Nonlinear essential maps of Mönch, 1-set contractive demicompact and monotone (S)

International Journal of Stochastic Analysis, Aug 2018

In this paper the notion of an essential map is extended to a wider class of maps. Here we show if F is essential and F≅G, then G has a fixed point.

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Nonlinear essential maps of Mönch, 1-set contractive demicompact and monotone (S)

Journal of Applied Mathematics and Stochastic Analysis NONLINEAR ESSENTIAL MAPS OF MONCH, 1-SET CONTRACTIVE DEMICOMPACT AND MONOTONE (S)+ TYPE 0 RAVI P. AGARWAL National University 1 DONAL O'REGAN National University In this paper the notion of an essential map is extended to a wider class of maps. Here we show if F is essential and F G, then G has a fixed point. 1. Introduction In this paper we extend the notion of an essential map introduced by Grands in [ 4 ] to a larger class of maps. The notion of essential is more general than the notion of degree, and in [ 4 ] it was shown that if F is essential and F G then G is essential. However, to be essential is quite general and as a result, Grands was only able to show this homotopy property for particular classes of maps (usually compact or more generally condensing maps [ 7 ]). Precup in [10] extended this notion to other maps by introducing a "generalized topological transversality principle". However, from an application point of view, the authors in [ 4, 10 ] were asking too much. What one needs usually in applications is the following question to be answered: If F is essential and F G, does G have a fixed point? In this paper we discuss this question and we show that for many classes of maps that arise in applications, this is in fact what happens. In particular, in Section 2, we discuss MSnch type maps, in Section 3, 1-set contractive demicompact maps and in Section 4, monotone maps of (S)+ type to illustrate the ideas involved. It is worth remarking as well that the ideas presented in this paper are elementary (in fact they only rely on Urysohn?s Lemma). This paper will only discuss single valued maps (the multivalued case will be discussed in a forthcoming paper). 2. M6nch Type Maps Throughout this section, E is a Banach space, U is an open subset of E, with 0 E U and F: UE is continuous (here U denotes the closure of U in E). Definition 2.1: We let M ou(U E) denote the set of all continuous maps F" UE, which satisfy Mhnch?s condition (i.e., if C C_ U is countable and C C_ -6({0} U F(C)) then C is compact) and with (I-F)(x)=/: 0 for x OU; here I is the identity map and OU the boundary of U in E. Pmark 2.1: Mhnch type maps were introduced in [ 6 ] see also [ 3 ]). Definition 2.2: A map F Mou(U,E) is essential if for every G Mou(U, E) with G Iou Flog, there exists x e U, with (I-G)(x) O. Theorem 2.1" Let E be a Banach space, U an open subset of E, and 0 U. Suppose F Mou(U,E is an essential map and H:U x[ 0,1 ]---E is a continuous map with the following properties: and o)u (I- Ht)(x 7 0 for any x OU and t (0, 1] (here Ht(x H(x,t)) for any continuous #:U---[ 0, 1 ] with #(OU)- 0 the map Rp?U--,E defined by Rt(x -H(x,#(x)) satisfies MiJnch?s condition (i.e. if C C U is countable and C C g-6({0} t2 Rt(C)) then C is compact). (2.1) (2.3) Then H 1 has a fixed point in U. Remark 2.2: It is possible to replace (2.3) in Theorem 2.1 with: countable and C C_ g-d({0} tO H(C x [ 0, 1 ])) then C is compact. Proof: Let ifCC_U is B {x e ?" (I- Ht)(x) When t O, I- H I- F and .since F M ou(U E) is essential, there exists x U, with (I-F)(x)- O. Thus B # The continuity of H imlilies t.hat B is closed. In addition, (2.2) (together with F Mou(U,E)) implies B C 0U Thus there exists a continuous #" U---,[ 0, 1 ], with #(OU) 0 and #(B) 1. Define a map R: U--,E by e [O,1]}. O for some R(x)-H(x,#(x)). Now, R is continuous and satisfies MSnch?s condition (see (2.3)). Moreover, for x G OU, (I- R)(x) (I- Ho)(X) (I- F)(x) Tk O and so R G Mou(5,E ). Also notice and since F E Mou(U,E is essential, there exists x E U with (I-R)(x)-0 (i.e., (I-Hu(z))(x)-0). Thus xGB and so #(x)-l. Consequently, (I-H 1)(x)-O and we are finished. Yl We now use Theorem 2.1 to obtain a nonlinear alternative of Leray-Schauder type for MSnch maps. To prove our result we need the following well known result from the literature [ 3 ]. Theorem 2.3: Let E be a Banach s_pace, U an open subset of E and 0 U. Theorem 2.2: Let E be a Banach space and D a closed, convex set of E, with 0 D. Suppose J?D---D is a continuous map, which satisfies MSnch?s condition. Then J has a fixed point in D. Suppose G: U--+E is a continuous map, which satisfies MSnch?s condition and assume addition, since co(G(C) .vjj-is convex and {0} toco(GC)U-0})- co(G(C) {0}), tG(x) # x for x OU and (0, 1). (2.4) Then G has a fixed point in U. Proof: We assume that G(x) 7 x for x 0U (otherwise we are finished). Then tG(x) :/: x for x OU and t [ 0, 1 ]. (2.) Let H(x,t) tG(x) for (x,t) U x[ 0,1 ] and F(x) 0 for x E U. Clearly, (2.1) and (2.2) hold. To see the validity of (2.3), let C C_ U be countable and C C_ --5({0} U Rt(C)). Now since ,R,@n #(x)G(x), we have R. (C) C co(G(C)tO {0}I In we have c c_ ({0} n.(c)) c_ (co(a(c) {0})) (a(c) {0}). Since G satisfies MSnch?s condition, we have C compact. Thus (2.3) holds. We can apply Theorem 2.1 if we show that F is essential. To see this, let 0 G Mou(U,E with 01ou-FIog-O. We must show that there exists xU with O(x)-x. Let D--C-d(O(U)) and let J" DD be defined by O(x), x e U (x) o, Now 0 G D and J" D--,D is continuous and satisfies MSnch?s condition. To see this, let C C_ D be countable with C C_ -6({0} t2 J(C)). Then C C_ --6({0} tO O(U Cl C)). Thus C N U C U is countable and ceu( c_ c)c_ ({0} 0(u ec)). Now since 0: U--,E satisfies MSnch?s condition, we have C n U compact. Thus since 0 is continuous, O(C )is compact and Mazur?s Theorem implies -5({0 tO 0(C C3 r)) is compact. Now since C C_-d-6({O}tAO(CVIU)), we have C compact. Consequently, J:D-D is continuous and satisfies MSnch?s condition. Theorem 2.2 implies that there exists xGD with J(x)-x. Now if xU, we have 0-J(x)-x, which is a contradiction, since 0 E U. Thus x E U; so x J(x) and we may apply Theorem 2.1 to deduce the result. O(x). Hence, F is essential (3.1) (3.2) (3.3) (3.4) and Let HI"U--E is a demicompact map. Then H 1 has a fixed point in U. Remark 3.2: It is enough to assume (3.2) for t G (0,1], since F:UE is demicompact with (I-F)(x)TO for xGOU; so there exists 6o>0 with [(I-F)(x)[ >_ 6o for x OU. P,oo : xi t ->0, wit _<" for Choose M > 0 so that 1 2- > 0. Fix k (1 -----2M,]. and consider Hk: V [ 0, 1 ]E defined by 3. 1-Set Contractive, Demicompact Maps Let E be a Banach space and U be an open, bounded subset of E, with 0 U. In this section we are interested in maps F:U-,E which are continuous, 1-set contractive and demicompact. Recall that F is k-set contractive (here k >_ 0 is a constant) if a(F()) <_ ka() for any f C_ U (here a denotes the Kuratowskii measure of noncompactness). F is demicompact if each sequence {xn} C_ U has a convergent subsequence {Xnk}, whenever {x n -F(xn) } is a convergent sequence in E. Definition 3.1" We let DMou(U E) denote the set of all continuous, 1-set contractive, demicompact maps F: U--.E, with (I- F)(x) :/: 0 for x c0U. Pmark 3.1: Demicompact 1-set contractive maps were discussed in detail in [8, Definition 3.2: A map F DMou(U,E is essential if for every G DMou(U,E), with G lou FLOG, there exists x E U, with (I-G)(x) O. Theorem 3.1: Let E be a Banach space and U be an open, bounded subset of E with 0 U. Suppose kF DMou(U,E is essential for every k [ 0, 1 ] (it is enough to assume this for k [e, 1] for some fixed e, 0 <_ e < 1). Let H: U x [ 0, 1 ]-E be continuous, 1-set contractive (i.e., a(U(A x [ 0,1 ]))<_ a(A) for any A C_ U) map with the following properties: H(x, O)- F(x) for x U there exists 6 > O with (I- Ht)(x) >_ 6 for x G OU and t G [O, 1] We first show that there exists an x G U, with Hk(x,t)-kH(x,t). x k Hkl(Xk)(here H1k kill). B -{x ?(I- Hkt)(x)--0 for some t [ 0,1 ]}. an xGU, .with (I-kF)(x)-O. Thus, B#O. Also B is closed. Next we claim When t- O, I-Hko- I-kF and since kF G DMou(5,E is essential, there exists BNOU- To see this, first notice that ](Hkt -Ht)(x)] ](1-k)Ht(x)l <_ (1-k)M for xU and t[ 0,1 ]. Thus for x0U and t[ 0,1 ], we have from (3.2) that [(I- Htk)(x) _> I(I- Ht)(x)l I(Ht -Ht)(x)l >_ 5- (1- k)M >_ -; note that k G (1- _8., 1. Thus our claim is true. As a result, there exists a contin21V1 uous #: U--[0,kl ]with #(OU)= 0 and #(B)- 1. Define the map R: UE by R(x)-Hk(x,#(x)). I is easy to see that R is continuous and k-set contractive (so automatically, 1-set contractive and demicompact [ 11 ]). Moreover, for x G 0U, and so R DMou(U, E). Also notice that (i.e. (I- H(kxk_))(xl -0). Thus, xk G B and so #(xk)- 1. Hence, (3.4) is true. and since kF DMou(U,E is essential, there exists x k U, with (I-R)(xk)- 0 We can apply the above argument for any k (1-2-LM, 1). Choose nO > 1, with n0{1,2,...} so that 1--d >1-2-" Let N + -ln0, n0+l,...}. For eachng0, there exists xn U with and so xn --(1-l)Hl(Xn) xn-Hl(Xn) -(l)Hl(Xn). (3.5) Now since Hi(x) _< M for x U, {xn- Hl(Xn) } is a convergent s,equence in E. Since H 1 is demicompact there exists a subsequence S of N + and x with Xn---*x as no in S. Let ncx in S in (3.5) (note HI is continuous) to deduce that x- H l(x) 0. In fact, x U from (3.2). Remark 3.3: We may replace (3.3) in Theorem 3.1 with (I H 1)(U is closed. Also, in Theorem 3.1, the boundedness of U could be replaced by the boundedness of the maps. We now use Theorem 3.1 to obtain a nonlinear alternative of Leray-Schauder type for 1-set contractive demicompact maps. We need the following result from the literature [8, pp. 326-327]. Theorem 3.2: Let E be a Banach space and D be a nonempty, bounded, closed, convex subset of E. Suppose J?D-+D is a continuous, l-set contractive, and dernicompact map. Then J has a fixed point in D. Theorem 3.3: Let E be a Banach space and U be an open bounded subset of E with 0 E U. Suppose G:U---+E is a continuous, l-set contractive, demicompact map, with tG(x) # x for x OU and (0, 1). Then G has a fixed point in U. Proof: We assume G(x) :/: x for x G 0U (otherwise we are finished). Then tG(x) 7k x for x OU and G [ 0, 1 ]. Let H(x, t) tG(x) for (x, t) E U x [ 0, 1 ] and F(x) 0 for x G U. Clearly, (3.1) and (3.3) hold. To see (3.2), suppose it is not true. Then there exist {Xn} C_ OU and a sequence {tn} C_ [ 0,1 ], with xn- tnG(xn)--+O as n--+oo. Without loss of generality, assume tn--+t. Then, (3.6) (3.7) ta(..) x.- t.a(x.)+ (t.If t-0, then xn-O; so 0GOU and this is a contradiction. IftG(0,1], then tG is demicompact (if t- 1 then tG- G, whereas if t G (0, 1) then tG is t-set contractive so demicompact [ 11 ]), so there exists a subsequence {xnl} and a x OU, with xnl x. Also since G is continuous, x-tG(x)- O. This contradicts (3.7). Thus (3.2) holds. We can apply Theorem 3.1 if we show kF is essential. To see this, let k G [ 0,1 ] be fixed and let 0GDMou(U,E with 01oU-kFIoU-O. We must show that there exists x G U with O(x)- x. Let n--d-6(O(U)) and let J" D-D be defined by 0, eU It is easy to see that J: D-D is continuous, 1-set contractive and demicompact. To see demicompactness, suppose {xn} C D and let {xn-J(xn) } be a convergent sequence in n. Then there exists a subsequence {x n } of {x,}, with x, U for each n/ (in whmh case since 0 is demmompact, {xn } las a convergent sdCbsequence) or x t for each n k (in which case {x n )= ?x n -J(xn )} is convergent by the asskumptmn). Theorem 3.2 implies that tere exists x G with J(x)= x. Now if x U we have 0 J(x) x, which is a contradiction, since 0 G U. Thus, x G U so x J(x) O(x). Hence, kF is essential and we may apply Theorem 3.1 to deduce the result. El 4. Demicontinuous (5?) + Maps In this section, E will be a Banach space. E* will denote the conjugate space of E and (., .) the duality between E* and E. Let X be a subset of E. Now (i) f: X-*E* is said to be monotone if (f(x)- f(y),x- y) >_ 0 for all x, y @ X, (ii) f:X-*E* is said to be of class (S)+ if for any sequence (xj) in X, for which xj we--a+kx and limsup(f(xj),xj- x) <_ O, we have xj--+x (hereWe-2k denotes weak convergence), (iii) f: X-+E* is said to be maximal monotone if it is monotone and maximal in the sense of graph inclusion among monotone maps from X to E*, (iv) f: X-*E* is called hemicontinuous if f(x + ty)W22" f(x) as t0, (v) f X+E* is called demicontinuous if y+x imphes f (y) f (x). Throughout this section, E will be a reflexive Banach space. We assume that E is endowed with an equivalent norm, with respect to which, E and E* are locally uniformly convex (this is always possible [ 1 ]). Then there exists a unique mapping (duality mapping) J:E--E* such that (J(x),x) Ix 2 Jx [2 for all x E E. Moreover, J is bijective, bicontinuous, monotone and of class (S)+ (see [1, p. 20]). Throughout this section, E, E* and J will be as above. Also, U will be a nonempty, bounded, open subset of E and T" E--,E* will be a fixed monotone, hemicontinuous, locally bounded mapping. Remark 4.1" From [5, p. 548], T is demicontinuous. Recall that T?X---+E* is locally bounded if un E X, u G X and Un+U imply that Tun is bounded. Remark 4.2: Recall that any monotone hemicontinuous mapping is maximal monotone. Moreover, since T" E-*E* is maximal monotone, then J + T is bijective and (J + T)- 1. E*+E is demicontinuous. Definition 4.1" We let EMou(],E denote the maps f-(J+T)-I(J-F): U-,E, where F?U--E* is demicontinuous, bounded (i.e., maps bounded sets into bounded sets) of class (S)+ with (T + F)(x) 75 0 for x OU. In this case we say f (J + T)- l(j_ F) Eiou(,E ). Definition 4.2: A map f (J + T)- l(j_ F) EMou(,E is essential if for every g (J + T)- l(j_ G) EMou(,E), with G[oU F[OU, there exists x G U with (T + G)(x) O. Theorem 4.1" Let E, E*, U, J and T be as above. Suppose f (J + T)-I(j-F) Eiou(r,E) is essential and H: x[ 0,1 ]E* is bounded with the following properties: and H(x, O)- F(x) for x U {x r?(T + Ht)(x 0 for some (0, 1]} does not intersect OU with tj-*t for which limsup(H .(xj),xj- x)_ 0 we have 3 xjwe--a,k x and H .(xj)+Ht(x (here Ht(x H(x, t)). 3 for any sequence (xj} C U with xj -- x and any sequence (tj} C [ 0,1 ] (4.1) (4.2) (4.3) Then T + H 1 has a fixed point in U. Proofi Let B {x e 5"(T + Ht)(x 0 for some e [ 0,1 ]}. When 0, we have T+ H0 T + F and since (J + T)- l(j etshsaetntBial,is tchleorseed.exisLtest a(nxj)x bEeUa wsietqhuen(cTe+inF)B(x)withO.xjTxhus,UB(#in 0p.artNiecuxltarw,exjswheDo.wk x). Thus, (T + I-I .)(xj)- 0 for some sequence (tj)in [ 0,1 ]. Without loss of generality, assume tj--t. Now since T is monotone, (H .(xj),xj- x) (- Txj + Tx, xj x) + (- Tx, xj x) <_ (- Tx, xj x) and this, together with x jx, gives limsup(Htj(xj),xj -x)<_ O. Now (4.3) implies Htj(xj) tt[x ). This together with Ut3(xJ)+ T(xj)- 0 and T demicontinuous gives Htlx + T(x)= 0. Consequently, B is closed. Also (4.3) and the fact that (J+T)- (g-F)e EMou(r,E) implies B V10G =O. Thus there exists a continuous #" V--.[ 0, 1 ], with #(OV) 0 and #(B) 1. Let R(x) H(x, #(x)). t--(x)]. (T + Ho)(X that We first show R is demicontinuous. Let _(xj) be a sequence in ,U with xjx. We can assume without loss of generality that #(xj),. In fact, #(x), since # is continuous. Since xj--,x, we have limsup(Hu(xj)(xj),x j- x) O. Now (4.3)implies R(xj)- Uv(xj)(xj)xweak,v(x)- R(x) and so R is demicontinuous. To show that R is of class (S)+, let(xj) be a sequence in U with xjWekx and limsup(U(xj)(xj), xj-x) O. Now we may assume without of generality that (xj)t. Now (4.3) implies xjx and Uv(xj)(xj)wkut(x) [since xjx and is continuous, we have Thus, R is of class (S)+. Also, if x e 0U we have (T+R)(x)(T + F)(x) O. Thus (J + T)- l(j R) e EMou(5,E). Next notice RIoU Ho ou F IOU, and since f (J + T)- l(j F) EMou(5,E is essential, there exists x U with (T+R)(x)-O (i.e., (T+Hulx))(x)=0). Thus, xB and so, #(x)-l. Consequently, (T + H1)(x 0 ana we are finished. V1 Theorem 4.2: Let E,E*,U,J and T be as above and suppose f (d + T)-l(j_ F) EMou(U E) is essential. In addition, assume that the following are satisfied: and G:U---E* is demicontinuous, bounded and of class (S) + T(x) + (1 t)F(x) + tG(x) :/: 0 for all (0, 1] and x OU. (4.4) (4.5) x Proof: Let H(x, t) + a)(x) o. t)F(x) + tG(x). Clearly (4.1), (4.2)and (4.3) (see [1, pp. 27, 28] hold. The result is immediate from Theorem 4.1. Yl Theorem 4.3: Let E, E* U,J and T be as above and let G:r+E * be demicontinuous, bounded and of class (S) +. In addition, assume that 0 E U and T(O) 0 (4.6) and T(x) +(1 are satisfied. Then there exists x U with (T + G)(x) O. Proof: Let F- J in Theorem 4.2. 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Ravi P. Agarwal, Donal O'Regan. Nonlinear essential maps of Mönch, 1-set contractive demicompact and monotone (S), International Journal of Stochastic Analysis, DOI: 10.1155/S1048953301000259