Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions
We present new fixed point theorems for weakly sequentially upper semicontinuous maps. These results are then used to establish existence principles for second order differential equations and inclusions.
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irk-123456789-1758322021-02-03T01:29:29Z Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions Agarwal, R.P. O'Regan, D. We present new fixed point theorems for weakly sequentially upper semicontinuous maps. These results are then used to establish existence principles for second order differential equations and inclusions. Наведено новi теореми про нерухому точку для слабко секвенцiйно напiвнеперервних вiдображень. Цi результати застосовано для доведення принципiв iснування для диференцiальних рiвнянь та включень другого порядку. 2002 Article Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions / R.P. Agarwal, D. O'Regan // Нелінійні коливання. — 2002. — Т. 5, № 3. — С. 287-296. — Бібліогр.: 13 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175832 517.988 en Нелінійні коливання Інститут математики НАН України |
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We present new fixed point theorems for weakly sequentially upper semicontinuous maps. These results
are then used to establish existence principles for second order differential equations and inclusions. |
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Article |
| author |
Agarwal, R.P. O'Regan, D. |
| spellingShingle |
Agarwal, R.P. O'Regan, D. Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions Нелінійні коливання |
| author_facet |
Agarwal, R.P. O'Regan, D. |
| author_sort |
Agarwal, R.P. |
| title |
Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions |
| title_short |
Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions |
| title_full |
Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions |
| title_fullStr |
Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions |
| title_full_unstemmed |
Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions |
| title_sort |
fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions |
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Інститут математики НАН України |
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2002 |
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http://dspace.nbuv.gov.ua/handle/123456789/175832 |
| citation_txt |
Fixed point theory for weakly sequentially upper semicontinuous maps with applications to differential inclusions / R.P. Agarwal, D. O'Regan // Нелінійні коливання. — 2002. — Т. 5, № 3. — С. 287-296. — Бібліогр.: 13 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT agarwalrp fixedpointtheoryforweaklysequentiallyuppersemicontinuousmapswithapplicationstodifferentialinclusions AT oregand fixedpointtheoryforweaklysequentiallyuppersemicontinuousmapswithapplicationstodifferentialinclusions |
| first_indexed |
2025-07-15T13:16:45Z |
| last_indexed |
2025-07-15T13:16:45Z |
| _version_ |
1837718995290030080 |
| fulltext |
UDC 517 .988
FIXED POINT THEORY FOR WEAKLY SEQUENTIALLY UPPER
SEMICONTINUOUS MAPS WITH APPLICATIONS
TO DIFFERENTIAL INCLUSIONS
ТЕОРIЯ НЕРУХОМОЇ ТОЧКИ ДЛЯ СЛАБКО СЕКВЕНЦIЙНО
НАПIВНЕПЕРЕРВНИХ ВIДОБРАЖЕНЬ IЗ ЗАСТОСУВАННЯМИ
ДО ДИФЕРЕНЦIАЛЬНИХ ВКЛЮЧЕНЬ
R. P. Agarwal
Florida Institute of Technology
Melbourne, Florida 32901, USA
D. O’Regan
National University of Ireland
Galway, Ireland
We present new fixed point theorems for weakly sequentially upper semicontinuous maps. These results
are then used to establish existence principles for second order differential equations and inclusions.
Наведено новi теореми про нерухому точку для слабко секвенцiйно напiвнеперервних вiдобра-
жень. Цi результати застосовано для доведення принципiв iснування для диференцiальних рiв-
нянь та включень другого порядку.
1. Introduction. The aim of this paper is twofold. First we present new fixed point results for
weakly sequentially upper semicontinuous maps. Secondly we use these results to obtain exi-
stence principles for second order differential inclusions. In the literature (see [1 – 3] and the
references therein) almost all papers establish existence for differential inclusions using the
theory of compact (strong) operators. However it is known [4] that the appropriate Niemytzki
operator in this situation is weakly completely continuous, and it is our opinion that it is more
natural to obtain existence criteria via the theory of weakly sequentially upper semiconti-
nuous maps. With this in mind we establish existence principles in Section 3 using a nonlinear
alternative of Leray – Schauder (or a Furi – Pera fixed point theorem) for weakly sequentially
upper semicontinuous maps. We present the details fully in Section 3, and the reader can see
that the results and ideas extend to higher order differential inclusions or indeed to operator
inclusions where the operator is weakly sequentially upper semicontinuous.
2. Fixed point theory. In this section we present the fixed point theory which will be needed
in Section 3. First we state a fixed point result due to Arino, Gautier and Penot [5].
Theorem 2.1. Let E be a metrizable locally convex linear topological space and let C be a
weakly compact, convex subset of E. Then any weakly sequentially upper semicontinuous map
F : C → K(C) has a fixed point (here K(C) denotes the family of nonempty, convex, weakly
compact subsets of C).
Remark 2.1. Recall F : C → K(C) is weakly sequentially upper semicontinuous if for any
weakly closed set A of C, F−1(A) is sequentially closed for the weak topology on C.
c© R. P. Agarwal, D. O’Regan, 2002
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3 287
288 R.P. AGARWAL, D. O’REGAN
Notice the proof of Theorem 2.1 is immediate from Himmelberg’s fixed point theorem and
the next result [6].
Theorem 2.2. Let E be a metrizable locally convex linear topological space with D a
weakly compact subset of E. If F : D → K(E) is a weakly sequentially upper semicontinuous
map, then F : D → K(E) is a weakly upper semicontinuous map.
Our next result replaces the weak compactness of the space C with a weak compactness
assumption on the operator F .
Theorem 2.3. Let E be a Banach space with C a closed, convex subset of E. Then any
weakly compact, weakly sequentially upper semicontinuous map F : C → K(C) has a fixed
point.
Proof. There exists a weakly compact subset K of C with F (C) ⊆ K ⊆ C. The Krein –
Šmulian theorem [7, p. 434] guarantees that co (K) is weakly compact. Notice also that F :
co (K) → co (K), so Theorem 2.1 guarantees that there exists x ∈ co (K) with x ∈ F (x).
Remark 2.2. In Theorem 2.3, E Banach can be replaced by any metrizable locally convex
linear topological space where the Krein – Šmulian theorem holds; for examples see [8, p. 553,
9, p. 82].
In applications to construct a set C so that F takes C back into C is very difficult and
sometimes impossible. As a result it makes sense to discuss maps F : C → K(E). Our first
result in this direction is the so called nonlinear alternative of Leray – Schauder (see [10]).
Theorem 2.4. Let E be a Banach space, C a closed convex subset of E, U a weakly open
subset of C, 0 ∈ U and Uw weakly compact (here Uw denotes the weak closure of U in C).
Suppose F : Uw → K(C) is a (weakly compact) weakly sequentially upper semicontinuous
map which satisfies the following property:
x /∈ λF x for every x ∈ ∂U and λ ∈ (0, 1); (2.1)
here ∂U denotes the weak boundary of U in C. Then F has a fixed point in Uw.
Proof. Suppose F does not have a fixed point in ∂U (otherwise we are finished), so x /∈
/∈ λF x for every x ∈ ∂U and λ ∈ [0, 1]. Consider
A = {x ∈ Uw : x ∈ t F (x) for some t ∈ [0, 1]}.
Now A 6= ∅ since 0 ∈ U . Also Theorem 2.2 guarantees that F : Uw → K(C) is weakly
upper semicontinuous. Thus A is weakly closed and in fact weakly compact since Uw is weakly
compact.
Also A ∩ ∂U = ∅ so there exists (since (E,w), the space E endowed with the weak
topology, is completely regular) a weakly continuous map µ : Uw → [0, 1] with µ(∂U) = 0
and µ(A) = 1. Let
J(x) =
{
µ(x)F (x), x ∈ Uw;
{0}, x ∈ C \Uw.
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FIXED POINT THEORY FOR WEAKLY SEQUENTIALLY UPPER SEMICONTINUOUS MAPS . . . 289
Clearly J : C → K(C) is a weakly compact, weakly sequentially upper semicontinuous map.
Theorem 2.3 guarantees that there exists x ∈ C with x ∈ J (x). Notice x ∈ U since 0 ∈ U .
As a result x ∈ µ(x)F (x), so x ∈ A. Thus µ(x) = 1 and so x ∈ F (x).
Next we present a Furi – Pera theorem for weakly sequentially upper semicontinuous maps.
This result can be found in [10] (Theorem 3.4); we note that one of the conditions is stated
incorrectly and that the proof there has to be adjusted slightly.
Theorem 2.5. Let E be a separable and reflexive Banach space, C and Q are closed bounded
convex subsets of E with Q ⊆ C and 0 ∈ Q. Suppose F : Q → K(C) is a weakly sequentially
upper semicontinuous map and assume the following condition is satisfied:
if {(xj , λj)}∞1 is a sequence in Q× [0, 1] with
xj ⇀ x ∈ ∂ Q and λj → λ and if x ∈ λF (x) for
0 ≤ λ < 1, then there exists j0 ∈ {1, 2, ...} with
{λj0 F (xj0)} ⊆ Q; here ∂ Q denotes the weak boundary
of Q relative to C and ⇀ denotes weak convergence.
(2.2)
Then F has a fixed point in Q.
Remark 2.3. A special case of (2.2) (which is all we need in Section 3) is the following
condition:
if {(xj , λj)}∞1 is a sequence in Q× [0, 1] with
xj ⇀ x and λj → λ and if x ∈ λF (x) for 0 ≤ λ < 1,
then there exists j0 ∈ {1, 2, ...} with {λj0 F (xj0)} ⊆ Q.
(2.3)
Proof. Let r : E → Q be a weakly continuous retraction (see [11]) and let
B = {x ∈ E : x ∈ F r (x) }.
Note B ⊆ C since F : Q → K(C). It is easy to see that B 6= ∅ is weakly closed and
weakly compact (note C is weakly compact since C is closed and convex (so weakly closed)
and bounded in the norm topology). It remains to show B ∩ Q 6= ∅. To do so we argue by
contradiction. Suppose B ∩Q = ∅. Also since E is separable we know from [7] that the weak
topology on C is metrizable; let d? denote the metric. With respect to (C, d?) note Q is closed,
B is compact, B ∩Q = ∅ so there exists ε > 0 with
d?(B,Q) = inf{d?(x, y) : x ∈ B, y ∈ Q} > ε.
For i ∈ {1, 2, .....} let
Ui =
{
x ∈ C : d?(x,Q) <
ε
i
}
.
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
290 R.P. AGARWAL, D. O’REGAN
Fix i ∈ {1, 2, ...}. Now Ui is d?-open in C, so Ui is weakly open in C. Also
Ui
w = Ui
d? =
{
x ∈ C : d?(x,Q) ≤ ε
i
}
and
∂ Ui =
{
x ∈ C : d?(x,Q) =
ε
i
}
.
Now B ∩ Ui
w = ∅ (since d?(B,Q) > ε) and Theorem 2.4 (with F = F r and U = Ui)
guarantees that there exists λi ∈ (0, 1) and yi ∈ ∂ Ui with yi ∈ λi F r (yi). We can do this
argument for each i ∈ {1, 2, ...}. Notice in particular since yi ∈ ∂ Ui that
{λi F r(yi)} 6⊆ Q for each i ∈ {1, 2, ...}. (2.4)
Now look at
D = {x ∈ E : x ∈ λF r (x) for some λ ∈ [0, 1] }.
Now D is weakly compact (so weakly sequentially compact by the Eberlein – Šmulian theorem).
This together with
d?(yj , Q) =
ε
j
, |λj | ≤ 1 for j ∈ {1, 2, ...},
implies that we may assume without loss of generality that
λj → λ? and yj ⇀ y? ∈ Qw ∩ C \Qw = ∂ Q.
Also since yj ∈ λj F r (yj) we have that y? ∈ λ? F r (y?) (recall F r : C → K(C) is weakly
upper semicontinuous). If λ? = 1 then y? ∈ F r (y?) which contradicts B ∩ Q = ∅. Thus
0 ≤ λ? < 1. But in this case (2.2), with
xj = r(yj) and x = y? = r(y?),
implies there exists j0 ∈ {1, 2, ....} with {λj0 F r (yj0)} ⊆ Q. This contradicts (2.4). Thus B ∩
∩ Q 6= ∅. As a result there exists x ∈ Q with x ∈ F r (x) = F (x).
3. Applications. In this section we present existence principles for the second order di-
fferential inclusion
y′′ ∈ f(t, y, y′) a.e. on [0, 1],
y(0) = y(1) = 0
(3.1)
where f : [0, 1] × R2 → CK(R) is a Lp-Carathéodory function (here p > 1 and CK(R)
denotes the family of nonempty, convex, compact subsets of R); by this we mean
(a) t 7→ f(t, x, y) is measurable for every (x, y) ∈ R2,
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FIXED POINT THEORY FOR WEAKLY SEQUENTIALLY UPPER SEMICONTINUOUS MAPS . . . 291
(b) (x, y) 7→ f(t, x, y) is upper semicontinuous for a.e. t ∈ [0, 1],
and
(c) for each r > 0, ∃ hr ∈ Lp[0, 1] with |f(t, x, y)| ≤ hr(t) for a.e. t ∈ [0, 1] and every
(x, y) ∈ R2 with |x| ≤ r and |y| ≤ r.
This section presents an existence principle for (3.1) using Theorem 2.4 and Theorem 2.5.
We remark that these existence principles could also be established using the theory of compact
(strong) operators (see [3], Chapter 3). However the proofs given in this section have the
advantage of automatically yielding new and general existence principles for nonlinear operator
equations where the operator is weakly sequentially upper semicontinuous. For notational
purposes, for appropriate functions u, let
‖u‖0 = sup
[0,1]
|u(t)|, ‖u‖1 = max{‖u‖0, ‖u′‖0} and ‖u‖Lp =
1∫
0
|u(t)|p dt
1
p
.
Recall W k,p[0, 1], 1 ≤ p < ∞, denotes the space of functions u : [0, 1] → Rn with u(k−1) ∈
∈ AC[0, 1] and u(k) ∈ Lp[0, 1]. Note W k,p[0, 1] is reflexive if 1 < p < ∞.
We begin by using Theorem 2.4 to establish an existence principle for (3.1).
Theorem 3.1. Let f : [0, 1] ×R2 → CK(R) be a Lp-Carathéodory function (1 < p < ∞)
and assume there exists a constant M0 (independent of λ) with ‖y‖1 6= M0 for any solution
y ∈ W 2,p[0, 1] to
y′′ ∈ λ f(t, y, y′) a.e. on [0, 1],
y(0) = y(1) = 0
for 0 < λ < 1. Then (3.1) has a solution in W 2,p[0, 1].
Proof. Since f is Lp-Carathéodory, there exists hM0 ∈ Lp[0, 1] with
|f(t, u, v)| ≤ hM0(t) for a.e. t ∈ [0, 1] and
every (u, v) ∈ R2 with |u| ≤ M0 and |v| ≤ M0.
(3.2)
Let
G(t, s) =
{
(t− 1) s, 0 ≤ s ≤ t ≤ 1;
(s− 1) t, 0 ≤ t ≤ s ≤ 1,
and N = max{N0, N1,M0} where
(
here
1
p
+
1
q
= 1
)
,
N0 = ‖hM0‖Lp sup
t∈[0,1]
1∫
0
|G(t, s)|q ds
1
q
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
292 R.P. AGARWAL, D. O’REGAN
and
N1 = ‖hM0‖Lp sup
t∈[0,1]
1∫
0
|Gt(t, s)|q ds
1
q
.
Also we let
N2 = ‖hM0‖Lp .
We will apply Theorem 2.4 with E = W 2,p[0, 1],
C =
{
u ∈ W 2,p[0, 1] : ‖u‖1 ≤ N and ‖u′′‖Lp ≤ N2
}
and
U =
{
u ∈ W 2,p[0, 1] : ‖u‖1 < M0 and ‖u′′‖Lp ≤ N2
}
.
Now let
F = L ◦Nf : C → 2E
where L : Lp[0, 1] → W 2,p[0, 1] and Nf : W 2,p[0, 1] → 2L
p[0,1] are given by
Ly(t) =
1∫
0
G(t, s) y(s) ds
and
Nf u =
{
y ∈ Lp[0, 1] : y(t) ∈ f(t, u(t), u′(t)) a.e. t ∈ [0, 1]
}
.
Note Nf is well defined since if x ∈ C then [1, 2] guarantees that Nf x 6= ∅.
Notice C is a convex, closed, bounded subset of E. We first show U is weakly open in
C. To do this we will show that C \U is weakly closed. Let x ∈ C \Uw. Then there exists
xn ∈ C \U (see [12, p. 81]) with xn ⇀ x (here W 2,p[0, 1] is endowed with the weak topology
and ⇀ denotes weak convergence). We must show x ∈ C \U . Now since the embedding
j : W 2,p[0, 1] → C1[0, 1] is completely continuous [13], there is a subsequence S of integers
with
xn → x in C1[0, 1] and x′′n ⇀ x′′ in Lp[0, 1]
as n → ∞ in S. Also
‖x‖1 = lim
n→∞
‖xn‖1 and ‖x′′‖Lp ≤ lim inf ‖x′′n‖Lp ≤ N2.
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FIXED POINT THEORY FOR WEAKLY SEQUENTIALLY UPPER SEMICONTINUOUS MAPS . . . 293
Note M0 ≤ ‖x‖1 ≤ N since M0 ≤ ‖xn‖1 ≤ N for all n. As a result x ∈ C \U , so C \Uw =
= C \U . Thus U is weakly open in C. Also
∂ U = {u ∈ C : ‖u‖1 = M0} and Uw = {u ∈ C : ‖u‖1 ≤ M0} .
To see this let x ∈ Uw. Then [12, p. 81] guarantees that there exists xn ∈ U with xn ⇀ x.
Essentially the same reasoning as above yields ‖x‖1 ≤ M0 and ‖x′′‖Lp ≤ N2, so Uw ⊆
⊆ {u ∈ C : ‖u‖1 ≤ M0} ≡ A. On the other hand if x ∈ A (note A is closed), then there
exists xn ∈ U with xn → x in W 2,p[0, 1], so in particular xn ⇀ x in W 2,p[0, 1]. Thus x ∈ Uw,
so Uw = {u ∈ C : ‖u‖1 ≤ M0}.
Next we note Uw is weakly compact (note W 2,p[0, 1] is reflexive). Notice also that F :
Uw → 2C since if y ∈ Uw then from (3.2) we have
‖F y‖0 ≤ ‖hM0‖Lp sup
t∈[0,1]
1∫
0
|G(t, s)|q ds
1
q
= N0,
‖(F y)′‖0 ≤ ‖hM0‖Lp sup
t∈[0,1]
1∫
0
|Gt(t, s)|q ds
1
q
= N1,
and
‖(F y)′′‖0 ≤ ‖hM0‖Lp = N2.
To apply Theorem 2.4 it remains to show F : Uw → K(C) is weakly sequentially upper
semicontinuous. Let A be a weakly closed set in C and let yn ∈ F−1(A) with yn ⇀ y in
W 2,p[0, 1]. To show F is weakly sequentially upper semicontinuous we must show y ∈ F−1(A).
Now since yn ∈ F−1(A), there exists xn ∈ A with xn = F (yn). Since C is weakly compact,
we have that {xn} ⊆ A is relatively weakly compact, and so the Eberlein – Šmulian theorem
[7, p. 430] (together with the fact that A is weakly closed) guarantees that there exists x ∈ A
and a subsequence S0 of integers with xn ⇀ x in W 2,p[0, 1] (as n → ∞ in S0). Now yn ⇀ y
in W 2,p[0, 1] together with the fact that j : W 2,p[0, 1] → C1[0, 1] is completely continuous
implies that there is a subsequence S of S0 with
yn → y in C1[0, 1] (note also x′′n ⇀ x′′ in Lp[0, 1]) (3.3)
as n → ∞ in S. Now xn = F (yn) = L ◦Nf (yn), so
x′′n(t) ∈ f(t, yn(t), y′n(t)) for a.e. t ∈ [0, 1]. (3.4)
Now (3.3), (3.4), together with a result of Pzusko [4] immediately yields
x′′(t) ∈ f(t, y(t), y′(t)) for a.e. t ∈ [0, 1].
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
294 R.P. AGARWAL, D. O’REGAN
As a result y ∈ F−1(A) (note x ∈ A from above). Thus F : Uw → 2C is weakly sequentially
upper semicontinuous, and it is immediate that F : Uw → K(C).
We now apply Theorem 2.4 to deduce our result; note (2.1) is satisfied since if there exists
x ∈ ∂ U and λ ∈ (0, 1) with x ∈ λF x then ‖x‖1 = M0 since x ∈ ∂ U and ‖x‖1 6= M0 by
assumption. Thus F has a fixed point in Uw.
Remark 3.1. It is easy to use the ideas in Theorem 3.1 to obtain an existence principle for
the operator inclusion y ∈ F y on [0, 1], when F is weakly sequentially upper semicontinuous;
we leave the details to the reader.
Our final theorem in this section shows how Theorem 2.5 can be applied to obtain an exi-
stence principle for differential equations and inclusions. For convenience we discuss the di-
fferential equation
y′′ = f(t, y, y′) a.e. on [0, 1],
y(0) = y(1) = 0.
(3.5)
Theorem 3.2. Let f : [0, 1] × R2 → R be a Lp-Carathéodory function (1 < p < ∞)
and assume there exists a constant M0 (independent of λ) with ‖y‖1 ≤ M0 for any solution
y ∈ W 2,p[0, 1] to
y′′ = λ f(t, y, y′) a.e. on [0, 1],
y(0) = y(1) = 0
for 0 < λ < 1. Then (3.5) has a solution in W 2,p[0, 1].
Proof. Since f is Lp-Carathéodory, there exists hr ∈ Lp[0, 1] with
|f(t, u, v)| ≤ hr(t) for a.e. t ∈ [0, 1] and
every (u, v) ∈ R2 with |u| ≤ r and |v| ≤ r;
(3.6)
here r = M0 or r = M0 + 1. Let G(t, s) be as in Theorem 3.1 and
N = max{N2 + 1,K2}, K = max{K0,K1,M0 + 1}
where
K0 = ‖hM0+1‖Lp sup
t∈[0,1]
1∫
0
|G(t, s)|q ds
1
q
,
K1 = ‖hM0+1‖Lp sup
t∈[0,1]
1∫
0
|Gt(t, s)|q ds
1
q
,
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FIXED POINT THEORY FOR WEAKLY SEQUENTIALLY UPPER SEMICONTINUOUS MAPS . . . 295
K2 = ‖hM0+1‖Lp ,
and
N2 = ‖hM0‖Lp .
We will apply Theorem 2.5 with E = W 2,p[0, 1],
Q =
{
u ∈ W 2,p[0, 1] : ‖u‖1 ≤ M0 + 1 and ‖u′′‖Lp ≤ N
}
and
C =
{
u ∈ W 2,p[0, 1] : ‖u‖1 ≤ K and ‖u′′‖Lp ≤ N
}
.
Let F : W 2,p[0, 1] → W 2,p[0, 1] be given by
F u(t) =
1∫
0
G(t, s) f(s, y(s), y′(s)) ds.
Essentially the same reasoning as in Theorem 3.1 guarantees that F : Q → C is weakly
sequentially continuous. We can apply Theorem 2.5 once we show (2.3) holds. Let {(xj , λj)}∞1
be a sequence in Q × [0, 1] with xj ⇀ x, λj → λ and x = λF x for 0 ≤ λ < 1. Now since
the embedding j : W 2,p[0, 1] → C1[0, 1] is completely continuous there is a subsequence S1
of integers with
xj → x in C1[0, 1] and x′′j ⇀ x′′ in Lp[0, 1] (3.7)
as j → ∞ in S1. Also we know F : Q → C is weakly sequentially continuous, so F xj ⇀ F x
in W 2,p[0, 1] . Now since the embedding j : W 2,p[0, 1] → C1[0, 1] is completely continuous
there is a subsequence S2 of S1 with
F xj → F x in C1[0, 1] and (F xj)
′′ → (F x)′′ in Lp[0, 1]
as j → ∞ in S2; note (F xj)
′′ → (F x)′′ in Lp[0, 1] as j → ∞ in S2 follows from (3.7) and
the Lebesgue dominated convergence theorem. Thus for any ε > 0 (with ε <
1
3
say), there
exists j0 ∈ S2 with
‖F xj‖1 ≤ ‖F x‖1 + ε and ‖(F xj)′′‖Lp ≤ ‖(F x)′′‖Lp + ε (3.8)
for j ∈ S2 and j ≥ j0. Also x = λF x together with ‖x‖1 ≤ M0 and (3.6) implies
‖λF x‖1 ≤ M0 and ‖λ (F x)′′‖Lp ≤ N2. (3.9)
Now (3.8), (3.9) and λj → λ implies that there exists j1 ≥ j0, j1 ∈ S2 with
‖λj F xj‖1 ≤ M0 + 1 and ‖λj (F xj)′′‖Lp ≤ N2 + 1 ≤ N
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
296 R.P. AGARWAL, D. O’REGAN
for j ∈ S2, j ≥ j1. Thus λj F xj ∈ Q for j ∈ S2 sufficiently large. The result now follows
from Theorem 2.5.
Remark 3.2. We refer the reader to [11] (Section 3) for another application of Theorem 2.5.
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Received 16.05.2002
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
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