1 − β P C 1 − t x βTx for some β ∈ 0, 1 .Since this contraction clearly maps C into C, it has a unique fixed point which is still denoted by x t , that is, x t 1 − β P C 1 − t x t βTx t .For Halpern's method, we consider the following iterative algorithm It is easily seen that the net {x t } and the sequence {x n } are well defined i.e., x t ∈ C and x n ∈ C .
The purpose of this paper is to prove that the above both implicit and explicit methods converge strongly to the minimum-norm fixed point x † of the nonexpansive mapping T. Some applications are also included.

Preliminaries
Let H be a real Hilbert space with inner product •, • and norm • , respectively.Let C be a nonempty closed convex subset of H. Recall that the nearest point or metric projection from H onto C is defined as follows: for each point x ∈ H, P C x is the unique point in C with the property Note that P C is characterized by the inequality Consequently, P C is nonexpansive.Below is the so-called demiclosedness principle for nonexpansive mappings.
We use the following notation: i Fix T stands for the set of fixed points of T; ii x n x stands for the weak convergence of x n to x; iii x n → x stands for the strong convergence of x n to x.

Main Results
The aim of this section is to introduce some methods for finding the minimum-norm fixed point of a nonexpansive mapping T. First, we prove the following theorem by using an implicit method.
Theorem 3.1.Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a nonexpansive mapping with Fix T / ∅.For β ∈ 0, 1 and each t ∈ 0, 1 , let x t be defined as the unique solution of fixed point equation

3.1
Then the net {x t } converges in norm, as t → 0 , to the minimum-norm fixed point of T.
Proof.First observe that, for each t ∈ 0, 1 , x t is well defined.Indeed, we define a mapping For x, y ∈ C, we have which implies that S t is a self-contraction of C. Hence S t has a unique fixed point x t ∈ C which is the unique solution of the fixed point equation 3.1 .
Next we prove that {x t } is bounded.Take u ∈ Fix T .From 3.1 , we have that is, Hence, {x t } is bounded and so is {Tx t }.
From 3.1 , we have 3.6 that is, Next we show that {x t } is relatively norm-compact as t → 0 .Let {t n } ⊂ 0, 1 be a sequence such that t n → 0 as n → ∞.Put x n : x t n .From 3.7 , we have x n − Tx n −→ 0.

3.8
Again from 3.1 , we get

3.9
It turns out that where M > 0 is some constant such that sup{ 1/2 x t 2 : t ∈ 0, 1 } ≤ M. In particular, we get from 3.10

3.11
Since {x n } is bounded, without loss of generality, we may assume that {x n } converges weakly to a point x * ∈ C. Noticing 3.8 we can use Lemma 2.1 to get x * ∈ Fix T .Therefore we can substitute x * for u in 3.11 to get However, x n x * .This together with 3.12 guarantees that x n → x * .The net {x t } is therefore relatively compact, as t → 0 , in the norm topology.Now we return to 3.11 and take the limit as n → ∞ to get

3.13
This is equivalent to Therefore, x * P Fix T 0. This is sufficient to conclude that the entire net {x t } converges in norm to x * and x * is the minimum-norm fixed point of T. This completes the proof.
Next, we introduce an explicit algorithm for finding the minimum norm fixed point of nonexpansive mappings.Theorem 3.2.Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a nonexpansive mapping with Fix T / ∅.For given x 0 ∈ C, define a sequence {x n } iteratively by where β ∈ 0, 1 and α n ∈ 0, 1 satisfying the following conditions: Then the sequence {x n } converges strongly to the minimum-norm fixed point of T.
Proof.First we prove that the sequence {x n } is bounded.Pick p ∈ Fix T .Then, we have

3.17
Next, we estimate x n 1 − x n .From 3.15 , we have

3.18
This together with Lemma 2.2 implies that lim Note that

3.20
Thus, We next show that lim sup where x P Fix T 0, the minimum norm fixed point of T. To see this, we can take a subsequence

3.24
Now since x * ∈ Fix T this is a consequence of Lemma 2.2 and 3.21 , we get by combining 3.22 and 3.23 Finally, we show that x n → x.As a matter of fact, we have

3.26
By C1 and 3.22 , it is easily found that lim n → ∞ δ n 0 and lim sup n → ∞ θ n ≤ 0. We can therefore apply Lemma 2.2 to 3.26 and conclude that x n 1 → x as n → ∞.This completes the proof.

Applications
We consider the following minimization problem where γ > 0 is any positive number.Note that the solution set S of 4.1 coincides with the set of fixed points of T γ for any γ > 0 .
If the gradient ∇ϕ is L-Lipschitzian continuous on C, then it is not hard to see that the mapping T γ is nonexpansive if 0 < γ < 2/L.Using Theorems 3.1 and 3.2, we immediately obtain the following result.The K-constrained pseudoinverse of A, A † K , is defined as where x † ∈ S b is the unique solution to 4.9 .i For each t ∈ 0, 1 , let x t be the unique solution of the fixed point equation

1 . 3 where 5 Let
P C is the metric or nearest point projection from H onto C. A typical example is the least-squares solution to the constrained linear inverse problem Ax b, x ∈ C, 1.4 where A is a bounded linear operator from H to another real Hilbert space H 1 and b is a given point in H 1 .The least-squares solution to 1.4 is the least-norm minimizer of the minimization problem min x∈C Ax − b 2 .1.S b denote the closed convex solution set of 1.4 or equivalently 1.5 .It is known that S b is nonempty if and only if P A C b ∈ A C .In this case, S b has a unique element with minimum norm equivalently, 1.4 has a unique least-squares solution ; that is, there exists a unique point x † ∈ S b satisfying x † min{ x : x ∈ S b }. 1.6

whereAssume 3 Here
C is a closed convex subset of a real Hilbert space H and ϕ : C → Ê is a continuously Fréchet differentiable convex function.Denote by S the solution set of 4.1 ; that is, S ∅.It is known that a point z ∈ C is a solution of 4.1 if and only if the following optimality condition holds: z ∈ C, ∇ϕ z , x − z ≥ 0, x ∈ C. 4.∇ϕ x denotes the gradient of ϕ at x ∈ C. It is also known that the optimality condition 4.3 is equivalent to the following fixed point problem, z T γ z, T γ P C I − γ ∇ϕ , 4.4

13 Then
x t βP K x t − γA * Ax t − b 1 − β P K 1 − t x t .4.{x t } converges in norm as t → 0 to A † K b .ii Define a sequence {x n } via the recursive algorithm x n 1 βP K x n − γA * Ax n − b 1 − β P K 1 − α n x n , 4.14where the sequence {α n } satisfies conditions C1 -C2 in Theorem 3.2.Then {x n } converges in norm to A † K b .
Theorem 4.1.Assume ϕ is continuously (Fréchet) differentiable and convex and its gradient ∇ϕ is L-Lipschitzian.Assume the solution set S of the minimization 4.1 is nonempty.Fix γ such that 0 < γ < 2/L.Then {x t } converges in norm as t → 0 to the minimum-norm solution of the minimization 4.1 .iiDefine a sequence {x n } via the recursive algorithmx n 1 βP C I − γ ∇ϕ x n 1 − β P C 1 − α n xn , 4.6 where the sequence {α n } satisfies conditions C1 -C2 in Theorem 3.2.Then {x n } converges in norm to the minimum-norm solution of the minimization 4.1 .Let S b denote the solution set of 4.8 .It is always closed convex but possibly empty .It is known that S b is nonempty if and only if P A K b ∈ A K .In this case, S b has a unique element with minimum norm; that is, there exists a unique point x † ∈ S b satisfying x † min{ x : x ∈ S b }. 4.9 Ax − b , x ∈ H 1 , 4.12where A * is the adjoint of A. Clearly ∇ϕ is Lipschitzian with constant L A * A A 2 .Therefore, applying Theorem 4.1, we obtain the following result. *