Fixed-point theory for closed inward set valued maps

MATI-IJXMATICAL COMPUTER MODELLING PERGAMON

Mathematical

and Computer Modelling 32 (2000) 1305-1310 www.elsevier.nl/locate/mcm

Fixed-Point Theory for Closed Inward Set Valued Maps R. P. AGARWAL Department of Mathematics, National University of Singapore Kent Ridge, Singapore 0511

D. O’REGAN Department of Mathematics, National University of Ireland Galway, Ireland Abstract-A new results of Browder-Fan type is presented for inward, contractive, closed maps. Also a nonlinear alternative of Leray-Schauder type for such maps is given. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Fixed-point

theorems,Inwardset valued maps, Measureof noncompactness.

1. INTRODUCTION The paper presents fixed-point results for closed, contractive, inward maps defined on FrCchet spaces. In the literature [l-5], upper semicontinuous (u.s.c.), contractive, inward maps were discussed. However, it was noted recently [6] that if we examine differential and integral inclusions in abstract spaces (with the dimension of the space infinite), the maps that arise are closed and contractive (of course if the maps were closed and compact, then they would automatically be U.S.C. (see [7, p. 4651)). It is of interest therefore from an application viewpoint to establish general fixed-point theorems for closed, contractive inward maps. Recently, some papers [8-lo] have appeared discussing particular cases of the maps described above. For example in [S,lO], closed, contractive (not necessarily inward) maps were studied, whereas in [9] u.s.c., contractive, inward maps were discussed. We now gather together some definitions and preliminary facts which will be used throughout this paper. Let X be a Hausdorff locally convex linear topological space and let 2x denote the family of nonempty subsets of X. Suppose C is a lattice with minimal element (which we will denote by 0). A mapping Q : 2x + C is called a measure of noncompactness [4] provided the following (i) (ii)

conditions

@(i$fl)) Q(0)

hold:

= Q(0)

for each Sz E 2x,

= 0 iff R is precompact,

(iii) @(A U B) = msx(@(A),

@(B)}

for each

A, B E 2x.

The above notion is a generalization of the set and ball measure of noncompactness; if {p, : a E A} is a family of seminorms which determines the topology on X, then for each (Y E A and

08957177/OO/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: SO8957177(00)00205-3

Typeset by &&‘I$$

1306

R.P. AGARWAL AND D. O'REGAN

R c 2x, we have the following: y,(R)

= inf {E > 0 : R can be covered by a finite number of sets each of

pa-diameter less than E} and

X,(0)

= inf { r > 0 : Cl can be covered by a finite number of pa-balls

with radius less than r). Then, letting C = {f noncompactness noncompactness

: A --f [O,oo]}, with C ordered pointwise, we define the set-measure of C by (G))(o) = ry&) for each (Y E A, and the ball measure of

y : 2x + X : 2x +

= ~~(02) for each (II E A and fi G X.

C by (x(n))(a)

precompact iff cya(s2) = X,(0) for each (Y E A. Now, let @ : 2x -+ C be a measure of noncompactness

Note, 0 is

and we assume additionally that C

is such that for each c E C and X E R, with X > 0, there is defined an element Xc E C.

If

k E R, with k > 0, then a mapping F : D ---t 2x (here D G X) is called a k - @contraction if @(F(R)) 5 @-condensing know [4] that @condensing

k@(a) for each 52 & D and F(D) is bounded. A mapping F : D -+ 2x is called provided that if R C D with @(F(R)) 2 @(a), then R is relatively compact. We if F : D -+ 2x is a k - @contraction with 0 5 k < 1, and Q, = y or X, then F is if either X is quasi-complete

(i.e., each closed, bounded subset of X is complete)

or D is complete. (h ere CD(X) A mapping F : D --+ CD(X) subsets of X) is said to be inward if

denotes the family of nonempty, closed, convex

F(x) n ID(X) # 0,

for each x E D,

(1.1)

here ID(Z) = {y E X : z + A(y - x) E D, for some X > 0). If D is convex, then (1.1)is equivalent to assuming for each z E D, there is a y E F(z)

and X E [0,1) such that Xz + (1 - X)g E D.

(1.2)

Finally, we state a fixed-point theorem due to Browder [I] ( more general formulations may be found in [2]). THEOREM 1.1. Let Q be a nonempty, compact, convex subset of a Hausdorff locally convex linear topological space X. Suppose F : Q + CD(X) is U.S.C.with

J’(z)n rQ(xc># 0,

for each x E Q.

(1.3)

Then, F has a fixed point. 2.

FIXED-POINT

THEORY

this section, X will be a Frechet space. Let n be a subset of X. We say F E ACG(S2,X) if F : R -+ CD(X) is a closed map (i.e., has closed graph). Our first two results extend well-known results in [l-5]. Throughout

THEOREM 2.1. Let Cp be either y or X. Suppose Q is a nonempty, closed, convex subset of a F&het space X. Assume F E ACG(Q, X) is a k - Cp-contraction with 0 5 k < 1 (in particular, F(Q) is a subset of a bounded set in X). In addition, suppose

F(s) * Then, F has a fixed point.

IQ&c)

# 0,

for each z E Q.

(2.1)

Fixed-Point Theory

1307

REMARK 2.1. In Theorem 2.1, X being Frechet can be replaced by X being a Hausdorff locally convex linear topological space provided either Q is complete or X is quasi-complete. PROOF. Let ze E Q. A standard result [3,4] guarantees that there exists a closed, convex set K with xc E K and K = m(F(Q In fact, K is compact.

n K)

u {x0}).

To see this, first recall [4] that @ is condensing. Also

@(F(K))

L @(F(Q

n K))

= @(w(F(Q

n K) u {LCO})) = a(K).

Now, since Cpis condensing we have that K is compact.

(Alternatively,

notice for each LYE A

(described in Section l),

@(K)(Q)= @@(F(Q n K) U {x0)))(a) =

@(F(Q

n K))(a)

and so @(K)(a) = 0 for each (Y E A.) We now concentrate our study on FIQ~K. Note, K REMARK 2.2.

nQ

I

k@(Qn K)(Q)

I k@(K)(a),

is compact.

If we use the assumptions in Remark 2.1, we show K

nQ

is compact

in the

following way @(F(K

n&))= @(~5(F(QnK)u{x~})) = @P(K) 2 @(Kn 4).

Notice, F~K~Q : K n Q + CD(X) and FIK”Q has closed graph. Also, [7, p. 4651 implies F~K~Q is u.s.c., [note for each a E A (as described in Section l), we have @(F(Q n K))(a) < rC(a(Qn K)(cu) = 0, i.e., @(F(Q n K))(a) = 0 for each (Y E A]. Consequently, FJK~Q : K n Q -+CK(X) is a U.S.C. map with K n Q convex and compact (here CK(X) denotes the family of nonempty, compact, convex subsets of X). Next, we claim

f’(x)

n IKnQ(x)

# 0,

for each 5 E K

n Q.

(2.2)

If our claim is true, then Theorem 1.1 implies FlKf-jQ has a fixed point and we are finished. It remains to prove the claim. Let z E K n Q. From (2.1), there is a 9 E F(z) and X E [0,1) with Xx + (1 - X)y E Q. Now, since y E F(x) C F(K n Q) C w(F(K

n Q)U (~0))

=

K,

we have (since K is convex) Xx +

(1- X)y E K.

Consequently,

Xa:+(l-X)yEKnQ, so(2.2)

holds.

I

THEOREM 2.2. Let G be either 7 or X. Suppose Q is a nonempty, closed, convex subset of a (in particular, F(Q) FHchet space X with 0 E Q. Assume F E ACG(Q, X) is a 1 -@-contraction is a subset of a bounded set in X) and (2.1) holds. In addition, suppose the following condition is satisfied: if(x,}~Qwithy,~F(z,)forallnandx,-y,-+Oasn--+oo,then there exists z E Q with 5c E F(z). Then, F has a fixed point.

(2.3)

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R.P. AGARWALAND D. O'REGAN

PROOF. Fir each n E (1,2,. ..}. bt F, = (Iand F, E ACG(Q, X). Next, we show

I/n)F.

&z(x)n IQ(x) # 0,

‘33len,F, is a (1 - l/n) - @contraction

for each zr E Q.

To see this, fix n E (2,3,. . . ) and fix I E &. Since (2.1) holds, there is a y E F(z) with Xz + (1 - X>y E Q. Since 0 E Q and Q is convex, we have Xz E Q. Thus,

and X E [0, 1)

,x+(1-A,((,-~),) = (1-~)[X5+(1--h)p]+~[h~lEQI and so since (1 - l/n)y

E (1-

l/n)F(z)

we have

Now, Theorem 2.1 guarantees that F, has a fixed point zfi E Q for each. n E (2,3,. . . ), Choose yn E F(zn) with zr,, = (1 - l/n)y,. Notice, z, - yn = -(l/n)y, --f 0 since F(Q) is bounded. Now, apply (2.3) to deduce that there exists z E Q with 2 E F(z). I REMARK

2.3. Conditions

so that (2.3) is satisfied, (e.g., sum of nonexpansive and strongly

continuous map) may be found in [4,11]. Next, we obtain two nonlinear alternatives of Leray-Schauder type for inward, condensing ACG maps. THEOREM 2.3. Let @ be either y or X. Suppose C is a closed, convex subset of a F&&et space X and VO is a d-bounded, (i.e., there exists K > 0 with Uo c (2 E X : d(0, z) < K}; here d is the metric associated with X), open subset of X. Let U = Uo n C and 0 t U. Assume F E ACG(o, X) is a k - @contraction with 0 5 k < 1 (in particular, F(o) is a subset of a bounded set in X). In addition, suppose for each x E U.

(2.4)

Then, either (Al) (A2)

F has a fixed point in 0, or there exists u E XJ and X E (0,l)

with u E M’(u).

REMARK 2.4. We could replace Wa d-bounded in Theorem 2.3 by, there exists a convex set CO with Ua c CO and F(U) c CO. PROOF.

Suppose (AZ) does not hold and F has no fixed points in 87. H = {z E 6 : z G XF(z),

Let

for some X E [0, 11).

Note, H(# 0) is closed. To see this let (z,) be a sequence in H (i.e., zn E X,F(s,) for some X, E [O,11) with zn --$ 20 E ti. Without loss of generality assume X, + XO E (O,l]. Since 5, E H, there exists yn E F(zn) with 5, = X,y,. Now, zn --) zo and y% + (l/&)zo. The closedness of F implies (I//\o)zo E F( 20 ) so 50 E H. Thus, H is closed. [In fact, H is compact. To see this, notice H 2 W(F(H) U (0)) so @(F(H)) = @E(F(H) U (0))) > (P(H). Since @ is condensing, we are finished]. Now, since H II NJ = 0, there is a continuous function ~1 : 0 --t [O,l] with p(H) = 1 and p(W) = 0. Since Vo is a d-bounded set and F(u) is a subset of a bounded set in X (metric d), there exists R > 0 with U&{x~X:d(0,x)
and

F(c)

c (x E X : d(O,s)

< R}.

Fixed-Point Theory

1309

Let52=Cfl{zEX:d(O,z)
Now, it is easy to see that N : fi + CD(X) k - @-contraction

and N(n)

has closed graph. In addition, N : !?l + CD(X)

is a

is a subset of a bounded set in X. We claim N(s) n Mu)

for each x E G.

# 0,

(2.5)

If the claim is true, then Theorem 2.1 implies that there exists 3c E fi with x E N(E). Also, z E U since 0 E U. Thus, z E p(z)F(z)

= XF( x ) w here 0 I X = p(x) < 1. Consequently, x E H, which

implies p(z) = 1 and so 2 E F(z). It remains to show (2.5). If x E fi \ 0, then N(x) = (0) E In(x) since 0 E 170fl C (so 0 E a). Next, fix x E 0. Now, since F(X) n

k(x)

#

and

0

0 E k(z),

then SF(X) r-lk(x)

with

# 0

s = P(Z),

see the ideas in Theorem 2.2. Consequently,

Thus, there exists y E b(z)F(x)

and X E [0, 1) with xx + (1 - A)y E c.

(2.6)

We must show Xs + (1 - X)y E a. Notice, first Y E P(Xv%)

+ (1 - P(X)){01

and

F(U) c {z E x : d(O,z) < R}

implies y~{t~X:d(0,z)
Thus, Xz + (1 - X)y E {z E X : d(0, z) I R}, and this together with (2.6) implies

Hence, Xx + (1 - X)y E fii.

I

2.4. Let Q be either y or x. Suppose C is a closed, convex subset of a Fr&het space X and Uo is a d-bounded, open subset of X. Let U = UO n C and 0 E U. Assume F E ACG(U, X) is a 1 - @contraction (in particular, F(u) is a subset of a bounded set in X) and (2.4) holds. In addition, suppose the following condition is satisfied: THEOREM

if {z~) C 6 with yn E F(x,) for all n and x,,‘- yn + 0, as n + DC),then there exists z E 0 with x E F(x). Then, either (Al) F has a fixed point in u, or (A2) there exists u E EKJ and X E (0,l)

with u E XF(u).

(2.7)

R. P.

1310

AGARWAL AND D. O’REGAN

PROOF. Suppose (A2) does not hold. For each 12E {1,2,. . .}, let F,, = (1 - l/n)F. a (1 - l/n)

- @contraction

Then, F, is

and F, E ACG(Q, X).

A similar argument to that in Theorem 2.2

J’s(z) II I&)

for each x E 8.

implies # 0,

We wish to apply Theorem 2.3. If there exists u E dU and X E (0,l) u E X

This is a contradiction

with u E XF,(u),

then

(1-i >

F(u).

since (A2) does not hold. Thus, Theorem 2.3 guarantees that F,, has a

fixed point xn E 6. Choose yn E F(zn) with x, = (l-l/n)&. to deduce that there exists x E a with x E F(x).

Now, z,-

yn + 0 and apply (2.7)

I

REMARK 2.5. In this paper, it would also be possible to consider maps F : D + AC(X) (here, denotes the family of nonempty, closed, acyclic (see [3]) subsets of X). Of course, instead of Theorem 1.1, one uses fixed-point theory for acyclic maps (see 131).

AC(X)

REFERENCES 1.

2. 3. 4. 5. 6.

F.E. Browder, The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann. 177, 283-301 (1968). K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112, 234-240 (1969). P.M. Fitzpatrick and W.V. Petryshyn, Fixed point theorems for multivalued noncompact acyclic mappings, Pacific Jour. Math. 54, 17-23 (1974). P.M. Fitzpatrick and W.V. Petryshyn, Fixed point theorems for multivalued noncompact inward mappings, Jour. Math. Anal. Appl. 46, 756-767 (1974). S. Reich, Fixed points in locally convex spaces, Math. Z. 125, 17-31 (1972). D. O’Regan, Multivalued integral equations in finite and infinite dimensions, Comm. Applied Analysis (to appear).

7. C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, Springer-Verlag, Berlin, (1994). 8. T, Cardinali and F. Papalini, Fixed point theorems for multifunctions in topological vector space, Jour. Math.

Anal. Appl.

186,

769-777

(1994).

9. D. O’Regan, A continuation theory for weakly inward mappings, Glasgow Math. Jour. (to appear). 10. D. O’Regan, Nonlinear alternatives for multivalued maps with applications to operator inclusions in abstract spaces, (to appear). Math. Applic. 35 (4), 27-34 11. D. O’Regan, New fixed-point results for l-set contractive mappings, Computers (1998).