Some random approximation theorems with ...

Nonlinear Analysis 35 (1999) 609 – 616

Some random approximation theorems with applications1 Ismat Beg a;∗ , Naseer Shahzadb a Department

of Mathematics, Kuwait University, P.O.Box 5969, Safat 13060, Kuwait of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan

b Department

Received 9 March 1995; accepted 17 July 1997

Keywords: Random operators; Random approximation; Random coincidence point; Random xed point; Banach space

1. Introduction Fan [8] proved a very interesting theorem which provides a tool to study xed point theory in connection with best approximation. Various aspects of that theorem have been studied by Fan [9], Ha [10], Singh and Watson [22], Lin [14], Lin and Yen [16] and many others. Fan’s theorem has been of great importance in nonlinear analysis, approximation theory, game theory and minimax theorems. The study of random approximations and random xed points have been a very active area of research in probabilistic functional analysis in the last decade (see Sehgal and Waters [21], Sehgal and Singh [20], Papageorgiou [18], Lin [15], Xu [25], Tan and Yuan [23, 24] and Beg and Shahzad [1–4]). Most random xed point theorems in Banach spaces deal with condensing or nonexpansive random operators. What about the random operators which are neither of the above cases?. The interesting case would be a 1-set-contractive random operator. The class of 1-set-contractive random operators includes condensing and nonexpansive random operators. Besides, it also includes other important random operators such as semicontractive random operators and LANE [locally almost nonexpansive] random operators. Recently, Beg and Shahzad [3] studied these random operators in details and gave many results regarding random approximations and random ∗ 1

Corresponding author. E-mail: [email protected]. This work is partially supported by Kuwait University research grant No. SM. 119.

0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 2 0 - 0

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xed point theorems in Hilbert spaces. The aim of this paper is to study random xed points in connection with random approximations. We prove random approximation theorems for more general random operators, i.e., 1-set-contractive random operators or continuous random operators. As applications of our theorems, we derive results regarding random coincidence points and random xed points. Our results improve and extend the results obtained by Bharucha-Reid [5] and Lin [15]. 2. Preliminaries Throughout this paper, ( ; ) denotes a measurable space. Let A be a subset of a Banach space E. Let 2A be the family of all subsets of A and WK(A) be the family of all nonempty weakly compact subsets of A. A mapping T : → A is called measurable if for any open subset C of A; T −1 (C) ∈ . If T is a multifunction, then T −1 (C) = {! ∈ : T (!) ∩ C 6= } ∈ . A mapping  : → A is said to be measurable selector of a measurable mapping T : → 2A if  is measurable and for any ! ∈ ; (!) ∈ T (!). A mapping T : × A → E is called a random operator if for each xed x ∈ A, the map T (:; x) : → E is measurable. A measurable map  : → A is a random xed point of the random operator T if T (!; (!)) = (!), for each ! ∈ . A measurable mapping  : → A is called random coincidence point of random operators g : × A → A and f : × A → E, if for every ! ∈ ; g(!; (!)) = f(!; (!)). A mapping T : A → E is called condensing if for any bounded subset B of A with (B)¿0; (T (B))¡(B), where (B) = inf {c¿0: B can be covered by a nite number of sets of diameter ≤ c}. This number (B) is called the (set-) measure of noncompactness of B. If there exists k; 0 ≤ k ≤ 1, such that for each nonempty bounded subset B of A we have (T (B)) ≤ k(B), then T is called k-set-contractive map. A mapping T : A → E is called nonexpansive if kT (x) − T (y)k ≤ kx − yk for x; y ∈ A; T is semicontractive if there exists a map V of A × A into E such that T (x) = V (x; x) for x ∈ A, while (a) For each xed v in A; V (:; v) is nonexpansive from A to E. (b) For each xed u in A; V (u; :) is completely continuous from A to E, uniformly for u in bounded subsets of A, (i.e., if vj converges weakly to v in A and {uj } is a bounded sequence in A, then V (uj ; vj ) − V (uj ; v) → 0, strongly in E). A continuous mapping T : A → E is called LANE (locally almost nonexpansive) if given x ∈ A and ¿0, there exists a weak neighborhood Nx of x in A (depending also on ) such that u; v ∈ Nx ; kT (u) − T (v)k ≤ ku − vk + . A map T : A → E is said to be demiclosed at y ∈ E if, for any sequence {xn } in A, the conditions xn → x ∈ A weakly and T (xn ) → y strongly imply T (x) = y. A random operator T : × A → E is continuous 1-set-contractive, condensing, nonexpansive, semicontractive, LANE, etc.) if the map T (!; :) : A → E is so, for each ! ∈ . We denote by I , the identity mapping of E. Let r¿0; then the retraction R of E onto S := {x ∈ E : kxk ≤ r} is de ned by  if kxk ≤ r; x rx R(x) := if kxk¿r:  kxk

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A random operator g : × A → E is said to be almost ane if kg(!; t) − yk ≤ kg(!; t1 ) − yk + (1 − )kg(!; t2 ) − yk; for all ! ∈ ; t1 ; t2 ∈ A; 0¡¡1; t = t1 + (1 − )t2 and y ∈ E. 3. Main results In the proof of our approximation theorem, we will need the following lemma due to Beg and Shahzad [3, Lemma 3.1]. Lemma 1. Let S be a nonempty closed convex separable subset of a re exive Banach space X; T : × S → S be a continuous 1-set-contractive random operator. Suppose that for any ! ∈ ; T (!; S) is bounded and I -T (!; :) is demiclosed. Then T has a random xed point. We now prove our main results. Theorem 2. Let S be a closed ball with center at origin and radius r in a separable re exive Banach space X; and T : × S → X be a continuous 1-set-contractive random operator. If for any ! ∈ ; I -RoT (!; :) is demiclosed; where R is a retraction of X onto S. Then there exists a measurable map  : → S such that k(!) − T (!; (!))k = d(T (!; (!)); S); for each ! ∈ . Proof. From [17], R is a continuous 1-set-contractive map. De ne F : × S → S by F(!; x) = R(T (!; x)). It is easy to see that F is a continuous 1-set-contractive random operator. By Lemma 1, there exists a random xed point  : → S of F, that is, F(!; (!)) = (!) for each ! ∈ . For each ! ∈

k(!) − T (!; (!))k = kF(!; (!)) − T (!; (!))k = kR(T (!; (!))) − T (!; (!))k  (!; (!))− T (!; (!))k = 0; if kT (!; (!))k ≤ r;  kT

= rT (!; (!))



kT (!; (!))k − T (!; (!)) = kT (!; (!))k − r; if kT (!; (!))k ≥ r: For any y ∈ S, we get kT (!; (!))k − r ≤ kT (!; (!))k − kyk ≤ kT (!; (!)) − yk. Therefore, k(!) − T (!; (!))k = d(T (!; (!)); S). Theorem 3. Let S; T and X be as in Theorem 2. Suppose for any ! ∈ ; I -RoT (!; :) is demiclosed; where R is a retraction of X onto S. Moreover T satis es one of the

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following conditions: (i) For each ! ∈ ; x ∈ S with x 6= T (!; x); there exists y (depending on ! and x), in Is (x) := {x + a(u − x): u ∈ S; a¿0} such that ky − T (!; x)k¡kx − T (!; x)k: (ii) T is weakly inward (i.e.; for each ! ∈ ; T (!; x) ∈ cl(Is (x)); for x ∈ S). (iii) For each ! ∈ ; each x in S; there is a z (real or complex number depending on whether the vector space is real or complex) such that |z|¡1 and zx + (1 − z)T (!; x) ∈ S. Then T has a random xed point. Proof. Let T satisfy (i). By Theorem 2, there exists a measurable map  : → S such that k(!) − T (!; (!))k = d(T (!; (!)); S); for each ! ∈ . If there exists ! ∈ such that (!) 6= T (!; (!)), then there exists a y ∈ IS ((!)) such that ky − T (!; (!))k¡k(!) − T (!; (!))k: Since y ∈ IS ((!)), there exists a u ∈ S such that y = (!) + a(u − (!)) for some a¿1. Then u = (y=a) + (1 − (1=a))(!). Therefore ku − T (!; (!))k¡k(!) − T (!; (!))k; which contradicts the choice of . Thus (!) = T (!; (!)) for each ! ∈ . If T satis es (ii). For each ! ∈ , each x ∈ S with x 6= T (!; x), since T (!; x) ∈ cl(Is (x)), there exists y in IS (x) such that ky − T (!; x)k¡kx − T (!; x)k;

and T satis es(i):

Assume that T satis es condition (iii). By Theorem 2, there exists a measurable map  : → S such that k(!) − T (!; (!))k = d(T (!; (!)); S); for each ! ∈ . Suppose there is some ! such that T (!; (!)) ∈= S, then 0¡k(!) − T (!; (!))k. From the assumption (iii), there is a number z such that |z|¡1 and z(!) + (1 − z)T (!; (!)) = x ∈ S. Therefore 0 ¡ k(!) − T (!; (!))k = d(T (!; (!)); S) ≤ kx − T (!; (!))k = |z|k(!) − T (!; (!))k ¡ k(!) − T (!; (!))k; which is a contradiction. Hence T (!; (!)) = (!); for each ! ∈ .

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Remark 4. (i) The conditions (i) and (iii) were rst considered by Browder [6] and Fan [8] in an attempt to extend xed point theorems to inward and weakly inward maps. (ii) Theorems 2 and 3 are still true; if we assume that S is separable (instead of X is separable): The proofs are exactly the same as Theorems 2 and 3: (iii) Theorems 1 and 4 in [15] do not follow from our Theorems 2 and 3; respectively. Since Lin [15] does not assume X re exive Banach space and I-RoT(!,.) demiclosed. However, our theorems strengthen his theorems; in one way: Lin assumes the random operator T is condensing. Theorem 5. Let S be a compact and convex subset of a Banach space E and g : × S → S be a continuous almost ane onto random operator. For each continuous random operator f : × S → E; there exists a measurable map  : → S satisfying kf(!; (!)) − g(!; (!))k = d(f(!; (!)); S); for each ! ∈ : Remark 6. Under the hypotheses of Theorem 5, it is clear that; the mappings p : × S → R+ and q : × S → R+ de ned by p(!; x) = kf(!; x) − g(!; x)k and q(!; x) = d(f(!; x); S) are measurable in ! and continuous in x. Proof follows on the same lines as in Lemmas 2 and 3 of Sehgal and Singh [20] and Lemma 6 of Engl [7]. Proof of Theorem 5. De ne a mapping T : →2S by T (!)= {x ∈S: kf(!; x)− g(!; x)k =d(f(!; x); S}; then by [19] and Remark 6, T (!) is nonempty and compact for any ! ∈ . Let G be a closed subset of S and D a countable dense subset of S: Denote L(G) =

∞ [  \

! ∈ : kf(!; x) − g(!; x)k¡d(f(!; x); S) +

n=1 x∈Dn

1 n

 ;

where Dn = {x ∈ D: d(x; G)¡1=n}. We show that T −1 (G) = L(G) if ! ∈ L(G); then for each n there exists xn ∈ Dn with kf(!; xn ) − g(!; xn )k ¡ d(f(!; xn ); S) + (1=n): Since {xn } ⊆ S; there exists a subsequence {xnj } converges to some x ∈G: This implies that kf(!; x) − g(!; x)k = d(f(!; x); S): Thus x ∈T (!) ∩G; i.e., ! ∈T −1 (G). Conversely, if ! ∈T −1 (G); then there is x ∈G with kf(!; x) − g(!; x)k = d(f(!; x); S): Since D is dense in S; we know by Remark 6, that for each xed n there exists xn in D such that d(xn ; G)¡

1 n

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and kf(!; xn ) − g(!; xn )k ≤ kf(!; x) − g(!; x)k + 1 2n 1 ≤ d(f(!; xn ); S) + : n

1 2n

= d(f(!; x); S) +

Hence ! ∈L(G): Therefore, T −1 (G) = L(G): By Himmelberg [11] and Remark 6, L(G) ∈: Hence T −1 (G) = L(G)∈  and T is measurable. By Kuratowski and Ryll– Nardzewski selection theorem [13], T has a measurable selector  : → S: This yields kf(!; (!)) − g(!; (!))k = d(f(!; (!)); S); for each ! ∈ : Theorem 7. Let S be a compact and convex subset of a Banach space E and g : ×S → S be a continuous almost ane onto random operator. For each continuous random operator f : × S → E which satis es one of the following conditions: (i) For each ! ∈ ; each x ∈ S with g(!; x) 6= f(!; x); there exists a y; depending on ! and x; in IS (g(!; x)) such that ky − f(!; x)k¡kg(!; x) − f(!; x)k: (ii) For each ! ∈ ; f(!; x)∈ cl(IS (g(!; x))) for x ∈ S: Then there exists a measurable map  : → S such that g(!; (!)) = f(!; (!)) for each ! ∈ : Proof. Let f satis es condition (i). From Theorem 5, there exists a measurable map  : → S such that kg(!; (!)) − f(!; (!))k = d(f(!; (!)); S); for each ! ∈ . If for some ! ∈ , g(!; (!)) 6= f(!; (!)): From the assumption (i), there exists y ∈ IS (g(!; (!))) such that ky − f(!; (!))k¡kg(!; (!)) − f(!; (!))k: Since y ∈ IS (g(!; (!))), there exists u ∈S; c¿0 such that y = g(!; (!)) + c(u − g(!; (!))): Therefore, y 6∈ S; otherwise it contradicts the choice of . We can assume that c¿1: Then   y 1 u= + 1 − g(!; (!)) = (1 − )y + g(!; (!)); c c where = 1 − (1=c); 0¡ ¡1:

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Therefore, d(f(!; (!)); S) ≤ kf(!; (!)) − [(1 − )y + g(!; (!))]k ¡ (1 − )kg(!; (!)) − f(!; (!))k + kg(!; (!)) − f(!; (!))k = kg(!; (!)) − f(!; (!))k; which contradict the choice of . Hence g(!; (!)) = f(!; (!)) for each ! ∈ : If f satis es condition (ii). For each ! ∈ ; each x ∈S with g(!; x) 6= f(!; x); since f(!; x)∈ cl(IS (g(!; x))); therefore there exists y in IS (g(!; x)) such that ky − f(!; x)k ≤ kg(!; x) − f(!; x)k; and f satis es condition (i). Remark 8. Theorem 5 generalizes Theorem 10 in Bharucha–Reid [5]. References [1] I. Beg, N. Shahzad, Random xed points of random multivalued operators on Polish spaces, J. Nonlinear Anal. (7) 20 (1993) 835–847. [2] I. Beg, N. Shahzad, Random approximations and random xed point theorems, J. Appl. Math. Stochast. Anal. 7 (2) (1994) 143 –148. [3] I. Beg, N. Shahzad, Applications of the proximity map to random xed point theorems in Hilbert spaces, J. Math. Anal. Appl. 196 (1995) 606–613. [4] I. Beg, N. Shahzad, Random xed points for random multivalued operators de ned on unbounded sets in Banach spaces, J. Stochast. Anal. Appl. 13 (3) (1995) 269 –278. [5] A.T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976) 641–657. [6] F.E. Browder, On a sharpened form of the Schauder xed point theorem, Proc. Natl. Acad. Sci. USA 74 (1977) 4749 – 4751. [7] W. Engl, Random xed point theorems for multivalued mappings, Paci c J. Math. 76 (1978) 351–360. [8] Ky Fan, Extensions of two xed point theorems of F.E. Browder, Math. Z. 112 (1969) 234 –240. [9] Ky Fan, Some properties of convex sets related to xed point theorems, Math. Ann. 266 (1984) 519 –537. [10] C.W. Ha, Extensions of two xed point theorems of Ky Fan, Math. Z. 190 (1985) 13 –16. [11] C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975) 53 –72. [12] S. Itoh, Random xed point theorems with an application to random dierential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979) 261–273. [13] K. Kuratowski, C. Ryll. Nardzewski, A general theorem on selector, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys. 13 (1965) 397– 403. [14] T.C. Lin, Approximation theorems and xed point theorems in cones, Proc. Amer. Math. Soc. 102 (1988) 502–506. [15] T.C. Lin, Random approximations and random xed point theorems for non-self maps, Proc. Amer. Math. Soc. 103 (1988) 1129 –1135. [16] T.C. Lin, C.L. Yen, Applications of proximity map to xed point theorem in Hilbert spaces, J. Approx. Theory 52 (1988) 141–148. [17] R.D. Nussbaum, The xed point index and xed point theorems for k-set contractions, Ph.D. Dissertation, Univ. of Chicago, 1969.

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[18] N.S. Papageorgiou, Random xed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986) 507–514. [19] J.B. Prolla, Fixed point theorems for set valued mappings and existence of best approximants, Numer. Funct. Anal. Optim. 5 (1982–1983) 449 – 455. [20] V.M. Sehgal, S.P. Singh, On random approximations and a random xed point theorem for set-valued mappings, Proc. Amer. Math. Soc. 95 (1985) 91–94. [21] V.M. Sehgal, C. Waters, Some random xed point theorems for condensing operators, Proc. Amer. Math. Soc. 90 (1984) 425– 429. [22] S.P. Singh, W. Watson, proximity maps and xed points, J. Approx. Theory 39 (1983) 72–76. [23] K.K. Tan, X.Z. Yuan, On deterministic and random xed points, Proc. Amer. Math. Soc. 119 (1993) 849 –856. [24] K.K. Tan, X.Z. Yuan, Random xed point theorems and approximation in cones, J. Math. Anal. Appl. 185 (1994) 378–390. [25] H.K. Xu, Some random xed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990) 395– 400.