Volume 2010, Article ID 323487,21pages doi:10.1155/2010/323487
Research Article
Some Variational Results Using Generalizations of
Sequential Lower Semicontinuity
Ada Bottaro Aruffo and Gianfranco Bottaro
Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Correspondence should be addressed to Ada Bottaro Aruffo,aruff[email protected]
Received 1 October 2009; Accepted 14 February 2010
Academic Editor: Mohamed Amine Khamsi
Copyrightq2010 A. Bottaro Aruffo and G. Bottaro. 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.
Kirk and Saliga and then Chen et al. introduced lower semicontinuity from above, a generalization of sequential lower semicontinuity, and they showed that well-known results, such as Ekeland’s variational principle and Caristi’s fixed point theorem, remain still true under lower semicon-tinuity from above. In a previous paper we introduced a new concept that generalizes lower semicontinuity from above. In the present one we continue such study, also introducing other two new generalizations of lower semicontinuity from above; we study such extensions, compare each other five conceptssequential lower semicontinuity, lower semicontinuity from above, the one by us previously introduced, and the two here definedand, in particular, we show that the above quoted well-known results remain still true under one of our such generalizations.
1. Introduction
In1Chen et al. proposed the following generalization1, Definitions 1.2 and 1.5.
Definition 1.1. LetX, τbe a topological space. Letx∈X.A functionf :X → −∞,∞is said to be sequentially lower semicontinuous from above atx“d-slscatx”ifxnn∈Nsequence
of elements of X for which xn → xand fxnn∈N weakly decreasing sequence, implies fx ≤ limn→∞fxn.Moreoverf is said to be sequentially lower semicontinuous from above
“d-slsc”if it is sequentially lower semicontinuous from above atxfor everyx∈X.
Moreover the authors of 1 conjectured that, for convex functions on normed spaces, sequential lower semicontinuity from above is equivalent to weak sequential lower semicontinuity from abovesee1, some rows below Definition 1.5. We exhibited some examples showing that such conjecture is falsesee4, Example 3.1 and Examples sketched in Remark 3.1.
In 4 we defined the following new concept, that generalizes sequential lower semicontinuity from above.
Definition 1.2see4, Definition 4.1. LetX, τbe a topological space. Letf be a function,
f:X → −∞,∞.Thenfis said to be
iinf-sequentially lower semicontinuous atx ∈ X “i-slscatx”ifxnn∈Nsequence of
elements ofX for whichxn → xand limn→∞fxn inff,impliesfx inff
equivalently, in the above condition the part limn→∞fxn inffcan be replaced
byfxninff;
iiinf-sequentially lower semicontinuous“i-slsc”if it isi-slscatxfor everyx∈X.
In particular we showed that for convex functions on normed spaces, such concept is equivalent to its weak counterpart4, Theorem 4.1.
Here, with the purpose to continue the study already begun in4, we define other two new concepts Definitions 3.1, called by us below sequential lower semicontinuity from above bd-slsc and uniform below sequential lower semicontinuity from above ubd-slsc, that generalize sequential lower semicontinuity from above and we show the following.
aAs it already happened fori-slsc,for convex functions on normed spaces, one of such new conceptsbd-slscis equivalent to its weak counterpartTheorem 4.1and parteof Remarks3.2; also by means of such result, it can be seen that, for convex functions on normed spaces and indifferently with respect to the topology induced by norm or to the weak topology, i-slscand bd-slscare each other equivalentpart
eof Remarks3.2, partbofTheorem 3.4andCorollary 4.2.
bSome results listed in1,2, such as Ekeland’s variational principle and Caristi’s fixed point theorem, remain still true under an hypothesis ofubd-slscSection 5.
Moreover we study the five concepts of sequential lower semicontinuity, lower semicontinuity from above, inf-sequential lower semicontinuity, below sequential lower semicontinuity from above and uniform below sequential lower semicontinuity from above, supplying further results and examples, with the purpose of getting a comparison between such five concepts, both in the general caseSection 3and in the case of convex functions
Section 4. In particular, inTheorem 4.10we prove that every convexubd-slscfunction on a Banach space is continuous in the points of the interior of its effective domain.
Finally, inSection 5, we also give some examples to show that, in the generalization of Ekeland’s variational principle by us proved, some hypotheses cannot be weakened.
2. Notations and Preliminaries
symbol. If Zis a linear space onRor onC, let dimZdenote the algebraic dimension ofZ,and, ifA ⊆ Z,let spAand coAdenote, respectively, the linear subspace ofZthat is generated byAand the convex hull ofA; ifx, y∈Zlet x, y:{λx 1−λy:λ∈0,1} and, ifx /y, letx, y:{λx1−λy:λ∈0,1}; if 0∈A, then theMinkowski functional (or gauge) of Ais the functiongA :Z → 0,∞ defined bygAx:inf{α∈R :x∈αA}for
everyx ∈ Z. IfZis a topological linear space, letZ denote the continuous dual ofZ; if
A⊆Z,let coAbe the closure of coA. IfZis a normed space, then|z|Zindicates the norm
inZof an elementz∈Z andSZa, r:{z∈Z :|z−a|Z < r}a∈Z, r ∈R. Let2 and c0,respectively, denote the real, or complex, Banach spaces of the sequences whose squares of
moduli of coordinates are summable, and of the infinitesimal sequences. IfAandBare sets, ifC ⊆ Aand f:A → Bis a function, then #Adenotes the cardinality ofA, f|C means the
restriction off toC; ifg :A → −∞,∞is a function, then domg :{x∈A:gx<∞}
denotes the effective domain ofg.IfZ is a topological space and ifA ⊆ Z, let∂Abe the boundary ofA. Let Edenote the integer part function. Ifτnn∈Nis a sequence of elements
of−∞,∞and if ∈ −∞,∞,thenτn means thatτnn∈N is a weakly decreasing
sequence with limn→∞τn . Henceforth we will shorten both lower semicontinuous and
lower semicontinuity in “lsc , both sequentially lower semicontinuous and sequential lower semicontinuity in “slsc”.
Definitions 2.1. LetXbe a linear space onF,A⊆X, y∈A.Thenin accordance with5
athe point yis said to be an internal point ofAif for every x ∈ X there exists an
αx∈Rsuch thatyλx∈Afor allλ∈0, αx;
b see5, page 8, above Exercise 1.1.20, and page 9, between the two ExamplesA
is said to beabsorbingif0is an internal point ofA,that is, if for everyx∈Xthere exists anαx∈Rsuch thatλx∈Afor allλ∈0, αx;
cthe setAis said to bebalancedifλx∈Afor everyx∈Aandλ∈Fsuch that|λ| ≤1;
dthe point y is said to be an extreme point of Aif x, z ∈ A, λ ∈ 0,1 for which
yλx 1−λzimpliesxzy.
Example 2.2. For every infinite dimensionalXnormed space onFthere existsCan absorbing balanced convex subset ofSX0,1, Cwithout interior points.
LetT:X → Xbe a linear bijective not continuous operatore.g., leten∈Xbe such that
|en|X 1n∈N, en/emifn, m∈N, n /m,{en :n∈N}linearly independent set of vectors,
Ba Hamel basis ofX such that{en : n ∈ N} ⊆ B, Ten : nen for everyn ∈ N, Tb : b
for everyb ∈ B\ {en : n ∈ N}, T extended for linearity to all X. LetC : T−1SX0,1∩
SX0,1.ThenCis a balanced convex as intersection of two balanced convex sets, is bounded
because contained inSX0,1, is absorbing as intersection of two absorbing setsT−1SX0,1
is absorbing because ifx ∈X\ {0}thenTx ∈X\ {0}and soλx∈T−1S
X0,1for every
λ∈Fwith|λ| ≤1/|Tx|X. Moreover, if by absurd there existed an interior point ofC, then,
beingCbalanced and convex, 0 should be an interior point ofCand thereforeTshould be a continuous operator, that is not possible.
a little change in the complex case for showing that “0 is not an interior point ofC”and so there are no interior points ofC, as above noted.
Leten ∈ X be such that|en|X ≤ 1n ∈ N, en/em ifn, m ∈ N, n /m,{en : n ∈ N}
linearly independent set of vectors,Ba Hamel basis ofXsuch that{en :n∈N} ⊆B,|b|X ≤1
for everyb∈B, D:{1/n1en:n∈N} ∪B\ {en:n∈N}.Then it is enough to define
C:co{αd:α∈F,|α|1, d∈D}. 2.1
Such aCis obviously a balanced set; for obtaining the remaining properties the same proof of Theorem 1 of6still works, with the unique following little change ifFCfor showing that
cd /∈Cwhenc >1 andd∈Di.e., one of the points of the demonstration of6: if by absurd such acd ∈ Cthen, using thatD is a Hamel basis, there should existm ∈ Z, λ1, . . . , λm ∈
0,1,withmj1λj 1, α1, . . . , αm ∈C,with|αj| 1 for everyj ∈ {1, . . . , m}such thatcd
m
j1λjαjd,thereforec
m
j1λjαj, that is impossible because|
m
j1λjαj| ≤
m
j1λj|αj| 1
whilec >1.
Theorem 2.3. Let X be a real normed space with algebraic dimension greater or equal to 2; then ∂SX0,1is an arcwise connected set.
Proof. Letx, y∈∂SX0,1, x /y.We will distinguish two cases:
ax / −y, bx−y.
In the caseathe arcγ :t ∈0,1→ 1−txty/|1−txty|Xconnectsxtoy:
in factγ0 x,γ1 y; moreoverγis defined and continuous, as, if by absurd there exists
t∈0,1such that1−tx−ty,then 1−t 1−t|x|X t|y|Xt, consequentlyt1/2 and
so1/2x−1/2y,that is in contradiction with the assumption ofa.
In the caseb,being dimX ≥2, there existsz∈∂SX0,1linearly independent from
x and y, hence z / −x and z / −y; then an arc connecting x to y can be found joining together two arcs, one connectingxtozand another connectingztoy,both of them existing in consequence ofa.
Lemma 2.4. LetI ⊆ Rbe an interval,x ∈ −∞,∞an extreme ofI, ∈ −∞,∞, f : I → −∞,∞ a convex function,xn ∈ I,xn/xn ∈ Nsuch that xn → xand fxn . Then
inf{fy:y∈I\ {x}}.
Proof. If −∞,then the conclusion is obvious; therefore we can suppose ∈ R. Letz ∈ I\{x}.Then there existnz∈Nsuch thatfxnz∈R,xnz∈x, zandmz∈N,mz> nzsuch that
xmz ∈x, xnz; therefore, being ≤fxmz≤ fxnz, we deduce thatfxmz∈Rand, for the convexity off,we getfxnz≤xmz−xnz/xmz−zfzxnz−z/xmz−zfxmz,whence, using in the second inequality thatfxnz ≥ fxmz,we obtain fz ≥ xmz −z/xmz −
xnzfxnz−xnz−z/xmz−xnzfxmz≥fxmz≥.
Theorem 2.5. LetX be an infinite dimensional normed space. Then there existsA⊆ X,Ainfinite, countable and linearly independent set such that, for everyM, N ⊆ A, M∩N ∅,it isspM∩
Proof. It is enough to use, with respect to an infinite dimensional closed separable subspaceY
ofX, metrizability and compactness ofSY0,1with respect to the weak∗topologysee, e.g.,
7, proof of Theorem V.5.1,8, Theorem III.10.2, and9, proof of Proposition 1.f.3.
3. New Weaker Concepts of Sequential Lower Semicontinuity
Definitions 3.1. Let X, τbe a topological space. Let f be a function,f : X → −∞,∞.
Then, iffis not∞identically,fis said to be
ibelow sequentially lower semicontinuous from above atx∈X “bd-slscatx”if there existsax∈infXf,∞such that:xnn∈Nsequence of elements ofXfor whichxn →
xandfxnlimn→∞fxn≤ax, impliesfx≤limn→∞fxn;
iibelow sequentially lower semicontinuous from above“bd-slsc”if it isbd-slscatxfor everyx∈X;
iiiuniformly below sequentially lower semicontinuous from above “ubd-slsc if there existsa∈infXf,∞such that:x∈X,xnn∈Nsequence of elements ofXfor which
xn → xandfxnlimn→∞fxn≤a,implyfx≤limn→∞fxn.
Whenfhas value∞constantly, all these properties are assumed to hold in a vacuous way.
In the above definitions, equivalently, we can replace the part “fxn
limn→∞fxn≤axresp.,≤a” with the following: “fxnn∈Nweakly decreasing sequence
andfxn ≤axresp.,≤afor everyn∈N.” Indeed one of the two implications is obvious
in both casesand the other can be proved, for example in the case ofi being the other case similar, in this way: if the above variant of definitioniis verified relatively to a certain value ofaxand ifbxis such that inff < bx< ax,then, for every sequencexnn∈Nof elements
of X for whichxn → x and fxn limn→∞fxn ≤ bx, there existsk ∈ Nsuch that
fxn≤axfor everyn > kand so we conclude, applying such variant to the sequencexnn>k.
Remarks 3.2. Here we will note some easy comparison between Definitions 3.1 and other previously considered generalizations of sequential lower semicontinuity Definitions 1.1 and1.2.
LetX, τbe a topological space. Letx∈ X.Letfbe a function,f : X → −∞,∞.
We note that
aiffis d-slscatx, thenfis bd-slscatx;
biffisbd-slscatx,thenfisi-slscatx,because iffis not constantly∞,ifxn → x
andfxninff,then limn→∞fxn≤axandfx≤inff;
ciffisd-slsc,thenfisubd-slsc,because iffis not constantly∞it follows thatiii of Definitions3.1is true with respect to an arbitrary value ofa >inff;
diffisubd-slsc,thenfisbd-slsc;
eiffisbd-slsc,thenfisi-slsc,forb;
Remarks 3.3. LetX, τ be a topological space. Letx ∈ X. Letf be a function,f : X → −∞,∞.We observe that:
aif lim infy→xfy > infXf, then f isbd-slsc at x,because it suffices to consider
ax∈infXf,lim infy→xfy;
bfor verifying thatfsatisfiesubd-slscwith respect to a certaina >infXfaas iniii
of Definitions3.1, it is enough to prove thatf|{z∈X:lim infy→zfy≤a}isubd-slsc.
Theorem 3.4. LetX, τbe a topological space satisfying the first axiom of countability. Letx∈X. Letfbe a function,f:X → −∞,∞. The following implications are true:
aiffisi-slscatx,thenfisbd-slscatx;
bif f is i-slsc,thenf isbd-slsc(so, under hypothesis of fulfillment of the first axiom of countability, a vice-versa of parteof Remarks3.2is true).
Proof. It is sufficient to show the partaof the theorem.
By absurd, we suppose thatf is notbd-slscat x; thenf is not∞ constantly and, choosing a sequenceakk∈N,withak>infffor everyk ∈N,such thatak → inff,for every
k ∈ Nthere exists a sequencexn,kn∈Nof elements ofX for which xn,k → x,fxn,kn∈N
is a weakly decreasing sequence with limn→∞fxn,k ≤ ak and fx > limn→∞fxn,k;
moreover, from the hypothesis, we deduce that inff < limn→∞fxn,k for every k ∈ N.
On the other hand, sinceτ verifies the first axiom of countability, there exists{Uh : h∈ N}
base for the neighbourhood system ofx,withUh1⊆Uhfor everyh∈N.
Now we will define a sequence xnn∈N by means of which we will produce a
contradiction. Letx0 xn0,0,wheren0min{n∈N:xn,0∈U0, inff < fxn,0< a01}and,
form∈N, chosenx0, . . . , xmwith inff < fxh h∈ {0, . . . , m},letxm1xnm1,km1,where km1 min{k∈N:k≥m1, ak< fxm}andnm1min{n∈N:xn,km1 ∈Um1,inff <
fxn,km1< akm11/m2, fxn,km1≤fxm}these choices are possible, because
fxn,k≥limp→∞fxp,k>infffor everyn, k∈N. For constructionxn → x, fxninff;
furthermorefx>inff,becausefx>limn→∞fxn,k>inff k∈N; but these facts are
in contradiction with thei-slscoffatx.
Theorem 3.5. LetXbe a topological linear space and letf:X → 0,∞be a function such that
fαx αfx for everyα∈0,∞and for every x∈X. 3.1
Suppose thatf isubd-slsc.Thenf isslsctoo. Therefore, for such a functionf, ubd-slsc, d-slsc, andslscare each other equivalent conditions (see partcof Remarks3.2).
Proof. Letabe relative to theubd-slscoff as iniiiof Definitions3.1. Then, since infXf
f0 0,we get thata >0.
Letx∈ Xand letxnn∈Nbe a sequence of elements ofX such thatxn → x; letα:
lim infn→∞fxn.We will conclude if we will prove thatfx≤α.
Now, since the desired conclusion is obvious ifα ∞, it is enough that we distinguish two cases:
In the casei,beingα0 inff,then there exists a subsequencefxnkk∈Noffxnn∈N
constantly 0 or strictly decreasing to 0: anyhow such subsequence is weakly decreasing and converging to 0< a; so, applyingubd-slscoff,the desired result follows.
In the second case, ii, let xnkk∈N be a subsequence of xnn∈N such that
limk→∞fxnk α; let k0 ∈ N be such that fxnk ∈ R for every k > k0 and let
yk : axnk/fxnkfor everyk ∈N, k > k0; then, beingX a topological linear space, it holds thatyk → ax/α, moreoverfyk a/fxnkfxnk afor everyk > k0; so, usingubd-slsc
off, we getfax/α≤awhencefx α/afax/α≤α.
Remark 3.6. Note that, ifX is a topological linear space, if A ⊆ X is an absorbing set and if f gA,then3.1is truesee5, Theorem 1.2.4iand definition foregoing.
Examples 3.7. aThere exist a bounded functiong:R → Rand a pointz ∈Rsuch thatgis
ubd-slsc,butgis notd-slscatz.
bThere exists a bounded function h : R → Rsuch thath isbd-slsc,but his not
ubd-slsc.
cLet W 2 on the fieldFendowed with its weak topology. Then there exist a
function k : W → 0,1 and a point w ∈ W such that k is i-slsc,but k is not bd-slsc
at w note that an analogue example on a topological space satisfying the first axiom of countability does not exist, in consequence ofTheorem 3.4.
aLet
gx
⎧ ⎨ ⎩
arctgx, ifx∈−∞,0,
−1, ifx∈0,∞,
3.2
andz0.Thengis lscin every point ofR\ {z}; thereforegis ubd-slsc: indeed it suffices to considera∈ −π/2,−1and to usebof Remarks 3.3. Besidesgis notd-slscatz,because ifzn:1/n1 z1/n1for everyn∈Nwe get thatzn → z,gznn∈Nis a weakly
decreasing sequence, butgz 0>−1limn→∞gzn.
bLet
hx
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
arctgx, if −2m−1< x≤ −2m m∈N,
0, if −2m−2< x≤ −2m−1 m∈N,
0, if x >0.
3.3
Thenhislscin every point ofR\{−2m−1 :m∈N}; moreover it isbd-slsc,as for everym∈N
it is sufficient to note that lim infx→ −2m−1hx arctg−2m−1>−π/2 infRhand to use aof Remarks 3.3. On the other handhis notubd-slsc,because for everya >infRh−π/2 there existma∈Nand a sequenceyn,an∈Nof real numbers for which limn→∞yn,a−2ma−1
andhyn,aa≤a,buth−2ma−1 0> ain fact it is enough to considermasuch that
cLet
kx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1
E|λ| 1 ifxλem for somem∈N and λ∈F\Z, 1
|λ| ifxλem for somem∈N and λ∈Z\ {0},
1 ifx∈ {0}
W\
m∈N
sp{em} ,
3.4
whereen : δn,mm∈Nfor everyn∈N,and letw0.We get thatkisi-slsc,as there does not
exist a converging sequencewnn∈Nof elements ofWwhich satisfies
lim
n→∞kwn 0infW k: 3.5
in fact, if a sequencewnn∈Nverifies3.5, then it follows that there existsn0 ∈Nsuch that
for every n > n0 there are λn ∈ F and mn ∈ N for whichwn λnemn and |λn| → ∞, hence|wn|2 → ∞,from here the sequence is unbounded and therefore cannot converge.
On the other hand k is not bd-slsc at w, because for every a > infWk 0 there exists a
sequencezn,an∈Nof elements ofW for which limn→∞zn,a 0 wandkzn,a a ≤a,
butk0 1 > a indeed it is sufficient to considersa ∈ Nsuch thatsa > 1,1/sa < aand
zn,asaenfor everyn∈N.
Example 3.8. It is possible to find an example of a Hilbert spaceY and of a function that is
lscwith respect to the topology induced onYby its norm, but that, with respect to the weak topology onY,is notbd-slscand so it is neitherubd-slscnord-slsc; indeed neitheri-slsc):
see4, Example 4.1and Remarks3.2.
4. Behavior of Some Weak Concepts of Sequential Lower
Semicontinuity with Respect to the Convexity
With respect to4, Theorem 4.1, a slightly more laborious demonstration allows to get a stronger result.
Theorem 4.1. LetXbe a normed space, letf :X → −∞,∞be a convex,i-slscfunction with respect to the topology induced onXby its norm. Thenfisbd-slscwith respect to the weak topology onX.
Proof. By absurd, we suppose that f is not bd-slsc with respect to the weak topology on
X. Thenf is not∞ constantly and there exists at least a pointx ∈ X such that for every
a >inffthere exists a sequenceyn,an∈Nof elements ofXfor whichyn,a x,fyn,an∈Nis
a weakly decreasing sequence,fyn,a≤afor everyn∈Nandfx>limn→∞fyn,a.Hence
fx>inffand we will reach an absurd if we will show that
there exists a sequencexnn∈N of elements ofX
for whichxn−→xandfxninff,
because, using the i-slscoff atxwith respect to the topology induced onX by its norm, relatively to such a sequence, it should befx inff.At first, in order to prove4.1, we will define a sequencebkk∈Nof real numbers and by induction we will define other four
sequences,ahh∈Nof real numbers,khh∈Nof natural numbers,jhh∈Nof integer numbers
andzhh∈Nof elements ofX such thatjh ∈ {0, . . . , h}andfzjh> inffiffzm >inff for somem∈ {0, . . . , h}h∈N,by means of the followings:
bk:
⎧ ⎨ ⎩
inff 1
k1 if inff∈R
−k if inff−∞
for every k∈N,
k0:0, a0:bk0, z0∈co
yn,a0:n∈N
such that|z0−x|X<1,
j0:
⎧ ⎨ ⎩
−1 iffz0 inff
0 iffz0>inff
4.2
and, ifh∈N,definedkp, ap, zpandjpfor everyp ∈ {0, . . . , h},we definekh1, ah1, zh1and
jh1in the following way:
kh1:
⎧ ⎨ ⎩
kh1 iffz0 · · ·fzh inff,
mink∈N:k > kh, bk≤f
zjh
iffzm>inff for somem∈ {0, . . . , h},
ah1:bkh1, zh1 ∈co
yn,ah1 :n∈N
such that|zh1−x|X <
1
h2,
jh1 :
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
−1, iffz0 · · ·fzh1 inff,
maxj ∈ {0, . . . , h1}:fzj
>inff, iffzm>inff
for somem∈ {0, . . . , h1}.
4.3
such definitions are possible, being x ∈ co{yn,a:n∈N} for every a > inff as a convex
closed subset ofXis weakly closed tooand thanks to choice’s axiom. By definition,bkk∈N
is strictly decreasing, with limk→∞bk inff,khh∈Nis strictly increasing and soahh∈Nis
a subsequence ofbkk∈Nand therefore is itself strictly decreasing, with limh→∞ah inff,
limh→∞zh xwith respect to the topology induced onXby its norm; moreoverjhh∈Nis
weakly increasing,jN\ {−1}{n∈N:fzn>inff}and hence, ifjNis a finite set, there
exists a subsequence ofzhh∈Nsuch that the values offin the elements of such subsequence
are constantly inff.
Now we will distinguish two cases:
ithere exists a subsequence ofzhh∈N such that the values off in the elements of
such subsequence are constantly inff, iithere does not exist a subsequence as ini.
In the caseiwe at once conclude, because, ifxnn∈Nis a subsequence ofzhh∈Nas
If we are in the case ii, thenjN is an infinite set; therefore by induction we can define a new sequencemnn∈Nin this way: letm0:minjN\ {−1}and, ifn∈N,defined
mpfor everyp∈ {0, . . . , n},letmn1:minjN\{−1} ∪ {m0, . . . , mn}; from here, defining
xn:zmn for everyn∈Nand taking into account the definition ofj,it follows thatxnn∈Nis
the subsequence ofzhh∈Nwhose elements are exactly all the “zh” such thatfzh>inff.
It will be enough to prove thatfxninff,because in such way we will have proved
4.1.
For everyn∈Nit holds:
fxn1 fzmn1≤amn1bkmn1 ≤f
zjmn1−1
fzmn fxn 4.4
where for obtaining the inequality between second and third terms we used thatzmn1 ∈
co{yp,amn1 :p∈N}by definition and for deducing the equality between fifth and sixth terms
we used that
jmn1−1max
j∈ {0, . . . , mn1−1}:f
zj
>inffmn 4.5
by definition.
At last and analogously as seen above, it is verified:
fxn fzmn≤amn for everyn∈N; 4.6
besides limn→∞amn inff,asamnn∈Nis a subsequence ofbkk∈Nand so we conclude.
Corollary 4.2. LetXbe a normed space, letf :X → −∞,∞be a convex function,i-slscwith respect to the weak topology onX;thenfis bd-slsc with respect to the same topology (so, in the present case, by the help of hypothesis of convexity, the conclusion ofTheorem 3.4is true, although the first axiom of countability may be not fulfilled).
Proof. From the easy observation that ifτandσare two topologies on a setY,withσ⊆τ, and if a function verifiesDefinition 1.2with respect toσthen it verifies the same condition with respect toτtoo, we get thatfisi-slscwith respect to the topology induced onXby its norm and so it is sufficient to useTheorem 4.1.
Examples 4.3. As it will be seen below, in examplesaandb,there exist examples of convex functions, satisfying “the same semicontinuity conditions” of Examples 3.7aandb,but that are not upperly bounded and whose values are in − ∞,∞ instead of inR; on the contrary, owing to Corollary 4.2, if W is as in cof Examples 3.7, an example of convex functionk:W → −∞,∞i-slscbut notbd-slsccannot exist.
aFor every X normed space having, as real space, algebraic dimension greater or equal to 2 and such that
the closed unitary sphere of X admits at least one extreme point, 4.7
there exist a convex functiong : X → 0,∞and a pointz ∈ X such thatg is
bFor every X normed space having, as real space, algebraic dimension greater or equal to 2 there exists a convex functionh:X → 0,∞such thathisbd-slsc,but
his notubd-slsc.
aLetz∈∂SX0,1be an extreme point ofSX0,1 such a point exists for4.7and
let
gx
⎧ ⎨ ⎩
|x|X, ifx∈SX 0,1\ {z},
∞, ifx∈ {z}X\SX0,1
4.8
Thengis convex, becauseSX0,1\ {z}is a convex setbeingzan extreme point of
SX0,1; moreovergislscin every point ofX\ {z}; thereforegisubd-slsc: indeed
it suffices to considera ∈ 0,1 and to usebof Remarks 3.3. Besides g is not
d-slscat z,because ifzn ∈ ∂SX0,1\ {z}n ∈ Nis chosen in such a way as to
converge tozthis choice is possible for hypothesis, usingTheorem 2.3we get that
gznn∈Nis a weakly decreasing sequenceit is constantly 1, butgz ∞>1
limn→∞gzn.
bLety∈∂SX0,1and let
hx
⎧ ⎨ ⎩
|x|X, ifx∈ {0}SX
y,1,
∞, ifx∈X\{0}SX
y,1. 4.9
Then his convex, because{0} ∪SXy,1 is a convex set; furthermore h islsc in
every point of {0} ∪ X \ ∂SXy,1; moreover it is bd-slsc, because for every
x ∈ ∂SXy,1\ {0}it is enough to note that lim infz→xhz |x|X > 0 infXh
and to use a of Remarks 3.3. On the other hand his notubd-slsc,because for every a > infXh 0 there exist xa ∈ X and a sequenceyn,an∈N of elements of X for which limn→∞yn,a xa, hyn,a a ≤ a,but hxa > a: in fact, being
∂SXy,1a connected set owing toTheorem 2.3and, chosen 0 < ba < min{a,2},
being ∂SXy,1∩X \SX0, baa nonempty open of ∂SXy,1 it contains 2y,
the nonempty subset∂SXy,1∩SX0, baof∂SXy,1 it contains 0must have at
least a not interior pointxa with respect to∂SXy,1; so there exists a sequence
zn,an∈N of elements of∂SXy,1with limn→∞zn,a xa,|zn,a|X > ba ≥ |xa|X for
everyn∈N, and it is enough to considery0,a∈SXy,1such that|y0,a−z0,a|X <1,
|y0,a|X>|xa|Xand, ifn∈N, givenyn,awith|yn,a|X>|xa|Xto selecthn> nsuch that
|zhn,a|X <|yn,a|X andyn1,a∈SXy,1such that|yn1,a−zhn,a|X <1/n2,|xa|X <
|yn1,a|X<|yn,a|X; with these choices, it results that limn→∞yn,axa,hyn,an∈N |yn,a|Xn∈Nis a strictly decreasing sequence with limn→∞hyn,a |xa|X ≤ba < a,
buthxa ∞>|xa|Xlimn→∞hyn,a.
Remark 4.4. Note that hypothesis4.7done inaof Examples 4.3 is verified for example by every reflexive Banach spacesee5, Theorem 2.4.5and use the weak topology, but there exist Banach spacese.g.,c0 andL1a, ba, b ∈R, a /b)which do not satisfy itsee10,
Remark 4.5. ForXRand also if functions with values in−∞,∞are considered, examples verifying conditions ofaorbof Examples 4.3 do not exist, because the following fact is true.
Iff :R → −∞,∞is a convex,i-slscfunction, thenfisd-slscand so, in this case, taking into account partsc,dandeof Remarks3.2, under hypothesis of convexity, the four conditions ofd-slsc, ubd-slsc, bd-slscandi-slscare each other equivalent.
Letx, xn ∈Rn ∈ N, xn → x,fxn ; then we will conclude, if we will prove
thatfx ≤ . It is not restrictive to suppose < ∞,xn/x,xn ∈ dom fn ∈Nandxan
extreme of domf otherwise, ifx ∈ domf◦, the functionf should be continuous inx; applyingLemma 2.4tof|domf it follows that inf{fy : y ∈ dom f\ {x}}; then, if by
absurdfx> ,it should beinff, that contradictsi-slscoff.
Example 4.6. For every infinite dimensional X normed space there exists a convex bd-slsc
functiong:X → Rthat is discontinuous in every point ofXand therefore, ifXis complete, is neither alscfunction onXsee, e.g.,5, Theorem 3.1.9.
Indeed, we will show that such a functiongcan be chosen as whatever a Minkowski functional of
an absorbing balanced convex subsetCofX such that
0 is not an interior point forC, C⊆SX0,1
4.10
for the existence of such a set, seeExample 2.2. Consequently, ifXis a Banach space, using Theorem 3.5,Remark 3.6and part cof Remarks 3.2, such a functiong neither isubd-slsc
nor isd-slsc.
LetCbe as in4.10and letg : gC.Theng is a real valued convex not continuous
functionsee, e.g.,8, Theorems II.12.1 and II.12.3, and foregoing Definition; hence, using a classical result of convex analysissee, e.g.,5, Theorem 3.1.8g is discontinuous in every point ofX.
Owing to partbofTheorem 3.4, for showing the remaining condition ong, namely thebd-slscofg,it suffices to prove thatgisi-slsc.
At first, note that
|x|X≤gx for everyx∈X. 4.11
In fact, beingC⊆SX0,1,for everyx∈Xit holds
|x|Xinf
α∈R:xα
X≤1
≤infα∈R :x
α ∈C
gx. 4.12
Now let x ∈ X, let xnn∈N be a sequence of elements of X for which xn → x and
limn→∞gxn 0 g0 infg; therefore, using 4.11, we get that x 0, whence
gx g0 0.
see, e.g.,11, Theorem 2.2.20band sentences afterward Proposition 1.1.11fails iflscis replaced bybd-slscor equivalentlyseeeof Remarks3.2andTheorem 3.4byi-slsc.
bOn the contrary, the above-mentioned classical result is still true if lscis replaced by
d-slscit is enough to use4, Theorem 3.2, and a classical result of convex analysissee, e.g.,
5, Theorem 3.1.9applied tof|dom f◦,also considering that, if there exists a pointx0 ∈ X
such thatfx0 −∞,thenfx −∞ for everyx ∈ dom f◦ see, e.g.,11, Proposition
2.1.4, or byubd-slscas we will show in the partcofTheorem 4.10.
Among other things, by means of such facts there is a different way to prove that, ifX
is a Banach space, the functiongofExample 4.6cannot be eitherubd-slscord-slsci.e., what already claimed in the last rows of the statement of number 4.6.
cThe aboveaandbshow a big difference between convex,bd-slscor convex,
i-slscfunctions on Banach spaces on the one hand and convex,d-slscor convex,ubd-slsc functions on Banach spaces on the other hand; such situation in a certain way renders, in the case in which convex functions on Banach spaces are considered, more meaningful those results, as for example4, Theorems 5.1, 5.3 and Corollary 5.1in which the classic hypothesis of “lower semicontinuity” can be replaced by an hypothesis of “bd-slsc”or of “i-slsc”.
Working with some properties of infinite dimensional Banach spaces and with the choice of the convex setC, we will able to exhibit an example as the following onethat may be regarded in a certain way as a refinement ofExample 4.6, because, also if it does not give a stronger conclusion than the one ofExample 4.6, it lets to define in a constructive way a sequence of points, by means of which ubd-slscis showed not to be true.
Moreover, with respect to Examples 4.3b, observe that inExample 4.8 we get an example of a function defined in a less general space, but having real values; hence the points by means of which we could prove that such a function is notubd-slsc,unlike that in partb of Examples 4.3, necessarily are all at the interior of its effective domain.
Example 4.8. For every infinite dimensionalXBanach space onF,there exists a convexbd-slsc
functiong:X → Rthat is notubd-slscand therefore, for partcof Remarks3.2, is not even ad-slscfunction.
Indeed, we will show that such a functiongcan be chosen as the Minkowski functional of a suitable setCsatisfying4.10; then infXg 0 and we will prove thatg is notubd-slsc,
exhibiting a sequence cqq∈Z of elements of X such that for every q ∈ Z there exists a sequence cq,kk∈N of points of X for which limk→∞cq,k cq, gcq,kk∈N is a weakly
decreasing sequence andgcq>1/qlimk→∞gcq,kfor everyq∈Z.
We will define the set C by means of the construction of Thorp 6, Theorem 1, already cited in the final part ofExample 2.2, but choosing a suitable Hamel basis, formed by elements having norm less or equal to 1 and verifying other suitable conditions that we are going to introduce.
ForTheorem 2.5there existsA⊆X, Ainfinite, countable and linearly independent set such that
M, N ⊆A, M∩N∅⇒spM∩spN{0}; 4.13
moreover let|a|X1 for everya∈Athis is not a restriction.
Consequently ifn∈Nαnan0 withαn∈Fandan∈A n∈N,thenαn0 for every
LetNq⊆A q∈Zbe such thatAq∈ZNq,eachNqis an infinite set andNq∩Np
∅ q, p∈Z, q /p.Further on, letλn∈Rn∈Nbe such thatn∈Nλn1.
Letbq :
n∈Nλneq,n for everyq ∈ Z,whereNq {eq,n : n ∈ N} witheq,n/eq,mif
n, m ∈ N, n /mq∈ Z such elementsbq q ∈ Zare existing asX is a Banach space.
Then|bq|X≤1 for everyq∈Z.
Hencebq ∈spNq\spNq, spNq∩spNt{0}ifq, t∈Z, q /tand thereforebq/btif
q, t∈ Z, q /t, A∩ {bq :q ∈Z} ∅andA∪ {bq :q∈ Z}is a linearly independent set of
vectors.
Therefore, we can considerBa Hamel basis of X such thatB ⊇ A∪D,where D :
{bq/q:q∈Z},with|b|X1 for everyb∈B\D,andC:co{αb:α∈F, |α|1, b∈B}.
Then C satisfies 4.10, because C is a particular case of the convex set defined, following Theorem 1 of6, in the final part ofExample 2.2.
Letg : gC.SinceC verifies4.10and g is its Minkowski functional, for the proof
already seen inExample 4.6we get thatgis a real valued convex bd-slscfunction.
At last we will prove thatg is not ubd-slsc, exhibiting a sequence as described in the statement.
Let
cq:
bq
q, p
q, k∈Nq\ {0, . . . , k},
bq,k: k
n0
λneq,n
n>k
λneq,pq,k,
cq,k:
bq,k
q
q∈Z, k∈N;
4.14
then for eachq ∈ Z, k ∈ Nit isbq,k ∈ co{eq,n : n ∈ N} ⊆ C, but there does not exist an
α > 1 such thatαbq,k ∈ Cbecause, beingB a linearly independent set of vectors, ifα > 1
the following one is the only way in whichαbq,k can be written as a linear combination of
elements ofB: αbq,k
k
n0αλneq,n
n>kαλneq,pq,k and, on the other hand,
k
n0αλn
n>kαλn α > 1; therefore gbq,k 1 and, being g positively homogeneous, gcq,k
1/q; whence, for each q ∈ Z, the sequencegcq,kk∈N is constant and therefore weakly
decreasing; furthermoregcq 1for a demonstration quite similar to the above proof that
gbq,k 1 and sogcq 1 > 1/q limk→∞gcq,k,besides limk→∞cq,k cq and we
conclude.
Lemma 4.9. LetY be a topological space and letXbe a topological linear space. The following facts hold
aif Ais a subset of Y,F is a closed subset ofY andG is an open subset of Y such that A∩GF∩G,thenA∩G⊆F(and thereforeA∩GF∩G);
bifUis an open subset ofX andCis a convex subset ofXsuch thatU∩C /∅andC /◦ ∅, thenU∩C /◦ ∅;
cif Ais a subset ofX,Cis a convex subset ofX andF is a closed subset of X such that
◦
Proof. aIf by absurdA∩G∩Y\F/∅,then, beingG∩Y\Fan open, it is∅/A∩G∩Y\F F∩G∩Y\F ∅,that is a contradiction.
bLetx ∈C◦andy ∈ U∩C.Ifx ywe have the desired result; otherwise, ifx /y, from7, demonstration inside of the proof of Theorem V.2.1it follows thatx, y\ {y} ⊆C◦, besides from the topological linear structure ofX it isx, y\ {y}∩U /∅and so we can conclude.
cApplyingbtoU: ◦
A,we get ◦
A∩ C /◦ ∅.Besides, fromaapplied withG:C◦,
we deduce ◦
A∩ C◦ ⊆A∩ C◦⊆ Fand so ◦
A∩ C◦⊆F ∩ C◦⊆A, namely, ◦
A∩ C◦is a not empty open set contained inAand we conclude.
Theorem 4.10. LetXbe a topological linear space and letf :X → −∞,∞be a convex,ubd-slsc function. Suppose thatfis not∞identically, leta∈Rbe such thata >infXfis relative tofas in
conditioniiiof Definitions3.1and letA:{x∈X:fx≤a}.Then
athe set A ∩ domf◦ is sequentially closed with respect to the relative topology of dom f◦;
henceforth suppose thatdomf◦/∅,then also the following facts hold:
bthere exists a pointx0∈ {x∈X:fx< a} ∩domf◦and thereforeA−x0is absorbing;
cifXis a Banach space, it results thatA /◦ ∅andfis continuous in the points ofdomf◦.
Proof. If there exists a pointz0 ∈ X such thatfz0 −∞, then fx −∞ for everyx ∈
domf◦see, e.g.,11, Proposition 2.1.4and all the parts of the desired result follow also ifXis simply a topological linear spacein fact in such casedomf◦ ⊆ {x∈X:fx< a}.
Henceforward we can suppose thatfz > −∞for everyz ∈ X; with such a further hypothesis, we will prove all the three parts of the desired result.
aLetxnn∈Nbe a sequence of elements ofA∩domf◦and letx∈domf◦be such
thatxn → x; letα lim infn→∞fxn. Thenα ≤ a; moreover there exists a subsequence
xnkk∈Nofxnn∈Nsuch thatfxnk → αand there exists a further subsequencexnkhh∈Nof
xnkk∈Nsuch thatfxnkhh∈Nis a weakly monotone sequence. Now, if by absurda < fx, we get α < fx and hence, using Lemma 3.1 of 4, it is not restrictive to suppose that
fxnkhh∈Nis a weakly decreasing sequence; so it is enough to use theubd-slscoffto obtain
a contradiction.
bLety0∈domf◦and letz∈Xbe such thatfz< a; thenz∈dom fand hence,
being dom fa convex set, it isy0, z⊆domf; thereforef|y0,zis an upper semicontinuous
function see12, Theorem 10.2 and so there exists a point x0 ∈ y0, z\ {z} such that
fx0 < a; besides, from7, demonstration inside of the proof of Theorem V.2.1it follows
thaty0, z\ {z} ⊆domf◦; consequentlyx0∈ {x∈X:fx< a} ∩dom f◦.
Now, ifw∈ {x∈X :fx< a} ∩dom f◦,thenwis an internal point ofA,because, being an interior point of dom f, for everyx ∈X there exists aβx ∈ R such thatw, w
βxx⊆dom f◦; thereforef|w,wβxxis a continuous function and so, beingfw< a,there exists anαx∈Rsuch thatfwλx< afor allλ∈0, αx.ThenA−wis absorbing.
cNow letXbe a Banach space. First of all we will prove that ◦
A /∅.
Frombthe setA−x0is absorbing; thereforeX n∈ZnA−x0and from Baire’s
lemma there exists an∈Zsuch thatnA−x0◦/∅,henceA−x0◦/∅,wherefore
◦
Now we will prove that partcofLemma 4.9can be applied withC:dom f. Since
A ⊆ dom f and ◦
A / ∅,we get∅ /
◦
A ⊆
◦
A ∩dom f. On the other hand, froma, the set
A∩ dom f◦is sequentially closedand hence closed, satisfying the topology ofXthe first axiom of countabilitywith respect to the relative topology ofdom f◦and therefore there existsFclosed subset ofXsuch thatA∩dom f◦F∩dom f◦. Consequently, from part
cofLemma 4.9, we have thatA /◦ ∅.
ThenA◦is a not empty open subset of dom fon whichfis bounded abovefrom the real elementaand so we can conclude, applying Theorem 3.1.8 of5.
Remark 4.11. The Example 3.1 of4and those cited in Remarks 3.1 of4give also examples of Banach spacesY and of convex,ubd-slsc,with respect to the weak topology, functions defined onYwith values in0,∞, that are notd-slscwith respect to the weak topology.
In fact it suffices to usebof Remarks 3.3 and, using the notations of the above cited examples, to note that inff 0 and that, relatively to whatever value ofa∈ 0,1, it is true thatD∩ {y∈Y :gy≤a}C∩ {y∈Y :gy≤a}is convex and closed,f|D∩{y∈Y:gy≤a}
g|D∩{y∈Y:gy≤a}and g is continuousfor Example 3.1 of 4,D∩SY0, a C∩SY0, ais
convex and closed,f|D∩S
Y0,a| |Y|D∩SY0,afor Example cited inaof Remarks 3.1 of4,
f|S
Y0,a| |Y|SY0,afor Example cited inbof Remarks 3.1 of4.
5. Ekeland’s and Caristi’s Theorems
Remark 5.1. In the following theorem we will show an extension of Ekeland’s variational principle see13, Theorem 1.1to the case in which the hypothesis of lscis replaced by
ubd-slsc.The proof is inspired by the demonstration of Theorem 2.1 of1: since we will add the resultb that was considered already in13, Theorem 1.1and for reading convenience, here we are writing the whole proof, removing some trivial mistakes of1.
The authors thank the referee for having pointed out to them that Theorem 2.1 of1 can be deduced also from14, Corollary 4 of Theorem 1that subsequently was generalized by15–17: in fact it is easy to prove that the hypotheses of such corollary are verified if we assumed-slscbut the same we cannot do if we assumeubd-slsc.
Moreover we wish here to point out another recent paper 18 in which other generalizations of Caristi-Kirk’s Fixed Point Theorem and Ekeland’s Variational Principle are given, in a different environment with respect to the one of the present paper.
Theorem 5.2. LetX, dbe a complete metric space and letf:X → − ∞,∞be a bounded from below and not∞identically function. Letε >0,λ >0andu∈Xbe such thatfu≤infx∈Xfxε.
Moreover,
suppose thatf isubd-slscand thatfu≤a,
where ais relative tof as in (iii) of Definitions 3.1. 5.1
Then there existsv∈Xsuch that afv≤fu,
bdu, v≤λ,
Proof. Let
ε0:a−inf
Xf. 5.2
Sincefu≤ainfXfε0and observing that, ifε > ε0, then a pointv∈Xverifyinga,b
andcwith respect toε0,λandualso satisfiesa,bandcwith respect toε,λandu, it is
not restrictive to supposeε≤ε0.
For everyz∈XletTz{x∈X :fx≤fz−ε/λdx, z}; thenTz/∅andTz{z}
if and only iffx> fz−ε/λdx, zfor everyx∈X\ {z}.
Ifz∈Xands∈Tz, thenTs⊆Tz, because ifx∈Tswe havefx≤fs−ε/λdx, s;
besidesfs≤fz−ε/λds, zass∈Tz; sofx≤fs−ε/λdx, s≤fz−ε/λds, z−
ε/λdx, s≤fz−ε/λdx, zand hencex∈Tz.
Let nowU:{X} ∪ {Tz:z∈X}. Then as a consequence of what was above noted,
Ts⊆U for everyU∈ U, s∈U. 5.3
There exists a functionh: {U, s : U∈ U, s ∈U} → Xsuch thathU, s∈ Tsand
fhU, s−infx∈Tsfx≤ 1/2fs−infx∈Ufxfor everyU∈ Uands∈U: for showing such a fact, ifU∈ U, s∈Uand iffs infx∈Ufxwe choosehU, s sin such a case we
must choosehU, s s, becausefs≤ fx ≤ fs−ε/λdx, sfor everyx ∈ Ts, being
Ts⊆Uon account of5.3, and soTs{s}, otherwise we use the characterization of greatest
lower bound and the choice’s axiom.
Then, for definition ofTs, we have that
fhU, s≤fs for everyU∈ U, s∈U. 5.4
Now we consider two recursive sequencesxnn∈NandSnn∈Ndefined byx0u, S0
X,xn1hSn, xn, Sn1Txn for everyn∈N; then, using5.4,5.3and definition ofh,we get
fxn1≤fxn≤fu<∞, Sn1⊆Sn for everyn∈N, 5.5
fxn1− inf x∈Sn1
fx≤ 1
2
fxn− inf x∈Sn
fx
for everyn∈N; 5.6
moreover, beingxn1∈Sn1Txn, it holds
ε
λdxn, xn1≤fxn−fxn1 for everyn∈N, 5.7
whencexnn∈Nis a Cauchy sequence, since there exists limn→∞fxn ∈Rbecause of5.5
Letv limn→∞xn; sincef is aubd-slscfunction,fu ≤ infXf ε0 aand using
5.5, we obtain that
fv≤ lim
n→∞fxn≤fu 5.8
and soais verified.
From5.7, using triangular inequality of distance, we deduce
ε
λdx0, xn≤
n
k1
ε
λdxk−1, xk≤
n
k1
fxk−1−fxk
fx0−fxn 5.9
for everyn∈Z; hence, considering thatx0u,vlimn→∞xn,fxn≥infx∈Xfxfor every
n∈Nand by means of a limit passage forn → ∞in the first and the last terms, we get
ε
λdu, v≤fu−nlim→∞fxn≤xinf∈Xfx ε−nlim→∞fxn≤ε, 5.10
from whencebfollows.
As an alternative, for showingb,we can observe thatTz⊆ {x∈X :dx, z≤λ}for
everyz ∈X such thatfz≤infx∈Xfx ε: indeed, in such hypothesis onz,ify ∈Tzand
if by absurddy, z> λthenfy≤ fz−ε/λdy, z< fz−ε≤ infx∈Xfx, that gives
a contradiction. Besides, using5.5, we have thatxn ∈Tu ⊆ {x∈X :dx, u≤ λ}for every
n∈N.Consequentlydv, u limn→∞dxn, u≤λ, that isb.
If, by absurd,cis not true, then
there existsx∈X\ {v} such thatfx≤fv− ε
λdx, v; 5.11
owing to5.5and5.8, it is
fv≤ lim
k→∞fxk≤fxm≤fxn−
ε
λdxm, xn for everyn, m∈N, m≥n 5.12
and hence, passing to the limit form → ∞in the first and the fourth terms, we getfv≤
fxn−ε/λdv, xnfor everyn∈N; therefore, using5.11, it holds:
fx≤fxn−
ε
λdv, xn− ε
λdx, v≤fxn− ε
λdxn, x for everyn∈N; 5.13
hencex∈Sn1for everyn∈N, from whence
fx≥inf
Sn
f for everyn∈N; 5.14
besides, since from5.5it follows thatinfSnfn∈Nis a weakly increasing sequence, we can do
1/2α, from whence α 0; then, using 5.11 and 5.8, we get that fx < fv ≤ limn→∞fxn limn→∞infSnf, that is in contradiction with5.14.
Example 5.3. In the hypotheses ofTheorem 5.2, but without the additional hypothesisfu≤ a,it is easy to verify that conclusionsaandcare still truein fact, ifε0a−infXfand if
fu>infXfε0,then it is enough to considert∈Xsuch thatft≤infXfε0and in such
case a pointvrelative to ε0, λandtas inTheorem 5.2solves the questionbut it can happen
that there existε, λandusatisfying the remaining hypotheses, for which there is not a point
vthat verifies conclusionsbandc.
OnRwe consider the equivalence relation∼defined byx∼yifx−y∈Qx, y∈R
and letR : {A∩0,2 : Aequivalence class with respect to ∼}; then #B ℵ0 for every
B ∈ R and∪R 0,2, hence #R 2ℵ0, so there exists a bijective functionϕ : R → 0,1;
moreoverB 0,2for everyB∈ R. Let nowf:R → Rbe defined by
fx
⎧ ⎨ ⎩
arctgx ifx∈−∞,02,∞,
ϕB ifx∈BB∈ R.
5.15
Becausef is real-valued and continuous in the pointsywherefy<0, thenfisubd-slsc,
consideringa0 >infRf; moreoverf is bounded from below. Letε π/2 1,λ 1 and
u1; thenfu∈ϕR≤1infRfε. If by absurd there existsv∈Rverifying conclusions bandc, then usingbwe get that|v−1| ≤1 and thereforev∈0,2; letB∈ Rsuch that
v ∈B. Consequently 1 ≥fv> 0 and, ifα ∈0, fv, we obtain thatϕ−1α ∈ R, whence
ϕ−1α 0,2and so there exists a sequencex
nn∈Nof elements ofϕ−1αconverging tov;
from this, usingc, we obtain thatαϕϕ−1α fx
n> fv−ε|xn−v|for everyn∈N
and, by means of a limit passage forn → ∞, we haveα≥fv, that is a contradiction.
Example 5.4. If inTheorem 5.2the hypothesis5.1is replaced by a hypothesis ofbd-slsconf,
then, in spite of what we noted at the beginning ofExample 5.3, conclusionccan be not true. Indeed here we will show an example of abd-slscfunctionf :1/2,∞ → 0,1for which there does not existv∈1/2,∞verifying conclusioncofTheorem 5.2with respect toε λ1with such a choice the hypothesisfu≤infx∈1/2,∞fxεofTheorem 5.2is satisfied
by everyu∈1/2,∞, that is for everyv∈1/2,∞there existsw∈1/2,∞\{v}such thatfw≤fv− |w−v|.
LetB:n∈Z3n25n−1/3n2, n∪n,3n27n1/3n2and let
fx ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1
n23n−x, if x∈
3n25n−1
3n2 , n
n∈Z,
1
n23x−n, if x∈
n,3n
27n1
3n2
n∈Z,
1, if x∈
1 2,∞
\B
i.e., if g : 1/2,∞ → 0,1 is the continuous piecewise affine function, with slope alternatively equal to −3 and 3 in the connected components of B and with value 1 in
1/2,∞\B∪Z,f coincides withg on1/2,∞\Z and has value 1 in the points of
Z.
Thenfis a bd-slscfunction, by the help ofaof Remarks 3.3.
On the other hand, now we will show that for everyv ∈1/2,∞there existsw ∈ 1/2,∞\{v}such thatfw≤fv− |w−v|:
iifv∈ B, it is enough to choosew ∈Bin the same connected component ofvand such thatfw< fv, because with such a choice it holds
fw fv−3|w−v|< fv− |w−v|; 5.17
iiifv∈Z,then a pointwsufficiently close tovsolves what is requested, because
lim
t→vft<1tlim→v
fv− |t−v|; 5.18
iiiif v ∈ 1/2,∞\B∪ Z, then it is sufficient to consider an element n ∈ Z
such that |n−v| ≤ 1/2 and to choosew in the interval with extremes nand v,
wsufficiently close ton, because
lim
t→nft
1
n2 ≤ 1 3 <
1 2 1−
1 2 ≤limt→n
fv− |t−v|. 5.19
Remark 5.5. In the following results we note that Caristi’s fixed point theorem see 19, Theorem2.1’and Caristi’s infinite fixed points theorem can be extended to the case in which the hypothesis oflscis replaced byubd-slscsee also the extensions given in the case
d-slscby2, Theorem 2.1and by1.
Theorem 5.6Caristi’s fixed point theorem; see19, Theorem2.1’. LetX, dbe a complete metric space and letϕ:X → Rbe a ubd-slsc and bounded from below function. LetT :X → Xbe a function such thatdx, Tx≤ϕx−ϕTxfor everyx∈X. Then there existsx0 ∈Xsuch that
Tx0 x0.
Proof. It is enough to repeat the same demonstration of Theorem 2.2 in 1, where Theorem 5.2has to be used instead of Theorem 2.1 in1.
Theorem 5.7Caristi’s infinite fixed points theorem. LetX, dbe a complete metric space and letϕ :X → Rbe a ubd-slsc and bounded from below function, that does not obtain its infimum on X. LetT :X → X be a function such thatdx, Tx ≤ϕx−ϕTxfor everyx∈X. ThenT admits infinite fixed points inX.
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