Convergence of approximating fixed points sets for multivalued nonexpansive mappings

Convergence of approximating fixed points sets for multivalued nonexpansive mappings

Paolamaria Pietramala

Abstract. LetKbe a closed convex subset of a Hilbert spaceH andT :K⊸Ka non- expansive multivalued map with a unique fixed pointzsuch that{z}=T(z). It is shown that we can construct a sequence of approximating fixed points sets converging in the sense of Mosco toz.

Keywords: multivalued nonexpansive map, fixed points set, Mosco convergence Classification: 47H09, 47H10

Let H be a Hilbert space, K a closed convex subset of H, T a multivalued nonexpansive map fromK in the family of non empty compact subsets ofK. It is our object in this paper to show that in a specific case it is possible to construct a sequence of approximant sets converging in the sense of Mosco to a fixed point ofT.

Our investigation is prompted by the papers of Browder [1], Reich [2], Singh and Watson [3], in which analogous problems are treated for singlevalued mappings.

In particular, in [1] it is shown that: if K is a closed convex bounded subset of a Hilbert space andT :K →K is a nonexpansive map, then, for any x₀ ∈K, the sequence {x_λ}_0≤λ<1 of the fixed points of the contraction maps T_λ,x0 defined by T_λ,x0(x) =λT(x) + (1−λ)x₀ converges, as x approaches 1, strongly in K to the fixed point ofT in K closest to x₀. The paper [3] extends this result to the case of not self-mappings (butT(∂K)⊆K) andK not necessarily bounded (but T(K) bounded).

The following example of multivalued self-map defined on a closed convex bound- ed subset of a finite-dimensional Hilbert space shows that the recalled results cannot be extended to genuine multivalued case.

LetH=R²,K= [0,1]×[0,1] andT the nonexpansive map defined by:

T(a, b) = triangle whose vertices are (0,0),(a,0),(0, b), ∀(a, b)∈K.

Thus, for (x0, y₀) ∈ K the point ((1−λ)x0,(1−λ)y0) is a fixed point of the map T_λ,(x0,y0) for all λ∈ [0,1) and we have ((1−λ)x0,(1−λ)y0) →(0,0) as λ approaches 1. If x₀ > y₀ (x₀ < y₀), then the fixed point ofT closest to (x₀, y₀) is (x₀,0) ((0, y₀)), but the net of the fixed points sets ofT_λ,(x0,y0) does not converge to (x₀,0) ((0, y₀)) even in the weaker convergence of sets, that is, the Kuratowski convergence.

In the setting of Hilbert spaces, our result is formulated for nonexpansive maps T that have a unique fixed point zand this point satisfies{z}=T(z). The precise generality of the class of functions satisfying this condition is not known but it has been studied, for example, in [4], [5], [6]. More recently the interest in optimiza- tion theory for such type of maps has prompted a corresponding interest in fixed point theory, since in [7] it has been shown that the maximization of a multivalued mapT with respect to a cone, which subsumes ordinary and Pareto optimization, is equivalent to a fixed point problem of determiningy such that{y}=T(y).

Now we introduce some necessary notations and definitions. LetK be a closed convex subset of a Hilbert spaceH. We denote byCB(H) the family of non empty closed bounded subsets ofHand byK(K) the family of non empty compact subsets ofK.

ForA∈ CB(H) we define

d(x, A) = inf{kx−yk:y∈A}.

For anyA, B∈ CB(H) we note withD(A, B) theHausdorff distance induced by the norm ofH, i.e.

D(A, B) = max{sup

a∈Ad(a, B), sup

b∈B

d(b, A)}.

Remark. IfB={b}andA∈ CB(H), we have that for alla∈A ka−bk ≤D(A, B).

We denote by→and⇀the strong and weak convergence, respectively.

Let {An} be a sequence of closed subsets of H. We define the inner limit (lim infAn) by

lim infAn={x∈H :∃ a sequence {xn}, xn∈An such that xn→x}

and theweak-outer limit(w−lim supAn) by

w−lim supAn={x∈H :∃ a subsequence {An_k} of {An} and a sequence {xnk}, xnk ∈A_nk such that x_nk⇀ x}.

We will say that {An} converges to A in the sense of Mosco (An

−−−→(M) A) if lim infAn=w−lim supAn=A.

A net {Aλ}_λ∈[0,1) of closed subsets of H converges to A in the sense defined before if every sequence{A_λn},λn→1 asn→ ∞, converges in such sense toA.

A multivalued mapT :K→ K(K) is said to be lipschitzianif D(T(x), T(y))≤Lkx−yk

for everyx, y ∈K,L≤0. T is said to be acontractionifL <1 andnonexpansive ifL= 1. A map T : K → K(K) is said to be demiclosed ifxn ⇀ x, yn→y and yn∈T(xn) implyy∈T(x).

LetT :K→ K(K) be a nonexpansive map. Forx₀∈Kandλ∈[0,1) we denote byT_λ,x0 the contraction map defined by

T_λ,x0(x) =λT(x) + (1−λ)x₀, ∀x∈K.

Finally, we denote by

F(T) ={x∈K:x∈T(X)}

and

F(T_λ,x0) ={x∈K;x∈T_λ,x0(x)}

the sets of fixed points ofT andT_λ,x0, respectively.

Theorem 1. LetKbe a closed convex subset of a Hilbert spaceH,T :K→ K(K) a nonexpansive map such thatF(T) ={z}, and let this pointzsatisfyT(z) ={z}.

Then, for everyx₀∈K,

F(T_λ,x0)−−−→^(M) F(T) as λ→1.

Proof: We have to prove that F(Tλn,x0)−−−→ {z}^(M) asn→ ∞for every sequence λn→1, 0≤λn<1.

Since we have always lim inf ⊆ w−lim sup, it remains to prove that w− lim supF(T_λn,x0)⊆ {z}and{z} ⊆lim infF(T_λn,x0).

Step 1.w−lim supF(Tλn,x0)⊂ {z}.

Letx∈w−lim supF(T_λn,x0), then there exist a subsequence{λnj}of{λn}and a sequence{y_λnj},y_λnj ∈F(T_λnj,x0) such thaty_λnj ⇀ x.

Sincey_λnj ∈F(T_λnj,x0), there existsw_λnj ∈T_λnj,x0(y_λnj) such that y_λnj =λnjw_λnj + (1−λnj)x₀.

Thus

ky_λnj −w_λnjk= (1−λnj)kw_λnj0x₀k →0 as λnj →1

because {w_λnj} is bounded. From the demiclosedness of I−T [8] it follows that 0∈(I−T)(x), hence x=z.

Step 2.{z} ⊆lim infF(T_λn,x0).

Letx_λn ∈F(T_λn,x0). We prove that x_λn →z. On the contrary, suppose that there existsε₀ >0 and a subsequence {λn_j} of{λn} such that

(1) kx_λnj −zk ≥ε₀.

Fromx_λnj =λn_jw_λnj + (1−λn_j)x0, w_λnj ∈T(xλ_nj) it follows that

x_λnj −(1−λnj)x₀ λn_j −z

=

w_λnj −z .

Furthermore, from the previous Remark, we have kw_λnj −zk ≤D(T(x_λnj), T(z)) and the nonexpansivity ofT yields

kw_λnj −zk ≤ kx_λnj −zk.

Hence

x_λnj −x₀ λnj

−(z−x₀)

2≤

(x_λnj −x₀) + (x₀−x)

2,

which implies

(2)

kxλ_nj −x₀k²≤2 λ_nj

1 +λ_njhxλ_nj −x₀, z−x₀i

≤ hx_λnj −x₀, z−x₀i

≤ kx_λnj −x₀k kz−x₀k.

If it werex_λ

nj =x₀ for a certainj, we should have x₀=λ_nj, w_λ

nj +x₀−λ_njx₀

=w_λnj ∈T(x_λnj)

=T(x₀).

Thenx₀ =z, contradicting (1).

Thus, from (2) it follows

kx_λnj −x₀k ≤ kz−x₀k,

which implies that the subsequence {xλ_nj} is bounded. Hence there exists a sub- sequence {x_λnj

k

} of {x_λnj} such that x_λnj

k

⇀ x. Proceeding as in the proof of Step 1, we obtainx=z. At this point, the well known relation

kz−x₀k ≤lim infkx_λnj

k −x₀k

and (see (2))

lim supkx_λnj

k

−x₀k ≤ kz−x₀k imply

kx_λnj

k

−x₀k → kz−x₀k.

Hence, we havex_λnj

k →z, contradicting (1).

Remark. In the following example, our theorem works.

LetH =R, K = [0,∞), T : K → K(K) be the nonexpansive map defined by T(x) = [0,^x₂]. Thus F(T) = {0}, {0} =T(0) and the net of fixed point sets of T_λ,x0, F(Tλ,x0) = [(1−λ)x0,2(1−λ)x0], converges toF(T) in the sense of Mosco.

References

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[2] Reich S.,Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl.75(1980), 287–292.

[3] Singh S.P., Watson B.,On approximating fixed points, Proc. Symp. Pure Math.45(part 2) (1986), 393–395.

[4] Reich S.,Fixed points of contractive functions, Boll. UMI5(1972), 26–42.

[5] ´Ciri´c L.B.,Fixed points for generalized multivalued contractions, Mat. Vesnik, N. Ser.9(24) (1972), 265–272.

[6] Is´eki K.,Multivalued contraction mappings in complete metric spaces, Math. Sem. Notes 2 (1974), 45–49.

[7] Corley H.W.,Some hybrid fixed point theorems related to optimization, J. Math. Anal. Appl.

120(1986), 528–532.

[8] LamiDozo E.,Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math.

Soc.38(1973), 286–292.

Universit´a della Calabria, Dipartimento di Matematica, Arcavacata di Rende, Cosenza, Italy

(Received November 5, 1990)