NONLINEAR ESSENTIAL MAPS OF MONCH , 1-SET CONTRACTIVE DEMICOMPACT AND MONOTONE ( S ) + TYPE

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


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 de- gree, 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 essen- tial 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).

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 M ou(U,E is an essential map and H:U x[0,1]---E is a continuous map with the following properties: o)-u (2.1) (I-Ht)(x 7 0 for any x OU and t (0, 1] (here Ht(x H(x,t)) and 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).(

2.3)
Then H 1 has a fixed point in U.

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 2.2: 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. Theorem 2.3: Let E be a Banach space, U an open subset of E and 0 U. Suppose G: U--+E is a continuous map, which satisfies MSnch's condition and assume 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]. ( ) 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 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 01o u-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) O(x).and we may apply Theorem 2.1 to deduce the result.
Hence, F is essential 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 contrac- tive 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 noncom- pactness).F is demicompact if each sequence {xn} C_ U has a convergent sub- sequence {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 contrac- tive, demicompact maps F: U--.E, with (I-F)(x) :/: 0 for x c0U.

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 con- tinuous, 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 (3.1) there exists 6 > O with (I-Ht)(x) >_ 6 for x G OU and t G [O, 1] (3.2) and HI"U--E is a demicompact map. (3.3) Then H 1 has a fixed point in U.
We first show that there exists an x G U, with x k Hkl(Xk)(here H1 k kill).
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 H I is continuous) to deduce that x-H l(x) 0. In fact, x U from (3.2).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 litera- ture [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 derni- compact 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 (3.6) tG(x) 7k x for x OU and G [0, 1]. (3.7) 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 x n-tnG(xn)--+O as n--+oo.Without loss of generality, assume tn--+t.Then, 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 0, 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, {x n } las a convergent sdCbsequence) or x t for each n k (in which case {x n )= 'xn -J(x n )} is convergent by the k assumptmn).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

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, f:X-*E* is said to be of class (S)+ if for any sequence (xj) in X, for which xj weak --+ x and limsup(f(xj),xj-x) <_ O, we have xj--+x (hereWe-2 k denotes weak convergence), 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*, f: X-*E* is called hemicontinuous if f(x + ty)W22 " f(x) as t0, 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 non- empty, bounded, open subset of E and T" E--,E* will be a fixed monotone, hemicon- tinuous, locally bounded mapping.Remark 4.1" From [5, p. 548], T is demicontinuous.Recall that T'X---+E* is locally bounded if u n E X, u G X and Un+U imply that Tu n 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[o U F[OU, there exists x G U with (T + G)(x) O. Let E, E*, U, J and T be as above.

(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+ H 0 T + F and since (J + T)-l(j F) EMou(/,E) is essential, there exists an x E U with (T + F)(x) O. Thus, B # 0. Next we show weD.k that B is closed.Let (xj) be a sequence in B with xjx U (in particular, xj x).