#### Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach

Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
RAVI P. AGARWAL
JEWGENI H. DSHALALOW
DONAL O'REGAN
New Leray-Schauder alternatives are presented for Mo¨nch-type maps defined between Fre´chet spaces. The proof relies on viewing a Fre´chet space as the projective limit of a sequence of Banach spaces. This paper presents new Leray-Schauder alternatives for Mo¨nch-type maps defined between Fre´chet spaces. Two approaches [1, 2, 3, 6, 7] have recently been presented in the literature both of which are based on the fact that a Fre´chet space can be viewed as a projective limit of a sequence of Banach spaces {En}n∈N (here N = {1, 2, . . .}). Both approaches are based on constructing maps Fn defined on subsets of En whose fixed points converge to a fixed point of the original operator F. Both approaches have advantages and disadvantages over the other [1] and in this paper, we combine the advantages of both approaches to present very general fixed point results. Our theory in particular extends and improves the theory in [3] (in [3], the single-valued case was discussed). Finally in this section, we gather together some definitions and a fixed point result which will be needed in Section 2. Now, let I be a directed set with order ≤ and let {Eα}α∈I be a family of locally convex spaces. For each α ∈ I, β ∈ I for which α ≤ β, let πα,β : Eβ → Eα be a continuous map. Then the set
1. Introduction
(1.1)
α∈I
is a closed subset of α∈I Eα and is called the projective limit of {Eα}α∈I and is denoted
by lim← Eα (or lim←{Eα, πα,β} or the generalized intersection [4, page 439] α∈I Eα).
Next, we recall a fixed point result from the literature [9] which we will use in Section 2.
Theorem 1.1. Let K be a closed convex subset of a Banach space X, U a relatively open
subset of K , x0 ∈ U, and suppose that F : U → CK (K ) is an upper semicontinuous map
(here CK (K ) denotes the family of nonempty convex compact subsets of K ). Also assume
that the following conditions hold:
M ⊆ U,
C ⊆ M
M ⊆ co
x0 ∪ F(M)
with M = C,
countable, implies M is compact,
x ∈/ (1 − λ) x0 + λFx for x ∈ U \ U, λ ∈ (0, 1).
Then there exist a compact set
of U and an x ∈
with x ∈ Fx.
Remark 1.2. In [9], we see that we could take
to be
y ∈ U : y ∈ (1 − λ) x0 + λF y for some λ ∈ [
0, 1
] .
We did not show that is compact in [9] but this is easy to see as we will now
show. First, notice that is closed since F is upper semicontinuous. Now let {yn}1∞
be a sequence in . Then there exists {tn}1∞ in [
0, 1
] with yn ∈ (1 − tn){x0} + tnF yn for
n ∈ N = {1, 2, . . .}. Without loss of generality, assume that tn → t ∈ [
0, 1
]. Let C = {yn}1∞.
Notice that C is countable and C ⊆ co({x0} ∪ F(C)). Now (1.2) with M = C guarantees
that C is compact (so sequentially compact). Thus there exist a subsequence N1 of N and
a y ∈ C with yn → y as n → ∞ in N1. This together with yn ∈ (1 − tn){x0} + tnF yn and the
upper semicontinuity of F guarantees that y ∈ (1 − t){x0} + tF y, so y ∈ = .
Consequently, is sequentially compact and hence compact. In fact, one could also of course
take to be
{y ∈ U : y ∈ F y}
for the compact set in Theorem 1.1.
2. Projective limit approach
Let E = (E, {| · |n}n∈N) be a Fre´chet space with the topology generated by a family of
seminorms {| · |n : n ∈ N}. We assume that the family of seminorms satisfies
|x|1 ≤ |x|2 ≤ |x|3 ≤ · · ·
for every x ∈ E.
To E, we associate a sequence of Banach spaces {(En, | · |n)} described as follows. For
every n ∈ N, we consider the equivalence relation ∼n defined by
x ∼n y iff |x − y|n = 0.
We denote by En = (E/∼n, | · |n) the quotient space, and by (En, | · |n) the completion of
En with respect to | · |n (the norm on En induced by | · |n and its extension to En are still
denoted by | · |n). This construction defines a continuous map µn : E → En. Now since
(2.1) is satisfied, the seminorm | · |n induces a seminorm on Em for every m ≥ n (again
this seminorm is denoted by | · |n). Also (2.2) defines an equivalence relation on Em
from which we obtain a continuous map µn,m : Em → En since Em/ ∼n can be regarded as
(1.2)
(1.3)
(1.4)
(1.5)
(2.1)
(2.2)
a subset of En. We now assume that the following condition holds:
for each n ∈ N, there exist a Banach space En, | · |n
and an isomorphism (between normed spaces) jn : En −→ En.
(2.3)
Remark 2.1. (i) For convenience, the norm on En is denoted by | · |n.
(ii) In our applications, En = En for each n ∈ N.
(iii) Note that if x ∈ En (or En), then x ∈ E. However if x ∈ En, then x is not necessarily
in E and in fact En is easier to use in applications as we will see in Theorem 2.3 (even
though En is isomorphic to En).
Finally, we assume that
E1 ⊇ E2 ⊇ · · ·
and for each n ∈ N,
|x|n ≤ |x|n+1
∀x ∈ En+1.
(2.4)
Let lim← En (or 1∞ En, where 1∞ is the generalized intersection [
4
]) denote the projective
limit of {En}n∈N (note that πn,m = jnµn,m jm−1 : Em → En for m ≥ n) and note that lim← En ∼=
E, so for convenience we write E = lim← En.
For each X ⊆ E and each n ∈ N, we set Xn = jnµn(X) and we let Xn and ∂Xn denote,
respectively, the closure and the boundary of Xn with respect to | · |n in En. Also the
pseudointerior of X is defined by [
2
]
pseudo − int(X) = x ∈ X : jnµn(x) ∈ Xn \ ∂Xn for every n ∈ N .
(2.5)
Our main result in this paper is the extension of Theorem 1.1 to an applicable result
in the Fre´chet space setting (we refer the reader to [
1
]; in applications, usually the set U
is bounded and as a result has empty interior in the nonnormable situation).
Theorem 2.2. Let E and En be as described above and let F : X → 2E, where X ⊆ E (here
2E denotes the family of nonempty subsets of E). Suppose that the following conditions are
satisfied:
x0 ∈ pseudo − int(X),
for each n ∈ N,
F : Xn −→ CK En is an upper semicontinuous map,
for each n ∈ N,
M ⊆ Xn with M ⊆ co
with M = C and C ⊆ M countable, implies that M is compact
Proof. Fix n ∈ N. Let n = {x ∈ Xn : x ∈ Fx in En}. Now Theorem 1.1 (note that (2.6)
implies that jnµn(x0) ∈ Xn \ ∂Xn) guarantees that there exists yn ∈ n with yn ∈ F yn. We
look at {yn}n∈N. Now y1 ∈ 1. Also yk ∈ 1 for k ∈ N \ {1} since yk ∈ X1 from (2.10)
(see also (2.4)). As a result, yn ∈ 1 for n ∈ N and since 1 is compact (see Remark 1.2),
there exist a subsequence N1 of N and a z1 ∈ 1 with yn → z1 in E1 as n → ∞ in N1 .
Let N1 = N1 \ {1}. Now yn ∈ 2 for n ∈ N1 so there exist a subsequence N2 of N1
and a z2 ∈ 2 with yn → z2 in E2 as n → ∞ in N2 . Note from (2.4) that z2 = z1 in E1
since N2 ⊆ N1. Let N2 = N2 \ {2}. Proceed inductively to obtain subsequences of
integers
N1 ⊇ N2 ⊇ · · · ,
and zk ∈ k with yn → zk in Ek as n → ∞ in Nk . Note that zk+1 = zk in Ek for k ∈ {1, 2, . . .}.
Also let Nk = Nk \ {k}.
Fix k ∈ N. Let y = zk in Ek. Notice that y is well defined and y ∈ lim← En = E. Now
yn ∈ F yn in En for n ∈ Nk and yn → y in Ek as n → ∞ in Nk (since y = zk in Ek)
together with the fact that F : Xk → CK (Ek) is upper semicontinuous (note that yn ∈ k
for n ∈ Nk) imply that y ∈ F y in Ek. We can do this for each k ∈ N so as a result, we have
y ∈ F y in E.
Next, we present an application of Theorem 2.2. We discuss the differential equation
y (t) = f t, y(t)
a.e. t ∈ [0, T),
y(0) = y0 ∈ R,
ρn(u) = sup
t∈[0,tn]
u(t) ,
(2.12)
(2.13)
where 0 < T ≤ ∞ is fixed. First we introduce some notation. If u ∈ C[0, T), then for every
n ∈ N, we define the seminorms ρn(u) by
where tn ↑ T. Note that C[0, T) is a locally convex linear topological space. The topology
on C[0, T), induced by the seminorms {ρn}n∈N, is the topology of uniform convergence
on every compact interval of [0, T).
Recall that a function g : [a, b] × R → R is an L1-Carathe´odory function if
(a) the map t → g(t, y) is measurable for all y ∈ R,
(b) the map y → g(t, y) is continuous for a.e. t ∈ [a, b].
Now, g : [a, b] × R → R is said to be an Lp-Carathe´odory function (1 ≤ p ≤ ∞) if g is a
Carathe´odory function and
(c) for any r > 0, there exists µr ∈ Lp[a, b] such that |y| ≤ r implies that |g(t, y)| ≤
µr (t) for a.e. t ∈ [a, b].
Finally, a function g : [0, T) × R → R is an Llpoc-Carathe´odory function if (a), (b), and
(c) above hold when g is restricted to [0, tn] × R for any n ∈ N.
Theorem 2.3. Suppose that the following conditions are satisfied:
for each n ∈ N, the problem
v (t) = g t, v(t) , a.e. t ∈ 0, tn ,
v(0) = y0
has a maximal solution rn(t) on 0, tn here rn ∈ C 0, tn .
Then (2.12) has at least one solution y ∈ C[0, T).
Remark 2.4. One could also obtain a multivalued version of Theorem 2.3 (with (2.12)
replaced by a differential inclusion) by using the ideas in the proof below with the ideas
in [
6
].
Proof. Here E = C[0, T), Ek consists of the class of functions in E which coincide on the
interval [0, tk], Ek = C[0, tk] with of course πn,m = jnµn,m jm−1 : Em → En for m ≥ n defined
by πn,m(x) = x|[0,tn]. We will apply Theorem 2.2 with
X = u ∈ C[0, T) : |u|n ≤ wn for each n ∈ N ;
here |u|n = supt∈In |u(t)|, where In = [0, tn] and wn = supt∈In rn(t) + 1. On any interval
In = [0, tn] (n ∈ N), we let F on C(In) be defined by
Fix n ∈ N. Notice that
F y(t) = y0 +
f s, y(s) ds.
t
0
Xn = u ∈ C 0, tn : |u|n ≤ wn .
Clearly, (2.6) holds with x0 = 0 and a standard argument from the literature [8]
guarantees that
F : Xn −→ En is continuous and compact,
so (2.7) and (2.8) hold.
To show that (2.9), fix n ∈ N and let y ∈ C(In) be such that y = λF y for λ ∈ (0, 1). We
claim |y|n < wn and if this is true, then y ∈/ ∂Xn and hence (2.9) is true. Let t ∈ In and we
now show that |y(t)| < wn. If |y(t)| ≤ |y0|, we are finished so it remains to discuss the
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
case when |y(t)| > |y0|. In this case, there exists a ∈ [0, t) with
Also
so
for s ∈ (a, t], y(a) = y0 .
y(s)
≤ y (s) ≤ g s, y(s)
a.e. on (a, t),
y(s)
≤ g s, y(s) , a.e. on (a, t),
y(a) = y0 .
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
Now a standard comparison theorem for ordinary differential equations in the real case
[5, Theorem 1.10.2] guarantees that |y(s)| ≤ rn(s) for s ∈ [a, t], so in particular |y(t)| ≤
rn(t) < wn, so (2.9) is true.
It remains to show that (2.10). To see this, fix n ∈ {2, 3, . . .} and suppose that y ∈ Xn
solves
Next, fix k ∈ {1, . . . , n − 1}. We must show that y ∈ Xk. Now since tn ↑ T, notice that
[0, tk] ⊆ [0, tn] so as a result,
y (t) = f t, y(t) , a.e. on 0, tn ,
y(0) = y0.
y (t) = f t, y(t) , a.e. on 0, tk ,
y(0) = y0.
Let t ∈ [0, tk] and essentially the same argument as above guarantees that |y(t)| < wk so
|y|k < wk. Thus y ∈ Xk and (2.10) holds.
The result now follows immediately from Theorem 2.2.
Our final result was motivated by Urysohn-type operators.
Theorem 2.5. Let E and En be as described in the beginning of Section 2 and let F : X → 2E,
where X ⊆ E. Suppose that the following conditions are satisfied:
x0 ∈ pseudo − int(X),
X1 ⊇ X2 ⊇ · · · ,
for each n ∈ N,
Fn : Xn −→ CK En is upper semicontinuous,
for each n ∈ N,
M ⊆ Xn
with M ⊆ co
y ∈/ (1 − λ) jnµn x0 + λFn y in En
∀λ ∈ (0, 1), y ∈ ∂Xn,
for each n ∈ N, the map
n : Xn −→ 2En , given by
n(y) =
∞
m=n
Fm(y) (see Remark 2.6), satisfies that
if C ⊆ Xn is countable with C ⊆
n(C), then C is compact,
if there exist a w ∈ X and a sequence yn n∈N with yn ∈ Xn and yn ∈ Fn yn in En
such that for every k ∈ N there exists a subsequence S ⊆ {k + 1, k + 2, . . .} of N
with yn −→ w in Ek as n −→ ∞ in S, then w ∈ Fw in E.
(2.31)
(2.32)
Then F has a fixed point in X.
Remark 2.6. The definition of n is as follows. If y ∈ Xn and y ∈/ Xn+1, then n(y) =
Fn(y), whereas if y ∈ Xn+1 and y ∈/ Xn+2, then n(y) = Fn(y) ∪ Fn+1(y), and so on.
Proof. Fix n ∈ N. Let n = {x ∈ Xn : x ∈ Fnx in En}. Now, Theorem 1.1 guarantees that
there exists yn ∈ n with yn ∈ Fn yn in En. We look at {yn}n∈N. Note that yn ∈ X1 for n ∈
N from (2.27). In addition with C = {yn}1∞, we have from assumption (2.31) that C(⊆ E1)
is compact; note that yn ∈ 1(yn) in E1 for each n ∈ N. Thus there exist a subsequence
N1 of N and a z1 ∈ X1 with yn → z1 in E1 as n → ∞ in N1 . Let N1 = N1 \ {1}. Proceed
inductively to obtain subsequences of integers
and zk ∈ Xk with yn → zk in Ek as n → ∞ in Nk . Note that zk+1 = zk in Ek for k ∈ N. Also
let Nk = Nk \ {k}.
Fix k ∈ N. Let y = zk in Ek. Notice that y is well defined and y ∈ lim← En = E. Now
yn ∈ Fn yn in En for n ∈ Nk and yn → y in Ek as n → ∞ in Nk (since y = zk in Ek) together
with (2.32) imply that y ∈ F y in E.
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D. O'Regan and R. Precup , Fixed point theorems for set-valued maps and existence principles for integral inclusions , J. Math. Anal. Appl . 245 ( 2000 ), no. 2 , 594 - 612 . Ravi P. Agarwal: Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901 -6975, USA E-mail address: Jewgeni H. Dshalalow: Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901 -6975, USA E-mail address: Donal O'Regan : Department of Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland E-mail address: donal. Volume 2014 Journal of Hindawi Publishing Corporation ht p:/ www .hindawi.com