Liminf and limsup contractions

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PII. S0161171201007086 http://ijmms.hindawi.com © Hindawi Publishing Corp.

LIMINF AND LIMSUP CONTRACTIONS

ALBERTO BRANCIARI

(Received 15 March 2001)

Abstract.We give some theorems related to the contraction mapping principle of Banach-Caccioppoli and Edelstein. The contractive conditions we consider involve the quantities lim inf_ξ_→·d(ξ, f ξ)and lim sup_ξ_→·d(ξ, f ξ)instead ofd(·, f·). Some examples are provided to show the diﬀerence between our results and the classical ones.

2000 Mathematics Subject Classiﬁcation. 54H25, 47H10.

1. Introduction. One of the simplest and most useful results in the ﬁxed point theory is the Banach-Caccioppoli contraction mapping principle (see [1,2]), which in the general setting of complete metric spaces reads as follows.

Theorem _1.1. _Let _{(X, d)}_{be a complete metric space,}_f _:_X →_X _{a mapping and}

c∈[0,1[such that

d(f x, f y)≤cd(x, y), ∀x, y∈X; (1.1)

then

(i) there exists a pointa∈Xsuch that for eachx∈X,limn→+∞fnx=a;

(ii) ais the unique ﬁxed point forf;

(iii) for eachx∈X,d(fn_{x, a)}_≤_cn_/(₁₋_{c)d(x, f x).}

In 1962, in the case of compact metric spaces, Edelstein in [3] has proved the fol-lowing generalization of the contraction mapping principle.

Theorem_1.2. _Let_{(X, d)}_{be a compact metric space and}_f _:_X→_X_{be a mapping}

such that

d(f x, f y) < d(x, y), ∀x, y∈X, x≠y; (1.2)

then there exists a unique ﬁxed point forf.

The main results of this paper are related to Theorems1.1and 1.2 in which we have replaced the contractive conditions above by similar ones using the mappings φ(·):=lim infξ→·d(ξ, f ξ)andψ(·):=lim supξ→·d(ξ, f ξ)instead ofd(·, f·).

Before stating our theorems we need some notations and deﬁnitions: by Z, Z+,

R, andR+ _{we denote, respectively, the sets of integers, nonnegative integers, real}

numbers and nonnegative real numbers; let nowXbe the cluster set ofX andϕ: X→R+ be a real-valued mapping, thenϕ is called (weak) lower semicontinuous at x∈Xif and only if

ϕ(x)≤lim inf ξ→x ϕ(ξ)

ϕ(x)≤lim sup ξ→x

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if this happens for allx∈X then we simply say thatϕis a (weak) lower semicon-tinuousmapping. Finally, taking a mappingf:X→Xthenϕis said to bef-orbitally (weak) lower semicontinuous at a∈X if and only if for eachx∈X and for each se-quence(ξn)n∈Z+ inO∞(x):= {fnx;n∈Z+}(theorbitofx) converging toa, one has

ϕ(a)≤lim inf n→+∞ ϕ

ξn ϕ(a)≤lim sup n→+∞ ϕ

ξn. (1.4)

2. Main results. We are now ready to state and prove our results; the ﬁrst two are related toTheorem 1.1, while the third one is related toTheorem 1.2.

Theorem2.1. Let(X, d)be a complete metric space such thatX≠∅and letf: X→Xbe a mapping such thatf (X)⊆X. Suppose that there exists a pointx∈X such that

lim inf

ξ→x d(ξ, f ξ) <+∞ (2.1)

and that the mappingϕ(·):=d(·, f·)is lower semicontinuous. Finally, letc∈[0,1[be such that

lim inf

η→f x d(η, f η)≤clim infξ→x d(ξ, f ξ) ∀x∈X

_; _(2.2)

then for allx∈Xsatisfying (2.1),

(i) there exists a pointa∈Xsuch thatfn_x_→_a_as_n_{→ +∞}_; (ii) f a=a;

(iii) d(fn_{x, a)}_≤_cn_/(₁₋_c)_{lim inf}

ξ→xd(ξ, f ξ).

Proof. _Let_x∈_X_{be such that (2.1) holds and consider the sequence}_(fn_x)n_∈_Z+, thus forn∈Z+_{we have}

lim inf

ξ→fn_xd(ξ, f ξ)≤c _ξlim inf_→_fn−1xd(ξ, f ξ)≤ ··· ≤c n_{lim inf}

ξ→x d(ξ, f ξ) <+∞, (2.3)

further, by the lower semicontinuity ofϕ, one has

dfn_{x, f}n+1_x_≤_{lim inf}

ξ→fn_xd(ξ, f ξ)≤c

n_{lim inf}

ξ→x d(ξ, f ξ), (2.4)

thus

+∞

n=0

dfnx, fn+1x≤lim inf

ξ→x d(ξ, f ξ) +∞

n=0

cn<+∞; (2.5)

this implies that(fn_x)n

∈Z+ is a Cauchy sequence so that, for the completeness of (X, d), there exists limn→+∞fnx=a∈X so that (i) is proved. Further, again by the

lower semicontinuity ofϕ, one has

d(a, f a)≤lim inf

ξ→a d(ξ, f ξ)≤nlim→+∞d

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thusf a=aand (ii) is proved. Finally, to see the validity of (iii), we note that

dfnx, fn+mx≤dfnx, fn+1x+···+dfn+m−1x, fn+mx

≤cn_+···+_cn+m−1_{lim inf}

ξ→x d(ξ, f ξ) =cn1+···+cm−1lim inf

ξ→x d(ξ, f ξ)

(2.7)

so that, lettingm→ +∞, one has

dfn_{x, a}_≤ cn

1−clim infξ→x d(ξ, f ξ), (2.8)

which proves item (iii).

Theorem 2.2. Let(X, d) be a complete metric space such that X ≠ ∅and let f:X→Xbe a mapping such thatf (X)⊆X. Suppose that there exists a pointx∈X such that

lim sup ξ→x

d(ξ, f ξ) <+∞ (2.9)

and that the mappingϕ(·):=d(·, f·)is weak lower semicontinuous. Finally, letc∈ [0,1[be such that

lim sup η→f x

d(η, f η)≤clim sup ξ→x

d(ξ, f ξ) ∀x∈X; (2.10)

then for allx∈Xsatisfying (2.9)

(i) there exists a pointa∈Xsuch thatfn_x_→_a_as_n_{→ +∞}_;

(ii) f a=aif and only ifϕisf-orbitally weak lower semicontinuous ata; (iii) d(fn_{x, a)}_≤_cn_/(₁₋_c)_{lim sup}

ξ→xd(ξ, f ξ).

Proof. We start with a pointx∈Xsuch that (2.9) holds and consider the

se-quence(fn_x)n

∈Z+, thus as in the previous proof we have

lim sup ξ→fn_x

d(ξ, f ξ)≤clim sup ξ→fn−1x

d(ξ, f ξ)≤ ··· ≤cn_{lim sup} ξ→x

d(ξ, f ξ) <+∞; (2.11)

using the weak lower semicontinuity ofϕone now has

dfnx, fn+1x≤lim sup

ξ→fn_x d(ξ, f ξ)≤c

n_{lim sup}

ξ→x

d(ξ, f ξ) (2.12)

thus

+∞

n=0

dfnx, fn+1x≤lim sup ξ→x

d(ξ, f ξ) +∞

n=0

cn<+∞; (2.13)

hence(fn_x)n

∈Z+ is a Cauchy sequence in the complete metric space(X, d), so that there existsa=limn→+∞fnx, thus (i) is proved. Now leta=f a, then

d(a, f a)=0≤lim sup k→+∞ d

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which is true for each subsequence(fnk_y)k_∈_Z₊_of_(fn_y)n_∈_Z₊ _{converging to}_a_as_k→

+∞, that is,ϕisf-orbitally weak lower semicontinuous ata. Conversely, suppose the f-orbitally weak lower semicontinuity ofϕata, then

d(a, f a)≤lim sup n→+∞ d

fn_{x, f}n+1_x₌₀_, _(2.15)

that is,f a=a, which guarantees (ii). Finally, as in the proof ofTheorem 2.1, we have

dfnx, fn+mx≤dfnx, fn+1x+···+dfn+m−1x, fn+mx

≤cn_+···+_cn+m−1_{lim sup} ξ→x

d(ξ, f ξ)

=cn₁_+···+_cm−1_{lim sup} ξ→x

d(ξ, f ξ)

(2.16)

and thus, asm→ +∞, one has

dfnx, a≤₁c₋n_clim sup ξ→x

d(ξ, f ξ), (2.17)

which proves item (iii).

Theorem_2.3. _Let_{(X, d)}_{be a compact metric space such that}_X≠∅_{and let}_f_: X→X be a mapping such thatf (X)⊆X. Suppose that for eachx∈Xsuch that lim infξ→xd(ξ, f ξ)≠0

lim inf

η→f x d(η, f η) <lim infξ→x d(ξ, f ξ) (2.18)

and that the mappingϕ(·):=d(·, f·)is lower semicontinuous; thenfhas a ﬁxed point.

Proof. We deﬁneφ:X→[0,+∞]by

φ(x)def=lim inf

ξ→x d(ξ, f ξ), (2.19)

thus we can observe that suchφis lower semicontinuous, in fact for eacha∈(X) one has

lim inf

x→a φ(x)=lim infx→a lim infξ→x d(ξ, f ξ)≥lim infx→a d(x, f x)=φ(a). (2.20)

Further,φis deﬁned on the compact setX, in fact it is a closed subset of the compact setX, thusφhas a minimum onX; we call ita, that is,

φ(a)=min

x∈Xφ(x). (2.21)

We now claim thatφ(a)=0, in fact suppose by contradiction that this is false, then by the hypotheses we havef a∈Xand

φ(f a)=lim inf

η→f a d(η, f η) <lim infξ→a d(ξ, f ξ)=φ(a), (2.22)

but this contradicts the minimality ofa∈X, thusφ(a)=0. Now for the lower semicontinuity ofϕone has

d(a, f a)≤lim inf

ξ→a d(ξ, f ξ)=φ(a)=0, (2.23)

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Mappings satisfying (2.2), (2.10), and (2.18) will be called in the next section, respec-tively,liminf contractions, limsup contractions, andweak liminf contractions.

3. Some remarks and examples. In this section, we give some remarks and ex-amples concerning liminf, limsup, and weak liminf contractions which are not clas-sical ones; we make use of the following notations: 1/2∞:=0 (for∞we mean+∞), P:= {2k|k∈Z}, andD:= {2k+1|k∈Z}.

Remark_3.1. _{All the contractive conditions we have considered have a local}

char-acter in the sense that they do not involve two generic points of the underlying space, but a single points and its orbit; under this aspects the results in this paper are related more to [4, Hicks-Rhoades theorem] rather than to Banach-Caccioppoli principle.

Remark_3.2. _{It is obvious that a Banach-Caccioppoli or an Edelstein contraction}_f

is also, respectively, a liminf and limsup or weak liminf contraction if it satisﬁes the additional hypothesisf (X)⊆X≠∅, so that our results are sometimes generaliza-tions of the classical ones; the opposite is not true as we will see in the sequel.

Example3.3. We consider the metric space(X, d)where

Xdef=

₁

2m, 1 2n

|m, n∈Z∪{+∞}, n≥m

(3.1)

and for each(2−m_,₂−n_{), (}₂−s_,₂−t₎_∈_X

d

₁

2m, 1 2n

,

₁

2s, 1 2t

def

= ₂1_m−₂1_s+₂1_n−₂1_t. (3.2)

This metric is equivalent to the Euclidean one inR2_{(restricted to}_X_{), and}_X_{is a closed} subset ofR2_{so that}_{(X, d)}_{is actually a complete metric space.}

Now consider the mappingf:X→Xdeﬁned by

f

₁

2m, 1 2n

def

=                                   

(0,0) ifm=n= ∞,

₁

2m+3,0

ifm∈P , n= ∞,

₁

2m−1,0

ifm∈D, n= ∞,

₁

2m+3, 1 2n+3

ifm∈P , n≠∞,

₁

2m−2, 1 2n+2

ifm∈D, n≠∞.

(3.3)

It is easy to see thatX= {(2−m_,₀₎_|_m_∈_Z_∪{+∞}}_{and that}_{f (X}₎_⊆_X_.

Let nowx=(2−2k_,₀₎_with_k_∈_Z_{, then}_{f x}₌₍₂−(2k+3)_,₀₎_and

lim inf

η→f x d(η, f η)=nlim→+∞

₂21k+3− 1 22k+1

+ 1

2n− 1 2n+2

= 3

22k+3,

lim inf

ξ→x d(ξ, f ξ)=nlim→+∞

₂12k− 1 22k+3

+ 1

2n− 1 2n+3

= 7

22k+3,

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thus

lim inf

η→f x d(η, f η)≤ 3

7lim infξ→x d(ξ, f ξ), (3.5)

d(x, f x)= 1 22k−

1 22k+3

= 7

22k+3=lim inf_ξ_→_x d(ξ, f ξ), (3.6)

while in the casex=(2−(2k+1)_,₀₎₍_k_∈_Z_{) we have}_{f x}₌₍₂−2k_,₀₎_and

lim inf

η→f x d(η, f η)=nlim→+∞

₂12k− 1 22k+3

+₂1_n−₂_n1₊₃ =₂₂7_k₊₃,

lim inf

ξ→x d(ξ, f ξ)=nlim→+∞

₂21k+1− 1 22k−1

+ 1

2n− 1 2n+2

= 3

22k+1= 12 22k+3,

(3.7)

thus

lim inf

η→f x d(η, f η)≤ 7

12lim infξ→x d(ξ, f ξ), (3.8)

d(x, f x)= 1 22k+1−

1 22k

= 1

22k+1< 3

22k+1=lim inf_ξ_→_x d(ξ, f ξ). (3.9)

In short for eachx∈Xone has

lim inf

η→f x d(η, f η)≤ 7

12lim infξ→x d(ξ, f ξ) <+∞ (3.10)

and, by (3.6) and (3.9), the mappingyd(y, f y)is lower semicontinuous, so that all the hypotheses ofTheorem 2.1are satisﬁed (in factf has(0,0)as ﬁxed point), but f is not a contraction in the sense of Hicks and Rhoades (see [4]), in fact for x=(2−(2k+1)_,₀₎₍_k_∈_Z_{) we have (}_{f x}₌₍₂−2k_,₀₎_,_f2_x₌₍₂−(2k+3)_,₀₎_):

df x, f2x=₂12k− 1 22k+3

=₂27k+3> 1 22k+1=

₂21k+1−

1 22k

=d(x, f x). (3.11)

The mapping of this example is actually both a liminf and a limsup contraction, but starting from it we give two other examples: in the ﬁrst one (Example 3.4) the mapping we give is a liminf but not limsup contraction, while the opposite is true in Example 3.5(all the details are left to the interested reader).

Example3.4. Let(X, d)be as above, we consider the following mapping:

f

₁

2m, 1 2n def =                                             

(0,0) ifm=n= ∞,

₁

2m+3,0

ifm∈P , n= ∞,

₁

2m−1,0

ifm∈D, n= ∞,

₁

2m+3, 1 2n+3

ifm∈P , n≠∞,

₁

2m−2, 1 2n+2

ifm∈D, n∈P ,

₁

2m−3, 1 2n+3

ifm, n∈D.

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It is easy to see thatfis a liminf but not limsup contraction, in fact forx=(2−2k_,₀₎ (k∈Z) one has

lim inf

η→f x d(η, f η)= 3 22k+3=

3 7·

7 22k+3=

3

7lim infξ→x d(ξ, f ξ),

lim sup η→f x

d(η, f η)= 7

22k+3=lim sup ξ→x

d(ξ, f ξ).

(3.13)

Example_3.5. _Let_{(X, d)}_{be as in Examples}_3.4_and_3.5_{and let}_f _{be the following}

mapping:

f

₁

2m, 1 2n def =                                             

(0,0) ifm=n= ∞,

₁

2m+3,0

ifm∈P , n= ∞,

₁

2m−1,0

ifm∈D, n= ∞,

₁

2m+3, 1 2n+3

ifm∈P , n≠∞,

₁

2m−2, 1 2n+2

ifm, n∈D,

₁

2m−1, 1 2n+1

ifm∈D, n∈P .

(3.14)

Nowfis a limsup but not liminf contraction, in fact forx=(2−(2k+1)_,₀₎₍_k_∈_Z_{) one has}

lim sup η→f x

d(η, f η)= 7 22k+3=

7 12·

12 22k+3=

7

12lim supξ→x

d(ξ, f ξ),

lim inf

η→f x d(η, f η)= 7 22k+3>

1

22k+1=lim inf_ξ_→_x d(ξ, f ξ).

(3.15)

The ﬁnal example shows that a weak liminf contraction may not be an Edelstein one (even in this case all the details are left to the interested reader).

Example3.6. LetXbe the following set:

Xdef=

₁

m, 1 n

|m, n∈Z+−{0}∪{∞}n≥m

, (3.16)

and letd:X2_→_R+_{be a mapping such that for each}_{(x, y), (u, v)}_∈_X_{one has}

d(x, y), (u, v)def=   

max{x, u}+|y−v| ifx≠u,

|y−v| ifx=u, (3.17)

with the deﬁnitions above it is easy to see that(X, d)becomes a compact metric space and thatX= {(1/m,0)|m∈(Z+_−{₀_}₎_∪{∞}}_{. We deﬁne the mapping}_f_:_X_→_X_in

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odd numbers and the set of positive even numbers):

f

₁

m, 1 n

def =

                                                      

(0,0) ifm=n= ∞,

₁

m−1,0

ifm∈D−{1}, n= ∞, ₁

3,0

ifm=1, n= ∞,

₁

m+3,0

ifm∈P , n= ∞,

₁

m−2, 1 n+2

ifm∈D−{1}, n∈Z+_−{₀_}_,

1, 1 n+1

ifm=1, n∈Z+−{0},

₁

m+3, 1 n+3

ifm∈P , n∈Z+−{0}.

(3.18)

This mapping is now a weak liminf contraction, that is, it satisﬁes (2.18) and all the hypotheses ofTheorem 2.3but it does not satisfy Edelstein condition (1.2); further it is not a liminf contraction in the sense ofTheorem 2.1, in fact one has

sup x∈X−{(0,0)}

lim infη→f xd(η, f η) lim infξ→xd(ξ, f ξ) =

1, (3.19)

so that for everyc∈[0,1[, (2.2) may not be satisﬁed.

References

[1] S. Banach,Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math.3(1922), 133–181 (French).

[2] R. Caccioppoli,Un teorema generale sull’ esistenza di elementi uniti in una trasformazione funzionale, Rend. Accad. Lincei11(1930), 794–799 (Italian).

[3] M. Edelstein,On ﬁxed and periodic points under contractive mappings, J. London Math. Soc.37(1962), 74–79.MR 24#A2936. Zbl 113.16503.

[4] T. L. Hicks and B. E. Rhoades,A Banach type ﬁxed-point theorem, Math. Japon.24(1979/80), no. 3, 327–330.MR 80i:54055. Zbl 432.47036.