Fixed points of asymptotically regular multivalued mappings

/. Austral. Math. Soc. (Series A) 53 (1992), 313-326 FIXED POINTS OF ASYMPTOTICALLY REGULAR MULTIVALUED MAPPINGS ISMAT BEG and AKBAR AZAM (Received 10 May 1990; revised 24 March 1991) Communicated by E. N. Dancer Abstract Some results onfixedpoint of asymptotically regular multivalued mapping are obtained in metric spaces. The structure of common fixed points and coincidence points of a pair of compatible multivalued mappings is also discussed. Our work generalizes known results of Aubin and Siegel, Dube, Dube and Singh, Hardy and Rogers, Hu, Iseki, Jungck, Kaneko, Nadler, Ray and Shiau, Tan and Wong. 1991 Mathematics subject classification (Amer. Math. Soc): 54 H 25, 47 H 10. Keywords and phrases: metric space, multivalued mapping, fixed point, coincidence point. 1. Introduction Let T be a single valued self mapping on a metric space X. A sequence {xn} in X is said to be asymptotically T-regular if d(xn , Txn) —> 0. The presence of a sequence {xn} for which d{xn , Txn) -> 0 is related to some property of T (see [3], [7], [8], [24], [25], and [27]) and hence is exploited to obtain fixed points of T. The aim of the present paper is to bring out the thrust of a similar assumption for multivalued mappings. The weakly dissipative multivalued mappings recently introduced by Aubin and Siegel in [3] satisfy such an assumption. As stated in Aubin and Siegel [3], such fixed point theorems have application to control theory, system theory and optimization problems. Moreover, such a sequence (for multivalued mappings) has been used by Itoh and Takahashi [12] and Rhoades, Singh and Kulshrestha [26]. © 1992 Australian Mathematical Society 0263-6115/92 $A2.00 + 0.00 313 https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press 314 Ismat Beg and Akbar Azam [2] Rhoades [24], [25] had compared these contractive conditions. Most of the contractive conditions used imply the asymptotic regularity of the mappings under consideration, so the study of such mappings play an important role in fixed point theory. In Section 2, some notation, definitions and facts, used in subsequent sections are listed. In Section 3 we prve the existence of a common fixed point of two multivalued mappings satisfying a contractive type condition in a metric space. In Section 4 a class of multivalued mappings is introduced which is larger (even in the case of single valued mappings) than those that Wong [32] refers to as Kannan mappings. The fixed point theorems therein are proved under less restrictive hypotheses and for wider classes than the results of Shiau, Tan and Wong [30]. In Section 5 we extend the idea of Jungck [13] to multivalued mappings and obtain a coincidence theorem for a pair of compatible multivalued mappings. The structure of common fixed points of these mappings is also studied. 2. Preliminaries Let (X, d) be a metric space and let CB(X) denote the family of all nonempty bounded closed subsets of X. For A, B e CB(X), let H(A, B) denote the distance between A and B in Hausdorff metric, that is f infE.A B „ if £ \ A„Br^ 0 , ' ,e ' H(A,B) = \ where N(e, A) = {x e X: d(x, A) < e} and E A,B = {e>O:ACN(e,B),BC N(e,A)}. Let T: X —> CB(X) be a mapping and {xn} a sequence in X. Then {xn} is said to be asymptotically T-regular if d(xn , Txn) -> 0. Let f: X ^ X be a mapping such that TX c / X . Then {xn} is called asymptotically Tregular with respect to f if d{fxn, 7"xn) —» 0. (cf. [27]). A point x is said to be a ybcerf point of a single valued mapping / (multivalued mapping T) provided x = fx (x € Tx). The point x is called a coincidence point of / and r if fx € Tx . We shall require the following well-known facts (cf. [21]). LEMMA 2.1. If A, B e CB(X) with H(A, B) <e, then for each a e A, there exists an element b € B such that d(a, b) < e. LEMMA 2.2. Let {An} be a sequence in CB(X) and limn^oc H(An, A) = 0 for Ae CB(X). If xne An and lim n ^ oo d{xn, x) = 0, then x e A . https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press [3] Fixed points of asymptotically regular multivalued mappings 315 3. Common fixed point of multivalued generalized contractions Wong [31] extended the result of Hardy and Rogers [9] by showing that two self mappings S and T on a complete metric space satisfying a contractive type condition have a common fixed point. In this section we extend this result of Wong to the case when S and T are multivalued and satisfy a more general contractive type condition. THEOREM 3.1. Let X be a complete metric space, S: X -* CB(X) and T: X —> CB{X). If there exists a constant a, 0 < a < 1, such that for each x,yeX, (1) H(Tx,Sy) < a max{d(x, y), d(x, Sx), d(y, Ty), (d(x, Ty) + d(y, Sx))/2} then there exists a common fixed point of S and T. PROOF. Let x0 Then Assume that /? = y/a. be an arbitrary but fixed element of X and choose x{ e H(Sx0 ,Txx)<p max{d(x0, x , ) , d(x0, Sx0), d{xx Sx0. ,Txx), Lemma 2.1 implies that there exists a point x2 e Txx such that max{^(x 0 , x , ) , d(x0, Sx0), d{xl, Txx), {d{xo,TXl) + d(Xl,Sxo < 0 max{d{xQ, x,), rf(x,, x 2 ), (rf(x0, x,) + d{xx, x 2 ))/2}. d(xl ,x2)<$ If d{xx, x2) > d{xQ, x , ) , then d(xx, x2) < ftd{x{, x 2 ), a contradiction. Thus d(x{, x2) < fid{x0, x , ) . Now H(Txx, Sx2) < 0 max{fi?(x,, x 2 ), d(xl, Txx), d(x2, Sx2), Again using Lemma 2.1, we obtain a point x 3 € Sx2 such that d(x2, x 3 ) < pmax{d(xx ,x2), d(xx ,Txx), d(x2 ,Sx2), (d(xx,Sx2) + d(x2,Txx))/2} < pd(xx, x2). https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press 316 Ismat Beg and Akbar Azam [4] By induction we produce a sequence {xn} of points of X, such that, for A:>0, X 2k+\ £ X 2k+2 e Sx2k , Tx2k+\ and d(xn,xn+l)<pd(xn_,,xn) <P"d(xo,Xl). Furthermore, for m> n , d(xn ,x m ) < d(xn, x n + x ) + d(xn+l, +1 x n + 2 ) + ••• + d{xm_x, xm) m 1 <Vr + fi* + ~. + ft - }d{x0,xl). It follows that {xn} is a Cauchy sequence and there exists a point t e X such that xn —> t. It further implies that x2k+ { —> t, and x2k+2 —> ?. Thus we have, d(t, St) < d(t, x2k+2) + ^ ( x 2 t + 2 , St) <d(t,x2k+2) + H(Tx2k+1,St) < d(t, x2k+2) + p max{d(x2k+l, t), d(t, St), d X ( 2k+l ' 50)/2}- > *2*+2)' ( r f ( ? ' X2k+l) + d(X2k+l Letting k -» oo, we have rf(?, 5/) < fid{t, St). Hence t e St. Similarly, d(t, Tt) < d(t, x2k+l) + H(Sx2k, Tt) < pd{t, Tt). Therefore t e Tt. COROLLARY 3.2. Let X be a complete metric space and T: X —> CB(X). If there exists a constant a, 0 < a < 1, such that for each x, y € X, H(Tx, Ty)<amax{d(x,y),d(x, Tx),d(y, Ty), (d(x,Ty) + d(y,Tx))/2} then there exists a sequence {xn} which is asymptotically T-regular and converges to afixedpoint of T. REMARK 3.3. Theorem 3.1 improved the results of Kaneko [20], which considered the mapping r of a reflexive Banach space X into the family of weakly compact subsets of X. The proximinality of the set Tx is a consequence of his assumption and it is used in his proof. No such assumption is required in Theorem 3.1. REMARK 3.4. In [22], Ray proved a fixed point theorem for a multivalued mapping T: X -• CB{X) satisfying H(Tx, Ty) < ad(x, y) + b(d(x, Tx) + d(y, Ty)) + c(d(x,Ty) + d(y,Tx)), https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press [5] Fixed points of asymptotically regular multivalued mappings 317 where a, b and c are non-negative real numbers and 0<a + 2b + c< 1. Theorem 3.1 (even in the particular case for S = T) is not a special case of the theorem of Ray [22] since T is not assumed to have closed graph. It also illustrates that the compactness of Tx is not necessary for the theorem of Aubin and Siegel [3]. Several other results may also be seen to follow as immediate corollaries to Theorem 3.1. Included among these are Dube [5, Theorem 1], Dube and Singh [6, Theorem 1], Iseki [11], Nadler [21, Theorem 5], Hardy and Rogers [9] and Wong [31]. 4. Fixed point of Kannan type multivalued mappings In this section we consider the mapping T: X -* CB{X) satisfying the condition H(Tx, Ty)<a{{d{x, Tx))d(x, Tx) + a2(d(y, Ty))d(y, Ty), where al;: E —> [0, 1) (/ = 1, 2). Such a mapping T is not a special case of the mapping considered in Section 1. In 1968 Kannan [17] had established a fixed point theorem for a single valued mapping T defined on a complete metric space X satisfying d(Tx, Ty) < a(d(x, Tx) + d(y, Ty)), where 0 < a < \ and x, y e X. Within the context of a complete metric space the assumption 0 < a < j is crucial even to the existence part of this result, but within a more restrictive yet quite natural setting, an elaborate fixed point theory exists for the case a = j . Mappings of this wider class were studied by Kannan in [18]. In recent years, Beg and Azam [4], Shiau Tan and Wong [29] and Wong [32] have also studied such mappings. THEOREM 4.1. Let X be a complete metric space and T: X -> CB(X) a mapping satisfying (2) H(Tx, Ty)<a,{d{x, Tx))d{x, Tx) + a2(d(y, Ty))d(y, Ty), for all x,y e X, where a(: E —> [0, 1) (/ = 1,2). If there exists an asymptotically T-regular sequence {xn} in X, then T has a fixed point x* in X. Moreover Txn -> Tx*. PROOF. By hypothesis, we have H(Txn , TxJ < ax{d{xn , Txn))d(xn , Txn) + a2(d{xm , Txm))d(xm , TxJ. https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press 318 Ismat Beg and Akbar Azam [6] Thus {Txn} is a Cauchy sequence. Since (CB(X), H) is complete (see [2]), there exists a K* e CB(X), such that H{Txn,K*)^0. Let x* € K*. Then d(x*, Tx*)<H(K*, Tx*) = lim H{Txn , Tx*) n—>oo " < \imo(al(d(xn , Txn))d{xn , Txn) + a2(d(x*, Tx*))d(x*, Tx*)) <a2(d(x*, Tx*))d{x*, Tx*). It further implies that (1 - a2(d{x*, Tx*)))d(x*, Tx*) < 0. Therefore d(x*, Tx*) = 0. Thus x* e Tx*. Now, H(K* ,Tx*)= lim H(Txn , Tx*) <a2(d(x*, <d(x*, Tx*))d{x*, Tx*) Tx*) = 0. It follows that Tx* = K* = lim Txn. 4.2. Let X be a complete metric space and T: X —> CB(X) a mapping satisfying (2). If there exists an asymptotically T-regular sequence {xn} in X and Txn is compact for each n, then each cluster point of {xn} is a fixed point of T. THEOREM PROOF. Let yn € Txn be such that d(xn, yn) = d{xn , Txn). Obviously, a cluster point of {xn} is a cluster point of {yn}. If y* is such a cluster point of {xn} and {yn} , then with x* (as in Theorem 4.1), d(yn,Tx*)<H(Txn,Tx*) < ax{d{xn , Txn))d{xn , Txn) + a2(d(x*, Tx*))d(x*, Tx*) <a{{d{xn,Txn))d(xn,Txn). Therefore y* e Tx*. Now d(y*, Ty*) < H(Tx*, Ty*) <ai(d(x*,Tx*))d(x*,Tx*) + a2(d(y*,Ty*))d(y*, Ty*). It follows that {1 - a2(d(y*, Ty*))}d(y*, Ty*) < 0. Hence y* e Ty*. https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press [7] Fixed points of asymptotically regular multivalued mappings 319 THEOREM 4.3. Let X be a complete metric space and T: X —> CB(X) a mapping satisfying (2) with ax(d(x, Tx)) + a2(d(y, Ty)) < 1. If inf{d(x, Tx): x € X) = 0 then T has a fixed point. PROOF. It is sufficient to show that there exists an asymptotically ^-regular sequence {xn} in X. Let x0 be an arbitrary but fixed element of X. Consider the sequence {xn}, xn e TxnX. The inequality (2) implies that d{xn,Txn)<H{Txn_x,Txn) + a2(d(xn,Txn))d(xn,Txn) <d{xn_x,Txn_x). It follows that the sequence {d{xn , Txn)} is decreasing. Therefore d{xn, Txn) -> inf{d{xn , Txn): n e N} and d(xn , Txn) - 0. Hence {xn} is asymptotically T-regular. Theorems 4.1, 4.2 and 4.3 generalize results of Shiau, Tan and Wong [30]. Here we desire to emphasize not only that our T belongs to a wider class of mappings but also that the hypothesis of compactness of Tx (in [30, Theorem 1]) is dropped. 5. Coincidence point of compatible multivalued mappings Jungck [14] introduced a contraction condition for single valued compatible mappings on a metric space. He also pointed out in [15] and [16] the potential of compatible mappings for generalized fixed point theorems. Subsequently a variety of extensions, generalizations and applications of this followed; for example, see [1], [28] and [29]. This section is a continuation of these investigations for multivalued compatible mappings. DEFINITION. Let X be a metric space. Mappings T: X —> CB(X), f:X—>X are compatible if, whenever there is a sequence {xn} c X satisfying lim fxn e lim Txn (provided lim fxn exists in X and lim Txn exists in CB(X)), then lim H(fTxn, Tfxn) = 0. If T is a single valued self mapping on X, this definition of compatibility becomes that of Jungck [14]. Let X = R, with Euclidean metric, Tx = https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press 320 Ismat Beg and Akbar Azam [8] [x2/4, x2/2], fx = x2/&. Then / and T are compatible but they do not commute. Let (p: (0, oo) —> [0, 1) be a function having the following property (cf. [10], [23]): (P) for t > 0, there exists d{t) > 0, s(t) < 1, such that 0 < r - t < d(t) implies <p{r) < s{t). The following Theorem is a generalization of Hu [10, Theorem 2], Jungck [13], Kaneko [19] and Nadler [21, Theorem 5]. THEOREM 5.1. Let T be a mapping from a complete metric space X into CB(X). Let f: X -* X be a continuous mapping such that TX C fX. If f and T are compatible and for all x, y € X, (3) H(Tx, Ty) < <p(d(fx, fy))d(fx, fy), then there exists a sequence {xn} which is asymptotically T-regular with respect to f, and fxn converges to a coincidence point of f and T. PROOF. Let x0 be an arbitrary, but fixed element of X. We shall construct two sequences {xn} and {yn} of points of X as follow. Let y0 = fxQ and Xj G X be such that y{ = fxl e Tx0 . Then inequality (3) implies that H(Tx0, Txx) < <p(d(fx0, fxx))d{fx0, fx,). Using Lemma 2.1 and the fact that TX c fX, we may choose x2e X such that y2 = fx2 e Txx and d{yx, y2) = d(fxx, fx2) < <p(d(fx0, fxx))d(fx0, <d(fxo,fxx). fxx) By induction we produce two sequences of points of X such that yn = fxn e Txn_x, n > 0. Furthermore, d(yn+l, yn+2) = d(fxn+l, fxn+2) < ?(d(fxn , fxn+l))d(fxn , fxn+l) = d(yn,yn+l). <d(fxn,fxn+l) It follows that the sequence {d(yn , y n+1 )} is decreasing and converges to its greatest lower bound which we denote by t. Now t > 0; in fact t = 0. Otherwise by property (P) of <p , there exists S{t) > 0, s(t) < 1, such that, 0 < r - t < S(t) implies (p{r) < s{t). https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press [9] Fixed points of asymptotically regular multivalued mappings 321 For this S(t) > 0, there exists a natural number N such that, 0 < d(yn , yn+i)) - t < 5{t), whenever n > N. Hence Let K = <P(d{yn, yn+x)) < s{t), whenever n > N. max{ip(d(yQ,yl)),(p(d(yl,y2)), ... ,9>(d{yN_l,yN)),s(t)}. T h e n , for n = 1 , 2 , 3 , . . . , d(yn, yn+l) < vidiy^ , yn))d{yn_,, <Kd{yn_x,yn) </(:V(y o ,y 1 ) ->0 as yn) n-^oo, which contradicts the assumption that t > 0. Consequently, which implies that d(fxn, Txn) —» 0. Hence the sequence {xn} is asymptotically T-regular with respect to / . Assume that {fxn} is not a Cauchy sequence. Then there exists a positive number t* and subsequences {«(*)} > {w(/)} of the natural numbers with «(/) < m(i) and such that d{yn(i),ym{i)) > t \ d{yn(i),ym(i)_x) < t* for / = 1, 2, 3 . . . . Then t* < d{yn(i), ym(l)) Letting /' -» oo and using the fact that d{y,i),ym(i)_l) < t*, we obtain l i m , . ^ r f O ^ , ym{i)) = t*. For this t* > 0, there exists 8{t*) > 0, s{t*) < 1, such that 0 < r - t* < S(t*) implies (p{r) < s(t*). For this S(t*) > 0, there exists a natural number NQ such that, / > No implies 0 < d(yn{i), ym{i)) - t* < S(t*). Hence <p{d{yn(i), ym(i))) < s{t") for i>N0. Thus {i), ym{i)))d(yn{i), ym(i) Letting / -> oo, we get t* < s(t*)t* < t*, a contradiction. Hence {/*„} is a Cauchy sequence. By completeness of the space, there exists an element p e https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press 322 Ismat Beg and Akbar Azam [10] X such that d(yn, p) —* 0 . Continuity of / implies that d(fyn , fp) —• 0 . Hence , fp) H(Tyn , Tp) < <p(d(fyn , fp))d{fyn Inequality (3) and the fact that {/*„} is a Cauchy sequence imply that there exists A e CB{X) such that Txn -> A . Furthermore, d(p,A)<limoH(Txn_l,Txn) = 0. Now d(fyn+l,Tyn)<H(fTxn,Tfxn). Letting n —» oo, we obtain d(fp, Tp) = 0. Hence fpeTp. Let X = [0, oo) with the Euclidean metric Tx = [0, x] Then / and T do not satisfy the condition of the theorems in [10], [13] and [21]. Considering the function <p(x) = c, where 10~4 < c < 1, it is easily seen that all the hypotheses of Theorem 5.1 are valid. Thus / and T have a coincidence point. EXAMPLE 5.2. and fx = 10 4 JC. COROLLARY 5.3. If, in addition to the hypotheses of Theorem 5.1 the mapping f satisfies, for all x, y € X, (4) d(fx ,fy)<y max{d(x, y), d(x, fx), d(y, fy), (d(x, fy)+d(y, fx))/2} where 0 < y < 1, then there exists a common fixed point of f and T. PROOF. Let /? = y/y. As in the proof of Theorem 5.1 there is a coincidence point p of f and T. Define the iterative sequence {tn} as follows: t0 = p and tn = ftn_l = ftQ, n = 0, 1 , 2 . . . . Now inequality (4) implies that d(tn,tn+l) = d(ftn_l,ftn) (fn_1, tn), d(tn , tn+l), d{tn_x, tn+l)/2} tn_x, tn) n <p d(t0,tx). It further implies that {tn} is a Cauchy sequence. By the completeness of X, we have fnt0 -> x* e X. Now consider a constant sequence {un} c X as follows: un = t0. Then lim fu = ft0 € TtQ = lim Tu . https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press [11] Fixed points of asymptotically regular multivalued mappings 323 Thus by the compatibility of / and T, H(fTt0, Tft0) = \w^H(fTun , Tfun) = 0. Hence f2t0 = fft0 vn = ft0 . Then and 6 fTt0 = TftQ. Choose another constant sequence, H(fTft0, Tf2t0) = \imoH(fTvn , TfvH) = 0. Thus ft0 = ff2t0 € fTftQ = Tf2t0. Consequently, we have f"+lt0 e TftQ. Using (3), we get l i m ^ ^ Txn = Tx*. Hence by Lemma 2.2, we obtain, x* € Tx*. Moreover, fx* = f lim ft0 = lim / " + ' r 0 = x*. Hence x* is a common fixed point of / and T. In Theorem 5.1 our hypothesis that / is continuous implies that T is continuous. And we use the continuity of / and T in our proof. In the next theorem we show that if fX is complete then the continuity and compatibility of / and T are not required. THEOREM 5.4. Let T be a mapping of a metric space X into CB{X). Let f: X -* X be a mapping such that TX c fX, fX is complete and the condition (3) is satisfied. Then (i) there exists a sequence {xn} which is asymptotically T-regular with respect to f and (ii) / and T have a coincidence point. PROOF. Examining the proof of Theorem 5.1, we see that the only change is that the completeness of fX allows us to obtain z e X such that fxn -* p — fz. Then d(fz, Tz) < d(fz , fxn+l) <d(fz,fxn+l) < d(fz, fxn+l) <d(fz,fxn+l) + d(fxn+i, Tz) + H(Txn,Tz) + <p(d(fxn , fz))d(fxn , fz) + d(fxn,fz). Letting n -> oo, we obtain d(fz, Tz) < d(fz ,p) + d(p, fz) = 0. Hence fz eTz. https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press 324 Ismat Beg and Akbar Azam [ 12] COROLLARY 5.5. Suppose that, in addition to the hypotheses of Theorem 5.4, / satisfies (4) and f and T are compatible. Then {fxn} converges to a coincidence point (say p) of f and T, and {fp} converges to a common fixed point of f and T. PROOF. By Theorem 5.4, there exists z e X such that fz e Tz. As in Corollary 5.3, compatibility of / and T implies that ffz — ftz — Tfz. Since fxn —* fz (see Theorem 5.4), fxn converges to a coincidence point of / a n d T. Now, inequality (4) implies that {f'z} is a Cauchy sequence. Let f"z —> x*. Since (as in Corollary 5.3) fn+lz € Tfz, we have d{x*, Tx*) < d(x*, f+lz) + H(Tf"z, Tx*) +l < d(x*, f z) + <p(d(fz, x*))d(fz, x*) <d(x*,f+lz) + d(fz,x*). Letting n -+ oo, we obtain d(x*, Tx*) = 0, i.e., x* e Tx*. Moreover, d(x*,fx*)<d(x*,f+lz) + d(f+lz,fx*) <d(x*,f+lz) + ymax{d(fz,x*),d(fz,f+lz),d(x*,fx*), ,fx*) + d(f+lz,x*))/2}. Letting « -» oo, we have d(x*, fx*) < yd{x*, fx*). Hence x* = fx*. We show that the assumption of TX C fX (Theorem 5.4) and compatibility of / and T (Corollary 5.5) cannot be dropped. EXAMPLE 5.6. Let X - R with the Euclidean metric Tx = [0, |JC|/3] , fx = (x + 3)/2 and <p(x) = 2/3 . Then all the hypotheses of Theorem 5.4 are satisfied and / ( - 2 ) e T(-2). Moreover / and T are not compatible, but the other assumptions of Corollary 5.5 are satisfied, f{-2) —> 3 and 3 is not a common fixed point of / and T. Acknowledgment Thanks are due to Professor B. E. Rhoades for providing us with a preprint of [28]. https://doi.org/10.1017/S1446788700036491 Published online by Cambridge University Press [13] Fixed points of asymptotically regular multivalued mappings 325 References [1] A. Asad and S. 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