Volume 2010, Article ID 183596, 13 pagesdoi:10.1155/2010/183596 Research Article Fixed Point Theorems for ws-Compact Mappings in Banach Spaces 1 Department of Mathematical Sciences, Flor
Trang 1Volume 2010, Article ID 183596, 13 pages
doi:10.1155/2010/183596
Research Article
Fixed Point Theorems for ws-Compact Mappings in Banach Spaces
1 Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, FL 32901, USA
2 Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia
3 Department of Mathematics, National University of Ireland, Galway, Ireland
4 Universit´e Cadi Ayyad, Laboratoire de Math´ematiques et de Dynamique de Populations,
Marrakech, Morocco
Correspondence should be addressed to Ravi P Agarwal,agarwal@fit.edu
Received 17 August 2010; Revised 21 October 2010; Accepted 4 November 2010
Academic Editor: Jerzy Jezierski
Copyrightq 2010 Ravi P Agarwal et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We present new fixed point theorems for ws-compact operators Our fixed point results are obtained under Sadovskii, Leray-Schauder, Rothe, Altman, Petryshyn, and Furi-Pera type conditions An example is given to show the usefulness and the applicability of our results
1 Introduction
Let X be a Banach space, and let M be a subset of X Following1, a map A : M → X is said to be ws-compact if it is continuous and for any weakly convergent sequencex nn∈Nin
naturally in the study of both integral and partial differential equations see 1 5 In this paper, we continue the study of ws-compact mappings, investigate the boundary conditions, and establish new fixed point theorems Specifically, we prove several fixed point theorems for ws-compact mappings under Sadovskii, Leray-Schauder, Rothe, Altman, Petryshyn and Furi-Pera type conditions Finally, we note that ws-compact mappings are not necessarily sequentially weakly continuousseeExample 2.14 This explains the usefulness of our fixed point results in many practical situations For the remainder of this section, we gather some
notations and preliminary facts Let X be a Banach space, let BX denote the collection of
Trang 2all nonempty bounded subsets of X and WX the subset of BX consisting of all weakly compact subsets of X Also, let B r denote the closed ball centered at 0 with radius r.
In our considerations, the following definition will play an important role
noncompactness if it satisfies the following conditions
1 The family kerψ {M ∈ BX : ψM 0} is nonempty and kerψ is contained
in the set of relatively weakly compact sets of X.
2 M1⊆ M2⇒ ψM1 ≤ ψM2
3 ψcoM ψM, where coM is the closed convex hull of M.
4 ψλM1 1 − λM2 ≤ λψM1 1 − λψM2 for λ ∈ 0, 1.
5 If M nn≥1is a sequence of nonempty weakly closed subsets of X with M1bounded
and M1 ⊇ M2 ⊇ · · · ⊇ M n⊇ · · · such that limn→ ∞ψ M n 0, then M∞ :∞
n1M n
is nonempty
The family ker ψ described in 1 is said to be the kernel of the measure of weak
noncompactness ψ Note that the intersection set M∞ from 5 belongs to ker ψ since
ψ M∞ ≤ ψM n for every n and lim n→ ∞ψ M n 0 Also, it can be easily verified that
the measure ψ satisfies
ψ
where M w is the weak closure of M.
A measure of weak noncompactness ψ is said to be regular if
ψ M 0 if and only if M is relatively weakly compact, 1.2
subadditive if
ψ M1 M2 ≤ ψM1 ψM2, 1.3
homogeneous if
ψ λM |λ|ψM, λ ∈ R, 1.4
set additive (or have the maximum property) if
ψ M1∪ M2 maxψ M1, ψM2. 1.5
Trang 3The first important example of a measure of weak noncompactness has been defined
by De Blasi7 as follows:
w M inf{r > 0 : there exists W ∈ WX with M ⊆ W B r }, 1.6
for each M ∈ BX.
Notice that w· is regular, homogeneous, subadditive, and set additive see 7.
In what follows, let X be a Banach space, C a nonempty closed convex subset of X,
F 1,x0 M FM,
F n,x0 M Fco
F n−1,x0 M ∪ {x0},
1.7
for n 2, 3,
Definition 1.2 Let X be a Banach space, C a nonempty closed convex subset of X, and ψ a
measure of weak noncompactness on X Let F : C → C be a bounded mapping that is
it takes bounded sets into bounded ones and x0 ∈ C We say that F is a ψ-convex-power condensing operator about x0and n0if for any bounded set M ⊆ C with ψM > 0, we have
ψ
Obviously, F : C → C is ψ-condensing if and only if it is ψ-convex-power condensing operator about x0and 1
Kuratowski measure of noncompactness
2 Fixed Point Theorems
Theorem 2.1 Let X be a Banach space, and let ψ be a regular and set additive measure of weak
noncompactness on X Let C be a nonempty closed convex subset of X, x0∈ C, and let n0be a positive integer Suppose that F : C → C is ψ-convex-power condensing about x0and n0 If F is ws-compact and F C is bounded, then F has a fixed point in C.
Proof Let
F {A ⊆ C, coA A, x0∈ A and FA ⊆ A}. 2.1
The setF is nonempty since C ∈ F Set M A∈FA Now, we show that for any positive
integer n we have
M coF n,x0 M ∪ {x0} Pn
Trang 4To see this, we proceed by induction Clearly M is a closed convex subset of C and FM ⊆
coFM ∪ {x0} Consequently, coFM ∪ {x0} ∈ F Hence, M ⊆ coFM ∪ {x0} As a result coFM ∪ {x0} M This shows that P1 holds Let n be fixed, and suppose that
Pn holds This implies Fn1,x0 M FcoF n,x0 M ∪ {x0} FM Consequently,
co
F n1,x0 M ∪ {x0} coFM ∪ {x0} M. 2.2
As a result
co
F n0,x0 M ∪ {x0} M. 2.3
Notice FC is bounded implies that M is bounded Using the properties of the measure of
weak noncompactness, we get
ψ M ψco
F n0,x0 M ∪ {x0} ψF n0,x0 M, 2.4
which yields that M is weakly compact Now, we show that FM is relatively compact To
see this, consider a sequencey nn∈N in FM For each n ∈ N, there exists x n ∈ M with
y n Fx n Now, the Eberlein-Smulian theorem 9, page 549 guarantees that there exists a
subsequence S of N so that x nn ∈S is a weakly convergent sequence Since F is ws-compact,
then Fx nn ∈S has a strongly convergent subsequence Thus, FM is relatively compact Keeping in mind that FM ⊆ M, the result follows from Schauder’s fixed point theorem.
As an easy consequence ofTheorem 2.1, we recapture10, Theorem 3.1
Corollary 2.2 Let X be a Banach space, and let ψ be a regular and set additive measure of weak
a bounded nonweakly compact subset of C, then F has a fixed point.
Theorem 2.3 Let X be a Banach space, and let ψ a measure of weak noncompactness on X Let C be a
i F has a fixed point in U, or
ii there is a u ∈ ∂U (the boundary of U in C) and λ ∈ 0, 1 with u λFu 1 − λp.
finished Then, u / λFu 1 − λp for u ∈ ∂U and λ ∈ 0, 1 Consider
A :x ∈ U : x tFx 1 − tp for some t ∈ 0, 1 . 2.5
Now, A / ∅ since p ∈ U In addition, the continuity of F implies that A is closed Notice that
A ∩ ∂U ∅, 2.6
Trang 5therefore, by Urysohn’s lemma, there exists a continuous μ : U → 0, 1 with μA 1 and
μ ∂U 0 Let
N x
⎧
⎨
⎩
μ xFx 1− μxp, x ∈ U,
It is immediate that N : C → C is continuous Now we show that N is ws-compact To
see this, letx nn∈Nbe a sequence in C which converges weakly to some x ∈ C Without loss
of generality, we may takex nn∈Nin U Notice that μx nn∈Nis a sequence in0, 1 Hence,
by extracting a subsequence if necessary, we may assume thatμx nn∈Nconverges to some
λ ∈ 0, 1 On the other hand, since F is ws-compact, then there exists a subsequence S of N
so thatFx nn ∈S converges strongly to some y ∈ C Consequently, the sequence Nx nn ∈S
converges strongly to λy 1 − λp This proves that N is ws-compact Our next task is to show that N is ψ-convex-power condensing about p and n0 To see this, let S be a bounded subset of C Clearly
N S ⊆ coF S ∪ p
By induction, note for all positive integer n, we have
N n,p S ⊆ coF n,p S ∪ p
Indeed, fix an integer n≥ 1 and suppose that 2.9 holds Then,
N n1,p S Nco
N n,p S ∪ p
⊆ Nco
F n,p S ∪ p
⊆ coF
co
F n,p S ∪ p
∪ p
coF n1,p S ∪ p
.
2.10
In particular, we have
N n0,pS ⊆ coF n0,pS ∪ p
Thus,
ψ
N n0,pS≤ ψco
F n0,pS ∪ p
ψF n0,pS< ψ S. 2.12
This proves that N is ψ-convex-power condensing about p and n0.Theorem 2.1guarantees
the existence of x ∈ C with x Nx Notice that x ∈ U since p ∈ U Thus, x μxFx
1 − μxp As a result, x ∈ A, and therefore μx 1 This implies that x Fx.
Trang 6Remark 2.4. Theorem 2.3is a sharpening of10, Theorem 4.1.
Lemma 2.5 see 11 Let Q be a closed convex subset of a Banach space X with 0 ∈ intQ Let μ
be the Minkowski functional defined by
μ x inf{λ > 0 : x ∈ λQ}, 2.13
for all x ∈ X Then,
i μ is nonnegative and continuous on X.
ii For all λ ≥ 0 we have μλx λμx.
iii μx 1 if and only if x ∈ ∂Q.
iv 0 ≤ μx < 1 if and only if x ∈ intQ.
v μx > 1 if and only if x /∈ Q.
Lemma 2.6 Let X be a Banach space, ψ a set additive measure of weak noncompactness on X, and Q
a closed convex subset of X with 0 ∈ intQ Let μ be the Minkowski functional defined in Lemma 2.5, and, r be the map defined on X by
r x x
max 1, μx for x ∈ X. 2.14
Then,
i r is continuous, rX ⊆ Q and rx x for all x ∈ Q.
ii For any subset A of X we have rA ⊆ coA ∪ {0}.
iii For any bounded subset A of X we have ψrA ≤ ψA.
Lemma 2.5ii, we get
μ rx μ x
max 1, μx ≤ 1. 2.15 This implies that rx ∈ Q The last statement follows easily fromLemma 2.5v Now, we proveii To this end, let A be a subset of X, and let x ∈ A Then,
r x x
max 1, μx
1 max 1, μx x
1− 1 max 1, μx
0
∈ coA ∪ {0}.
2.16
Thus, rA ⊆ coA ∪ {0} Using the properties of a measure of weak noncompactness, we get
ψ rA ≤ ψcoA ∪ {0} ψA ∪ {0} ψA. 2.17 This provesiii
Trang 7Theorem 2.7 Let X be a Banach space, and let ψ a regular set additive measure of weak
noncompactness on X Let Q be a closed convex subset of X with 0 ∈ Q, and let n0a positive integer.
is bounded and
if x j , λ j
is a sequence in ∂Q × 0, 1 converging to x, λ with
x λFx and 0 < λ < 1, then λ j F
x j
∈ Q for j sufficiently large 2.18
holding Also, suppose the following condition holds:
Then, F has a fixed point.
B {x ∈ X : x Frx}. 2.20
We first show that B / ∅ To see this, consider rF : Q → Q First, notice that rFQ
is bounded since FQ is bounded and rFQ ⊆ coFQ ∪ {0} Clearly, rF is continuous, since F and r are continuous Now, we show that rF is ws-compact To see this, let x nn∈Nbe
a sequence in Q which converges weakly to some x ∈ Q Since F is ws-compact, then there exists a subsequence S of N so that Fx nn ∈S converges strongly to some y ∈ X The continuity
of r guarantees that the sequence rFx nn ∈S converges strongly to ry This proves that rF is ws-compact Our next task is to show that rF is ψ-convex-power condensing about 0 and n0
To do so, let A be a subset of Q In view of2.19, we have
rF 1,0 A rFA rF 1,0 A ⊆ coF 1,0 A ∪ {0}. 2.21 Hence,
rF 2,0 A rFco
rF 1,0 A ∪ {0}
rFco
rF 1,0 A ∪ {0}
⊆ rFco
F 1,0 A ∪ {0}
rF 2,0 A,
2.22
Trang 8and by induction
rF n0,0A rFco
rF n0−1,0 A ∪ {0}
⊆ rFco
rF n0−1,0 A ∪ {0}
⊆ rFco
F n0−1,0 A ∪ {0}
rF n0,0A.
2.23
Taking into account the fact that F is ψ-convex-power condensing about 0 and n0 and using
2.19, we get
ψ
rF n0,0A≤ ψrF n0,0A≤ ψco
F n0,0A ∪ {0}
≤ ψF n0,0A< ψ A,
2.24
whenever ψA > 0 InvokingTheorem 2.1, we infer that there exists y∈ Q with rFy y Let z Fy, so Frz FrFy Fy z Thus, z ∈ B and B / ∅ In addition, B is closed, since Fr is continuous Moreover, we claim that B is compact To see this, first notice
B ⊆ FrB ⊆ FB
F 1,0
where B coB ∪ {0} Thus,
B ⊆ FrB ⊆ FrF
⊆ Fco
F
∪ {0}
F 2,0
,
2.26
and by induction
B ⊆ FrB ⊆ FrF n0−1,0
⊆ Fco
F n0−1,0
∪ {0}
F n0,0
,
2.27
Now, if ψB / 0, then
ψ B ≤ ψF n0,0
< ψ
ψB, 2.28
which is a contradiction Thus, ψB 0 and so B is relatively weakly compact Now, 2.19 guarantees that rB is relatively weakly compact Now, we show that FrB is relatively
Trang 9compact To see this, let y nn∈N be a sequence in FrB For each n ∈ N, there exists
x n ∈ rB with y n Fx n Since rB is relatively weakly compact, then, by extracting a
subsequence if necessary, we may assume that x nn∈N is a weakly convergent sequence
Now, F is ws-compact implies that y nn∈N has a strongly convergent subsequence This
proves that FrB is relatively compact From 2.25, it readily follows that B is relatively compact Consequently, B B is compact We now show that B ∩ Q / ∅ To do this, we argue
by contradiction Suppose that B ∩ Q ∅ Then, since B is compact and Q is closed, there exists δ > 0 with distB, Q > δ Choose N ∈ {1, 2, } such that Nδ > 1 Define
U i
x ∈ X : dx, Q <1
i
for i ∈ {N, N 1, }, 2.29
here dx, Q inf{x − y : y ∈ Q} Fix i ∈ {N, N 1, } Since distB, Q > δ, then
B ∩U i ∅ Now, we show that Fr : U i → X is ws-compact To see this, let x nn∈Nbe a weakly
convergent sequence in U i Then, the set S : {x n : n∈ N} is relatively weakly compact and so
ψ S 0 In view of 2.19, we infer that ψrS 0 and so rS is relatively weakly compact.
By extracting a subsequence if necessary, we may assume thatrx nn∈Nis weakly convergent
Now, F is ws-compact implies that Frx nn∈Nhas a strongly convergent subsequence This
proves that Fr is ws-compact Our next task is to show that Fr is ψ-convex-power condensing about 0 and n0 To see this, let A be a bounded subset of U i and set A coA ∪ {0} Then,
keeping in mind2.19, we obtain
Fr 1,0 A ⊆ FA
,
Fr 2,0 A Frco
Fr 1,0 A ∪ {0}
⊆ FrcoFA ∪ {0}
⊆ FcoFA ∪ {0}
F 2,0 A ,
2.30
and by induction,
Fr n0,0A Frco
Fr n0−1,0 A ∪ {0}
⊆ Frco
F n0−1,0
∪ {0}
⊆ Fco
F n0−1,0
∪ {0}
F n0,0
.
2.31
Thus,
ψ
Fr n0,0A≤ ψF n0,0
< ψ
ψA, 2.32
Trang 10whenever ψ A / 0 ApplyingTheorem 2.3to Fr : U i → X, we may deduce that there exists
y i , λ i ∈ ∂U i × 0, 1 with y i λ i Fr y i Notice in particular since y i ∈ ∂U i × 0, 1 that
λ i Fr
y i
/
∈ Q for i ∈ {N, N 1, }. 2.33
We now consider
D {x ∈ X : x λFrx, for some λ ∈ 0, 1}. 2.34
Clearly, D is closed since F and r are continuous Now, we claim that D is compact To
see this, first notice
D ⊆ FrD ∪ {0}. 2.35 Thus,
D ⊆ FrD ∪ {0} ⊆ FrcoFrD ∪ {0} ∪ {0} Fr 2,0 ∪ {0}, 2.36 and by induction
D ⊆ FrD ∪ {0} ⊆ Frco
Fr n0−1,0 D ∪ {0}∪ {0} Fr n0,0∪ {0}, 2.37 consequently
ψ D ≤ ψFr n0,0∪ {0}≤ ψFr n0,0
. 2.38
Since Fr is ψ-convex-power condensing about 0 and n0, then ψD 0, and so D is relatively
weakly compact Now,2.19 guarantees that rD is relatively weakly compact Now, we
show that FrD is relatively compact To see this, let y nn∈Nbe a sequence in FD For each
n ∈ N, there exists x n ∈ rD with y n Fx n Since rD is relatively weakly compact then,
by extracting a subsequence if necessary, we may assume thatx nn∈Nis a weakly convergent
sequence Now, F is ws-compact implies that y nn∈Nhas a strongly convergent subsequence
This proves that FrD is relatively compact From 2.35, it readily follows that D is relatively compact Consequently, D D is compact Then, up to a subsequence, we may assume that
λ i → λ∗ ∈ 0, 1 and y i → y∗ ∈ ∂U i Hence, λ i Fr y i → λ∗Fr y∗, and therefore y∗
λ∗Fr y∗ Notice λ∗Fr y∗ /∈ Q since y∗ ∈ ∂U i Thus, λ∗/ 1 since B ∩ Q ∅ From assumption
2.18, it follows that λi Fr y i ∈ Q for j sufficiently large, which is a contradiction Thus,
B ∩ Q / ∅, so there exists x ∈ Q with x Frx, that is, x Fx.
Lemma 2.6 Clearly rz ∈ ∂Q for z ∈ X \ Q and rD ⊆ coD ∪ {0} for any bounded subset
D of X.