A fixed point theorem for non-self multi-maps in metric spaces

A fixed point theorem for non-self multi-maps in metric spaces

B.C. Dhage

Abstract. A fixed point theorem is proved for non-self multi-valued mappings in a met- rically convex complete metric space satisfying a slightly stronger contraction condi- tion than in Rhoades [3] and under a weaker boundary condition than in Itoh [2] and Rhoades [3].

Keywords: metrically convex metric space, multi-valued non-self map, fixed point Classification: 47H10, 54H25

Let (X, d) be a metric space. ThenXis said to be metrically convex if for every pairx, y∈X,x6=y, there is a pointz∈X such thatd(x, y) =d(x, z) +d(z, y).

We need the following lemma in the sequel.

Lemma 1([1]). LetK be a non-empty and closed subset of a metrically convex metric spaceX. Then for anyx∈ K and y /∈ K, there exists a point z ∈ ∂K such thatd(x, y) =d(x, z) +d(z, y), where ∂Kdenotes the boundary of K.

Let CB(X) denote the family of all non-empty, closed and bounded subsets ofX. Denote forA, B∈CB(X)

D(A, B) = inf{d(a, b)|a∈A, b∈B} and

δ(A, B) = sup{d(a, b)|a∈A, b∈B}.

Note thatD(A, B)≤H(A, B)≤δ(A, B), whereH(A, B) denotes the Hausdorff distance ofAandB.

In [2] Itoh proved a fixed point theorem for the non-self mapsF :K→CB(X) satisfying certain contraction condition in terms of Hausdorff metricHonCB(X) under the boundary condition F(∂K) ⊂ K. Recently Rhoades [3] generalized this result to a wider class of non-self multi-maps onK. In this paper we prove a fixed point theorem for non-self multi-maps onK satisfying a slightly stronger contraction condition than that in Rhoades [3] and under a weaker boundary condition.

Theorem 1. Let(X, d) be a metrically convex complete metric space andK a non-empty closed subset of X. LetF :K→CB(X)be a multi-map satisfying (1) δ(F x, F y)≤αmax{d(x, y), D(x, F x), D(y, F y)}+β[D(x, F y) +D(y, F x)]

for allx, y∈K, whereα≥0,β≥0 satisfy

(2) 2α+ 3β <1.

Further, if F x∩K6=∅for eachx∈∂K, thenF has a unique fixed pointp∈K such thatF p={p}and F is continuous atpin the Hausdorff metric onX.

Proof: Letx∈Kbe arbitrary and consider a sequence{x_n}in Kas follows:

Letx0 =xand take a pointx1∈F x0∩K ifF x0∩K 6=∅. Otherwise choose a pointx₁ ∈∂Ksuch that

d(x₀, x^′₁) =d(x₀, x₁) +d(x₁, x^′₁) for somex^′₁∈F x₀ ⊂X\K.

Similarly pickx₂ ∈F x₁∩Kif F x₁∩K6=∅, otherwise choose a pointx₂ ∈∂K such that

d(x₁, x₂) +d(x₂, x^′₂) =d(x₁, x^′₂) for somex^′₂∈F x1 ⊂X\K.

Continuing in this way we have

x_n+1∈F xn∩K if F xn∩K6=∅, orx_n+1 ∈∂Ksatisfying

d(xn, x_n+1) +d(x_n+1, x^′_n+1) =d(xn, x^′_n+1) for somex^′_n+1 ∈F xn⊂X\K.

By the construction of{xn}, we can write

{xn}=P∪Q⊂K, where

P ={xn∈ {xn}:xn∈F x_n−1} and

Q={xn∈ {xn}:xn∈∂K, xn∈/F xn−1}.

Then for any two consecutive termsxn,x_n+1of the sequence{x_n}, we observe that there are only the following three possibilities:

(i) xn, xn+1∈P,

(ii) xn∈P,x_n+1∈Q, and (iii) xn∈Qandx_n+1∈P.

First we show that{xn} is a Cauchy sequence inK. Now for anyxn, x_n+1∈ {xn}, we have the following estimates:

Case I. Suppose thatxn, x_n+1∈P, then we have d(xn, x_n+1)≤δ(F xn−1, F xn)

≤αmax{d(xn−1, xn), D(xn−1, F xn−1), D(xn, F xn)}

+β[D(x_n−1, F xn) +D(xn, F x_n−1)]

≤αmax{d(x_n−1, xn), d(xn, x_n+1)}

+β[d(xn−1, xn+1) +d(xn, xn)]

=αmax{d(x_n−1, xn), d(xn, x_n+1)}+βd(xn−1, x_n+1)

≤αmax{d(xn−1, xn), d(xn, x_n+1)}

+β[d(xn−1, xn) +d(xn, x_n+1)]

= max{(α+β)d(x_n−1, xn) +βd(xn, x_n+1), (α+β)d(xn, x_n+1) +βd(x_n−1, xn)}

and hence

d(xn, x_n+1)≤kd(xn−1, xn), wherek= max{^α+β_1−β,₁ ^β

−(α+β)}<1, since 2α+ 3β <1.

Case II. Letxn∈P andx_n+1∈Q. Then

d(xn, xn+1) +d(xn+1, x^′_n+1) =d(xn, x^′_n+1) for somex^′_n+1 ∈F xn. Clearly,

(3)

d(xn, xn+1)≤d(xn, x^′_n+1) d(xn, x^′_n+1)≤δ(F x_n−1, F xn).

Now following arguments similar to those in Case I, we obtain (4) d(xn, x^′_n+1)≤kd(xn−1, xn),

where againk= max{^α+β_1−β,₁ ^β

−(α+β)}<1.

From (3) and (4) it follows that

(5) d(xn, x_n+1)≤kd(x_n−1, xn).

Case III. Suppose that xn ∈Q andx_n+1 ∈P. Note that then xn−1 ∈P and there is a pointx^′_n∈F xn−1 such that

(6) d(xn−1, xn) +d(xn, x^′_n) =d(xn−1, x^′_n).

Now,

d(xn, x_n+1)≤d(xn, x^′_n) +d(x^′_n, x_n+1)

≤d(xn, x^′_n) +δ(F x_n−1, F xn)

≤d(xn, x^′_n) +αmax{d(xn−1, xn), D(xn−1, F xn−1), D(xn, F xn)}

+β[D(xn−1, F xn) +D(xn, F xn−1)]

≤d(xn, x^′_n) +αmax{d(xn−1, xn), d(xn−1, x^′_n), d(xn, x_n+1)}

+β[d(xn−1, xn+1) +d(xn, x^′_n)]

≤d(xn, x^′_n) +αmax{d(x_n−1, x^′_n), d(xn, x_n+1)}

+β[d(x_n−1, xn) +d(xn, x_n+1) +d(xn, x^′_n)]

=d(xn, x^′_n) +αmax{d(xn−1, x^′_n), d(xn, x_n+1)}

+β[d(xn−1, x^′_n) +d(xn, x_n+1)]

≤d(x_n−1, x^′_n) +αmax{d(x_n−1, x^′_n), d(xn, x_n+1)}

+β[d(x_n−1, x^′_n) +d(xn, x_n+1)].

From (4) of Case II applied ton−1, we haved(xn−1, x^′_n)≤kd(xn−2, xn−1) and hence

d(xn, x_n+1)≤kd(x_n−2, x_n−1) + max{kd(x_n−2, x_n−1), d(xn, x_n+1)}

+β[kd(x_n−2, x_n+1) +k(xn, x_n+1)]

= max{(1 +α+β)kd(x_n−2, x_n−1) +βd(xn, x_n+1), (1 +β)kd(xn−2, xn−1) + (α+β)d(xn, xn+1)}.

This implies

d(xn, xn+1)≤max{(1 +α+β)k/(1−β),(1 +β)k/[1−(α+β)]}d(x_n−2, xn−1)

=qd(xn−2, xn−1), where

q= max{(1 +α+β)k/(1−β),(1 +β)k/[1−(α+β)]}

=kmax{(1 +α+β)/(1−β),(1 +β)/[1−(α+β)]}=k(1 +β)/[1−(α+β)]

= (1 +β)/[1−(α+β)] max{(α+β)/(1−β), β/[1−(α+β)]}

= max{(1 +β)(α+β)/[(1−β)(1−(α+β))], β(1 +β)/[1−(α+β)]²}

<1.

To see this, the inequality (2) yields α+β <1−2β−α

⇒α+β+αβ+β² <1−2β−α+αβ+β²

⇒(α+β+αβ+β²)/(1−2β−α+αβ+β²)<1

⇒(1 +β)(α+β)/[(1−β)(1−α−β)]}<1.

Similarly again from (2) we have

2α+ 3β < α²+ 2αβ+ 1

⇒β+β²<1−2α−2β+α²+ 2αβ+β²

⇒β(1 +β)<1−2(α+β) + (α+β)²

⇒β(1 +β)<[1−(α+β)]²

⇒β(1 +β)/[1−(α+β)]² <1.

Now for anyn∈N, we have

(7) d(x_2n, x_2n+1)≤qd(x_2n−2, x_2n−1)≤qⁿd(x₀, x₁).

Sincenis arbitrary, one has

(8) d(xn, x_n+1)≤qⁿd(x₀, x₁).

Then from Cases I–III, it easily follows that{x_n} is a Cauchy sequence inK. As Kis closed it is complete and hence limnxn=pexists. We show thatpis a fixed point ofF. Without loss of generality we may assume thatxn+1∈F xnfor some n∈N. Then

D(p, F p) = lim

n D(x_n+1, F p)

≤ lim

n δ(F xn, F p)

≤ lim

n max{d(xn, p), D(xn, F xn), D(p, F p)}

+βlim

n [D(xn, F p) +D(p, F xn)]

=αlim

n max{d(x_n, p), d(xn, xn+1), D(p, F p)}

+βlim

n [D(xn, F p) +d(p, x_n+1)]

= (α+β)D(p, F p) which is possible only whenp∈F p.

Further, we have δ(p, F p)≤δ(F p, F p)

≤αmax{d(p, p), D(p, F p), D(p, F p)}+β[δ(p, F p) +D(p, F p)]

=βδ(p, F p) and henceF p={p}.

To show the uniqueness ofp, letq(6=p) be another fixed point of F. Then d(p, q)≤δ(F p, F q)

≤αmax{d(p, q), D(p, F p), D(q, F q)}+β[D(p, F q) +D(q, F p)]

= (α+ 2β)d(p, q).

This is a contradiction sinceα+ 2β <1 and hencep=q.

Finally we prove the continuity ofF atp. Let{z_n} ⊂X by any sequence such thatzn→pas n→ ∞. Now

limn H(F zn, F)≤ lim

n δ(F zn, F p)

≤αlim

n max{d(zn, p), D(zn, F zn), D(p, F p)}

+βlim

n [D(zn, F p) +D(p, F zn)]

≤αlim

n max{d(z_n, p), D(zn, F zn)}

+βlim

n [d(zn, p) +D(p, F zn)]

= (α+β)H(F zn, F p)

whereα+β <1. Therefore limnH(F zn, F p) = 0, showing thatF is continuous

atp. This completes the proof.

The following fixed point theorem for non-self multi-maps on a complete con- vex metric space satisfying a slightly weaker contraction condition and under a stronger boundary condition than ours has been proved by Rhoades [3].

Theorem 2([3]). Let(X, d)be a metrically convex metric space andKa non- empty closed subset of X.

LetF:K→CB(X)satisfy

(9) H(F x, F y)≤αd(x, y) +βmax{D(x, F x), D(y, F y)}

+γ[D(x, F y) +D(y, F x)]

for allx, y∈X where α, β, γ≥0 such that (10)

1 +α+γ 1−β−γ

α+β+γ 1−γ

<1.

Further if F x ⊂ K for each x ∈ ∂K, then there exists ap ∈ K such that p∈F pandF is upper semi-continuous atp.

Proof: The existence of such a fixed point p ∈ K follows from Theorem 1 of Rhoades [3]. We only show the upper semi-continuity ofF atp.

Let{zn} ⊂K be any sequence such thatzn→pas n→ ∞.

Let{yn}be a sequence in Ksuch thatyn∈F xnfor eachn∈N andyn→q.

To finish, we shall prove thatq∈F p. Now d(q, p) = lim

n d(yn, p)≤lim

n H(F zn, F p)

= lim

n d(zn, p) +βlim

n max{D(z_n, F zn), D(p, F p)}

+γlim

n [D(zn, F p) +D(p, F zn)]

=βlim

n max{d(zn, yn),0}+γlim

n d(p, yn)

=βd(p, q) +γd(p, q) = (β+γ)d(p, q)

which is possible only whend(q, p) = 0 asβ+γ <1. Henceq∈F pand the proof

ot the theorem is complete.

Next we prove two fixed point theorems for multi-maps on a metric space satis- fying a contractive condition more general than (1) and under certain compactness type conditions.

Theorem 3. Let(X, d) be a complete metrically convex metric space andK a non-empty compact subset of X. Suppose thatF :K→CB(X)is a continuous multi-map satisfying

(11) δ(F x, F y)< αmax{d(x, y), D(x, F x), D(y, F y)}+β[D(x, F y) +D(y, F x)]

for all x, y ∈ K, x /∈ F x, y /∈ F y, where α, β > 0 satisfy 2α+ 3β ≤ 1. If F x∩K6=∅ for eachx∈∂K then the multi-mapF has a unique fixed point.

Proof: First we note that if the multi-mapF has a fixed point then from con- dition (11) it follows that the fixed point is unique.

SinceKis compact, both sides of the inequality (11) are bounded onK. Now there are two possibilities:

Case I. Suppose that the right hand side of (11) is zero for some (x, y)∈K×K, then we havex=y∈F y. ThusF has a fixed point and so it is unique.

Case II. Suppose that the right hand side of (11) is positive for all x, y ∈ K.

Denote for brevity

M(x, y) =αmax{d(x, y), D(x, F x), D(y, F y)}+β[D(x, F y) +D(y, F x)].

Now in the case when 2α+ 3β <1, the conclusion of Theorem 3 follows from Theorem 1. Therefore we treat only the case when 2α+ 3β= 1.

Define a functionT :K²→R⁺ by

(12) T(x, y) =δ(F x, y)

M(x, y) .

Clearly the functionT is well defined sinceM(x, y)6= 0 for allx, y∈K.

SinceF, D andδ are continuous, T is continuous and from the compactness ofKit follows that there is a point (u, v)∈K² such thatT attains its maximum at this point. Call the value c. From (11) we get 0 < c <1. By the definition ofT, we obtain

δ(F x, F y)≤cM(x, y)

=α^′max{d(x, y), D(x, F x), D(y, F y)}+β^′[D(x, F y) +D(y, F x)]

for allx, y ∈K, where 2α^′+ 3β^′=c(2α+ 3β)<1. As Kis compact, it is closed and so the desired conclusion follows by an application of Theorem 1. The proof

is complete.

Theorem 4. Let(X, d) be a complete metrically convex metric space andK a compact subset of X. Suppose thatF :K→CB(X)is a continuous multi-map satisfying

(13) H(F x, F y)< αd(x, y) +βmax{D(x, F x), D(y, F y)}

+γ[D(x, F y) +D(y, F x)]

for allx, y∈X,x /∈F x,y /∈F y, whereα, β, γ >0satisfy(^1+α+γ_1−β−γ) (^α+β+γ_1−γ )≤1.

If F x⊂K for eachx∈∂K then the multi-mapF has a fixed point.

Proof: The proof is similar to Theorem 3 and now the desired conclusion follows

by an application of Theorem 2.

Acknowledgment. The author expresses sincere thanks to the learned referee for his careful reading and suggesting some improvement upon the original version of this paper. He is also thankful to Prof. B.E. Rhoades for providing the reprints of his papers.

References

[1] Blumenthal L.M.,Theory and Applications of Distance Geometry, Clarendon Press, Ox- ford, 1943.

[2] Itoh S.,Multi-valued generalized contraction and fixed point theorems, Comment. Math.

Univ. Carolinae18(1977), 247–248.

[3] Rhoades B.E.,A fixed point theorem for a multi-valued non-self mappings, Comment. Math.

Univ. Carolinae37(1996), 401–404.

Mathematics Research Centre, Mahatma Gandhi Mahavidyalaya, Ahmedpur 413 515 (Maharashtra), India

(Received June 9, 1997,revised September 4, 1998)