Volume 2010, Article ID 246382,16pages doi:10.1155/2010/246382
Research Article
Critical Point Theorems for Nonlinear Dynamical
Systems and Their Applications
Wei-Shih Du
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan
Correspondence should be addressed to Wei-Shih Du,[email protected]
Received 9 February 2010; Accepted 18 April 2010
Academic Editor: Lai Jiu Lin
Copyrightq2010 Wei-Shih Du. 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 some new critical point theorems for nonlinear dynamical systems which are generalizations of Dancˇs-Heged ¨us-Medvegyev’s principle in uniform spaces and metric spaces by applying an abstract maximal element principle established by Lin and Du. We establish some generalizations of Ekeland’s variational principle, Caristi’s common fixed point theorem for multivalued maps, Takahashi’s nonconvex minimization theorem, and common fuzzy fixed point theorem forτ-functions. Some applications to the existence theorems of nonconvex versions of variational inclusion and disclusion problems in metric spaces are also given.
1. Introduction
In 1983, Dancˇs et al.1proved the following existence theorem of critical pointor stationary point or strict fixed pointfor a nonlinear dynamical system.
Dancˇs-Heged ¨us-Medvegyev’s Principle [
1
]
LetX, dbe a complete metric space. LetΓ:X → 2Xbe a multivalued map with nonempty
values. Suppose that the following conditions are satisfied:
ifor eachx∈X, we havex∈ΓxandΓxis closed,
iix,y∈Xwithy∈Γximplies thatΓy⊆Γx,
iiifor eachn∈Nand eachxn1∈Γxn, we have limn→ ∞dxn, xn1 0.
Then there existsv∈Xsuch thatΓv {v}.
of these results have been investigated by several authors; for example, see 2, 3 and references therein.
In 1963, Bishop and Phelps4proved a fundamental theorem concerning the density of the set of support points of a closed convex subset of a Banach space by using a maximal element principle in certain partially ordered complete subsets of a normed linear space. Later, the famous Br´ezis-Browder’s maximal element principle 5 was established and applied to deal with nonlinear problems. Many generalizations in various different directions of maximal element principle have been studied in the past; for example, see2,3,6–10and references therein. However, few literatures are concerned with how to define a sufficient condition for a nondecreasing sequence on a quasiordered set to have an upper bound. Recently, Du 7 and Lin and Du 3 defined the concepts of sizing-up function and μ -bounded quasiordered setsee Definitions1.1and1.3belowto describe a rational condition for a nondecreasing sequence on a quasiordered set to have an upper bound.
Definition 1.1see3,7. LetX be a nonempty set. A functionμ: 2X → 0,∞defined on
the power set 2XofXis calledsizing-upif it satisfies the following properties:
μ1μ∅ 0,
μ2μA≤μBifA⊆B.
Definition 1.2see3,7. LetXbe a nonempty set andμ: 2X → 0,∞a sizing-up function.
A multivalued mapT :X → 2X with nonempty values is said to be oftype μif, for each
x∈Xandε >0, there exists ayyx, ε∈Txsuch thatμTy≤ε.
Definition 1.3 see 3, 7. A quasiordered set X, with a sizing-up function μ : 2X →
0,∞, in shortX,, μ, is said to beμ-boundedif every nondecreasing sequencex1 x2
· · ·xnxn1· · · inXsatisfying
lim
n→ ∞μ{xn, xn1, . . .} 0 1.1
has an upper bound.
In7 see also3, Lin and Du established the following abstract maximal element principle in aμ-bounded quasiordered set with a sizing-up functionμ.
Theorem LD [see [
3
,
7
]]
LetX,, μbe aμ-bounded quasiordered set with a sizing-up functionμ: 2X → 0,∞. For
eachx∈X, letS:X → 2Xbe defined bySx {y∈X:xy}. IfSis of typeμ, then, for
eachx0 ∈X, there exists a nondecreasing sequencex0 x1 x2 · · · inXandv∈Xsuch
that
ivis an upper bound of{xn}∞n0, iiSv⊆∞n0Sxn,
iiiμ∞n0Sxn μSv 0.
to the petal theoerm. Many generalizations in various different directions of these results in metricor quasimetricspaces and more general in topological vector spaces have been investigated by several authors in the past; for detail, one can refer to 2, 3,7–9, 11–23. By applying Theorem LD, Du 7 gave a generalized Br´ezis-Browder principle, system
vectorial versions of Ekeland’s variational principle and maximal element principle and a vectorial version of Takahashi’s nonconvex minimization theorem. Moreover, the author investigated the equivalence between scalar versions and vectorial versions of these results. For more detail, one can see7.
The paper is divided into four sections. InSection 3, we establish some new critical point theorems for nonlinear dynamical systems which are generalizations of Dancˇs, Heged ¨us and Medvegyev’s principles in uniform spaces and metric spaces by applying an abstract maximal element principle established by Lin and Du. We also give some generalizations of Ekeland’s variational principle, Caristi’s common fixed point theorem for multivalued maps, Takahashi’s nonconvex minimization theorem, and common fuzzy fixed point theorem forτ-functions. Some existence theorems of nonconvex versions of variational inclusion and disclusion problems in metric spaces are also given inSection 4. Our techniques and some results are quite original in the literatures.
2. Preliminaries
Let us begin with some basic definitions and notation that will be needed in this paper. Let
Xbe a nonempty set. A fuzzy set inXis a function ofXinto0,1. LetFXbe the family of all fuzzy sets inX. A fuzzy map onX is a map fromX intoFX. This enables us to regard each fuzzy map as a two-variable function ofX×Xinto0,1. LetFbe a fuzzy map onX. An elementxofXis said to be a fuzzy fixed point ofFifFx, x 1see, e.g.,11,12,16,24–26. LetΓ :X → 2X be a multivalued map. A pointx ∈X is called acritical pointorstationary
pointor strict fixed point 1,3,8,27–29ofΓifΓv {v}.
Let “” be a quasiorderpreorder or pseudoorder, that is, a reflexive and transitive relationonX. ThenX,is called a quasiordered set. In a quasiordered setX,, recall that an element v in X is called a maximal element of X if there is no element x of X, different fromv, such thatv x. Denote by RandNthe set of real numbers and the set of positive integers, respectively. A sequence{xn}n∈N inX is callednondecreasing resp.,
nonincreasingifxnxn1resp.,xn1xnfor eachn∈N.
LetXbe a nonempty set andU,V any subsets ofX×X. Denote by
Δ {x, x:x∈X} the diagonal ofX×X,
Ux y∈X:x, y∈U the entourage ofx∈X,
U−1x, y∈X×X:y, x∈U,
U◦V x, y∈X×X:z, y∈U, x, z∈V for somez∈X.
2.1
Recall that auniformspace X,Uis a nonempty setXendowed of a uniformityU, with the latter being a family of subsets ofX×Xand satisfying the following conditions:
u1 Δ⊆V for anyV ∈ U,
u3IfV ∈ U, then there existsW ∈ Usuch thatW◦W−1⊂V,
u4IfV ∈ UandV ⊂W⊂X×X, thenW∈ U.
Two pointsxandyofXare said to beV-closewheneverx, y∈V andy, x∈V. A sequence{xn}n∈NinXis called aCauchy sequenceforUU-Cauchy sequence, for short
if, for anyV ∈ U, there exists ∈Nsuch thatxnandxmareV-close forn,m≥. A nonempty
subsetC ofX is said to besequentiallyU-completeif everyU-Cauchy sequence in C
converges. A uniformityUdefines a unique topologyτUonX. A uniform spaceX,Uis said to beHausdorff if and only if the intersection of all theV ∈ Ureduces to the diagonalΔ ofX, that is, ifx, y∈V for allV ∈ Uimplies thatxy. This guarantees the uniqueness of limits of sequences.
LetX, dbe a metric space. A real-valued functionf : X → −∞,∞is said to be proper iff /≡ ∞. Recall that a functionp:X×X → 0,∞is called aτ-function9,18, if the following conditions hold:
τ1px, z≤px, y py, zfor allx, y, z∈X,
τ2if x ∈ X and {yn} in X with limn→ ∞yn y such that px, yn ≤ M for some
MMx>0, thenpx, y≤M,
τ3for any sequence{xn}inXwith limn→ ∞sup{pxn, xm:m > n}0, if there exists
a sequence{yn}inXsuch that limn→ ∞pxn, yn 0, then limn→ ∞dxn, yn 0,
τ4forx, y, z∈X,px, y 0 andpx, z 0 imply thatyz.
It is known that any w-distance 15,18, 19, 21,22, 30,31 is a τ-function; see 18, Remark 2.1.
The following result is crucial in this paper.
Lemma 2.1. LetX, dbe a metric space and letp :X×X → 0,∞be a function. Assume that
psatisfies conditionτ3. If a sequence{xn}inX withlimn→ ∞sup{pxn, xm:m > n} 0, then
{xn}is a Cauchy sequence inX.
Proof. Let {xn} in X with limn→ ∞sup{pxn, xm : m > n} 0. We claim that {xn} is a
Cauchy sequence. For eachn∈N, letnsupi,j≥ndxi, xj. Then{n}is nonincreasing and so
limn→ ∞n : ∞exists. If∞ > 0, then there exist sequences{in}and{jn}within < jnsuch
thatdxin, xjn> 1/2∞forn∈N. On the other hand, since limn→ ∞pxin, xjn 0, byτ3,
we have limn→ ∞dxin, xjn 0, a contradiction. Therefore∞ 0 which shows that{xn}is a
Cauchy sequence inX.
Remark 2.2. Notice that the functionpwas assumed aτ-function in18,Lemma 2.1and the
proof of18,Lemma 2.1was incomplete since only limn→ ∞dxn, xn1 0 was demonstrated
if any sequence{xn}inXsatisfied limn→ ∞sup{pxn, xm:m > n}0.
3. New Critical Point Theorems in Uniform Spaces and Metric Spaces
Theorem 3.1. LetX be a nonempty set, and letψ : X×X → −∞,∞andτ :X → −∞,∞
be functions. LetDbe a nonempty subset ofX andΓ :D → 2Da multivalued map with nonempty
values. Suppose the following:
H10≥ψx, y≥τxfor allx∈ Dand ally∈Γx,
H2for anyx ∈ Dand ε > 0, there existsy yx, ε ∈ Γxsuch thatτz ≥ −εfor all
z∈Γy.
Then there exists a sizing-up functionμ: 2D → 0,∞such thatΓis of typeμ.
Proof. Defineμψ,Γ: 2D → 0,∞by
μψ,ΓA
⎧ ⎨ ⎩
0, if A∅,
sup−ψx, y:x∈A, y∈Γx, if A /∅.
3.1
Thenμψ,Γis a sizing-up function. We will claim thatΓis of typeμψ,Γ. Letx∈ Dandε >0
be given. ByH1andH2, there existsyyx, ε∈Γxsuch that
μψ,ΓΓysup−ψa, b:a∈Γy, b∈Γa ≤sup−τa:a∈Γy
≤ε.
3.2
HenceΓis of typeμψ,Γ.
Theorem 3.2. Let X,U be a uniform space, and let ψ : X ×X → −∞,∞ and τ : X →
−∞,∞be functions. LetDbe a sequentiallyU-complete nonempty subset ofXandΓ:D → 2Da
multivalued map with nonempty values. Suppose that conditions (H1) and (H2) inTheorem 3.1hold
and further assume that
H3for eachx∈ D,x∈ΓxandΓxis closed inD,
H4x,y∈ Dwithy∈Γximplies thatΓy⊆Γx,
H5for each V ∈ U, there existsδ δV > 0 such that x,y ∈ D with y ∈ Γxand
ψx, y>−δimplies thatx, y∈V.
Then there exist a quasiorderonDand a sizing-up functionμ: 2D → 0,∞such thatD,, μ
is aμ-bounded quasiordered set.
Proof. Put a binary relationΓonDby
xΓy⇐⇒y∈Γx 3.3
and letSΓ : D → 2D be defined bySΓx {y ∈ D: xΓy}. Clearly,SΓx Γxfor eachx ∈ D andΓ is a quasiorder from H3 andH4. Letμψ,Γ : 2D → 0,∞be the
same as inTheorem 3.1. From the proof ofTheorem 3.1, we know thatμψ,Γ is a sizing-up
quasiordered set. Letc1Γc2Γ· · ·ΓcnΓcn1Γ· · · be a nondecreasing sequence inD
satisfying
0 lim
n→ ∞μψ,Γ{cn, cn1,· · · }
lim
n→ ∞sup
−ψx, y:x∈ {cn, cn1,· · · }, y∈SΓx
.
3.4
Sincecm ∈ SΓcnform,n∈Nwithm ≥n, limn,m→ ∞ψcn, cm 0. LetV ∈ Uand choose
W ∈ Usuch thatW◦W−1 ⊂V. ByH5, there existsδ δW >0 such thatx,y∈ Dwith
y ∈Γxandψx, y > −δimply thatx, y∈ W. Since limn,m→ ∞ψcn, cm 0, there exists
n0∈Nsuch thatψcn, cm>−δfor allm,n∈Nwithm≥n≥n0. It implies thatcn, cm∈W
and hencecm, cn ∈ W−1for allm,n ∈ Nwith m ≥ n ≥ n0. SinceW◦W−1 ⊂ V, we have cn, cm ∈ V andcm, cn ∈ V form ≥ n ≥ n0. Therefore,{cn}n∈N is a nondecreasingU
-Cauchy sequence inD. By the sequentialU-completeness ofD, there existsξ∈ Dsuch that
cn → ξasn → ∞. For eachn∈N, sinceSΓcnis closed fromH3and
cm∈SΓcm⊆SΓcn ∀m≥n, 3.5
we obtainξ∈SΓcnorcnΓξ. Henceξis an upper bound of{cn}. ThereforeD,Γ, μψ,Γ
is aμψ,Γ-bounded quasiordered set.
Theorem 3.3. LetX,Ube a Hausdorffuniform space, and letψ:X×X → −∞,∞andτ:X →
−∞,∞be functions. LetDbe a sequentiallyU-complete nonempty subset ofX,g : D → Da
map, andΓ : D → 2D a multivalued map with nonempty values. LetI be any index set. For each
i∈I, letFibe a fuzzy map onD. Suppose the conditions (H1), (H2), (H3), and (H5) inTheorem 3.2
hold and further assume
H4Sx,y∈ Dwithy∈Γximplies thatgy∈ΓxandΓy⊆Γx;
H6for anyi, x∈I× D, there existsyi,x∈Γxsuch thatFix, yi,x 1.
Then there existsz0∈ Dsuch that
aFiz0, z0 1for alli∈I, b Γz0 {gz0}{z0}.
Proof. ApplyingTheorem 3.1andTheorem 3.2,Γ≡SΓis of typeμψ,ΓandD,Γ, μψ,Γ
is aμψ,Γ-bounded quasiordered set, whereΓ,SΓ, andμψ,Γare the same as in Theorems 3.1and3.2. By Theorem LD, for eachx∈ D, there existsvx∈Γxsuch thatμψ,ΓΓvx 0.
Then it follows from the definition ofμψ,Γ,vx∈Γvx, andμψ,ΓΓvx 0 thatψvx, z 0
for allz∈Γvx. We want to prove thatΓvx {vx}. Sinceψvx, z>−δfor allz∈Γvxand
allδ >0, byH5, we havevx, z∈Vfor allz∈Γvxand allV ∈ U. SinceUis a Hausdorff
uniformity,
vx, z∈
V∈U
V Δ ∀z∈Γvx, 3.6
and hence we haveΓvx {vx}. For eachi∈I, byH6,Fivx, vx 1. On the other hand,
byH4S, we have gvx ∈ Γvx {vx}. ThereforeΓvx {gvx} {vx}. The proof is
Theorem 3.4. LetX,U,D,g,Γ,ψ, andτbe the same as inTheorem 3.3. Assume that the conditions
(H1), (H2), (H3),H4S, and (H5) in Theorem 3.3hold. LetI be any index set. For eachi ∈ I, let
Ti:D → 2Dbe a multivalued map with nonempty values. Suppose that, for eachi, x∈I× D, there
existsyi,x∈Tix∩Γx. Then there existsz0∈ Dsuch that
az0is a common fixed point for the family{Ti}i∈I(i.e.,z0∈Tiz0for alli∈I); b Γz0 {gz0}{z0}.
Proof. For eachi∈I, define a fuzzy mapFionDby
Fi
x, yχTix
y, 3.7
whereχA is the characteristic function for an arbitrary setA ⊂ X. Note that y ∈ Tix ⇔
Fix, y 1 fori∈I. Then for anyi, x∈I× D, there existsyx∈Γxsuch thatFix, yx 1.
SoH6inTheorem 3.3holds and hence all conditions inTheorem 3.3are satisfied. Therefore the result follows fromTheorem 3.3.
Remark 3.5. LetX, dbe a complete metric space. For each >0, let
V x, y∈X×X:dx, y< . 3.8
It is easy to see that the familyUd :{V: >0}is a Hausdorffuniformity onX andXis
Ud-complete.
Lemma 3.6. LetX, dbe a metric space,g :X → X a map andΓ :X → 2X a multivalued map
with nonempty values. Suppose that
h1for eachx∈X,x∈Γx,
h2x,y∈Xwithy∈Γximplies thatgy∈ΓxandΓy⊆Γx,
h3if a sequence {xn} in X satisfies gxn1 ∈ Γxn for each n ∈ N, then
limn→ ∞dxn, xn1 0.
Then there exist functionsψ :X×X → −∞,∞andτ :X → −∞,∞such that the conditions
(H1) and (H2) inTheorem 3.1hold.
Proof. Defineψ:X×X → −∞,∞andτ :X → −∞,∞by
ψx, y−dx, y,
τx inf
y∈Γx
−dx, y. 3.9
ThenH1inTheorem 3.1holds withD ≡X.
Let us verifyH2. Letx∈Xandε >0 be given. Then there existsα∈Nsuch that
2−α< ε
Note first thatτu>−∞for someu∈Γx. Indeed, on the contrary, suppose thatτz −∞
for allz ∈ Γx. Takez1 ∈ Γx. Thusτz1 infy∈Γz1−dz1, y < −1. Hence there exists
z2∈Γz1such thatdz1, z2>1. Sinceτz2<−2, there existsz3∈Γz2such thatdz2, z3>
2. Continuing in the process, we can obtain a sequence{zn} ⊂Xsuch that, for eachn∈N,
izn1∈Γzn,
iidzn, zn1> n.
So, we have limn→ ∞dzn, zn1 ∞which contradicts conditionh3. Therefore there exists
u∈Γxsuch thatτu>−∞. Letv1u. Choosev2∈Γv1 Γu⊆Γxsuch that
−dv1, v2≤τv1
1
2. 3.11
Letk∈Nand assume thatvk∈Xis already known. Then, by induction, we obtain a sequence
{vn}inXsuch thatvn1∈Γvnand
−dvn, vn1≤τvn
1
2n, for eachn∈N. 3.12
It follows that
vn1∈Γvn⊆Γvn−1⊆ · · · ⊆Γv1 Γu⊆Γx, for eachn∈N. 3.13
Byh2andh3, we have limn→ ∞dvn, vn1 0. So there existsβ∈Nsuch that
dvn, vn1<
ε
8, ∀n∈Nwithn≥β. 3.14
SinceΓvn1⊆Γvnfor eachn∈N, we have
τvn inf y∈Γvn
−dvn, y
≤ inf
y∈Γvn1
−dvn1, y
dvn1, vn
τvn1 dvn, vn1.
3.15
From3.12and3.15, we obtain
−τvn1≤2dvn, vn1
1
2n, n∈N. 3.16
Letγmax{α, β}. Hence, combining3.10,3.14, and3.16, we have
0≤ −τvn1<
ε
Letyxvγ1. Thus, by3.13and3.17,yx∈Γxand 0≤ −τyx< ε/2. On the other hand,
from the definition ofτ, we have
0≤dyx, a
≤ −τyx
, ∀a∈Γyx
. 3.18
Finally, in order to complete the proof, we need to show thatτz≥ −εfor allz∈Γyx. Let
z∈Γyx. ThenΓz⊆Γyxanddyx, z≤ −τyx. For anyw∈Γz, sinceΓz⊆Γyx, we
get
dz, w≤dz, yx
dyx, w
≤ −2τyx
< ε, 3.19
and hence it implies thatτz≥ −ε. ThereforeH2can be satisfied.
Theorem 3.7. Let X, d be a complete metric space,g : X → X a map, andΓ : X → 2X a
multivalued map with nonempty values. LetIbe any index set. For eachi∈I, letFibe a fuzzy map
onX. Suppose that conditions (h2) and (h3) inTheorem 3.4hold and further assume
h1Sfor eachx∈X,x∈ΓxandΓxis closed,
h4for anyi, x∈I×X, there existsyi,x∈Γxsuch thatFix, yi,x 1.
Then there existsz0∈Xsuch that
aFiz0, z0 1for alli∈I, b Γz0 {gz0}{z0}.
Proof. For each >0, define
V x, y∈X×X:dx, y< ,
Ud{V: >0}.
3.20
ThenUd is a Hausdorffuniformity onX andX isUd-complete. Clearly, conditionsH3,
H4S, andH6inTheorem 3.3hold. ByLemma 3.6,H1andH2inTheorem 3.1holds. Let
V∈ Udfor >0. TakeδV: >0. Ifx,y∈Xwithy∈Γxandψx, y>−δV,
thendx, y< which means thatx, y∈V. SoH5inTheorem 3.2holds. Therefore the conclusion follows fromTheorem 3.3.
Theorem 3.8. LetX, d,g,I, andΓ be the same as in Theorem 3.7. Assume that the conditions
h1S, (h2) and (h3) inTheorem 3.7hold. LetI be any index set. For eachi ∈ I, letTi : X → 2X
be a multivalued map with nonempty values. Suppose that for eachi, x∈ I×X, there existsyx ∈
Tix∩Γx. Then there existsz0∈Xsuch that
az0is a common fixed point for the family{Ti}i∈I,
b Γz0 {gz0}{z0}.
Remark 3.9. Theorems 3.3–3.8 all generalize and improve the primitive Dancˇs-Heged
4. Some Applications to Nonlinear Problems
The following result is a generalization of Ekeland’s variational principle and Takahashi’s nonconvex minimization theorem forτ-functions with common fuzzy fixed point theorem.
Theorem 4.1. LetX, dbe a complete metric space,f :X → −∞,∞a proper l.s.c. and bounded
from below function,ϕ:−∞,∞ → 0,∞a nondecreasing function, andpaτ-function onXwith
px,·being l.s.c. for eachx∈X. LetI be any index set. For eachi∈I, letFibe a fuzzy map onX.
Suppose that, for eachi, x∈I×Xand anyε >0, there existsyi,x,ε∈Xsuch thatFix, yi,x,ε 1
andεpx, yi,x,ε≤ϕfxfx−fyi,x,ε. Then for eachε >0and anyu∈Xwithfu<∞
andpu, u 0, there existsv∈Xsuch that
aεpu, v≤ϕfufu−fv,
bεpv, x> ϕfvfv−fxfor allx∈Xwithx /v,
cFiv, v 1for alli∈I.
Moreover, if one further assumes that
Hfor eachε >0and anyx∈Xwithfx>infz∈Xfz, there existsy∈Xwithy /xsuch
thatεpx, y≤ϕfxfx−fy,
thenfv infz∈Xfz.
Proof. Takeg ≡ idas an identity map. Letε >0 be given and letu ∈Xwithfu< ∞and
pu, u 0. Put
Wx∈X :εpu, x≤ϕfufu−fx. 4.1
By the lower semicontinuity off andpu,·,Wis a nonempty closed set inX. SoW, dis a complete metric space. DefineΓ:W → 2Wby
Γx y∈ W:xyorεpx, y≤ϕfxfx−fy. 4.2
Then for eachx∈ W, we havex∈ΓxandΓxis closed. It is easy to see that ifx,y∈ Wwith
y∈Γx, thenΓy⊆Γx. By our hypothesis, for eachi, x∈I× W, there existsyi,x∈Γx
such thatFix, yi,x 1.
We will prove that if a sequence{xn}inWsatisfiesxn1 ∈Γxnfor eachn∈N, then
limn→ ∞dxn, xn1 0. Let{xn} ⊂ Wsatisfyxn1 ∈Γxnfor eachn ∈N. Then{fxn}is a
nonincreasing sequence. Sincef is bounded below,r ≡ limn→ ∞fxn infn∈Nfxnexists.
We claim that limn→ ∞sup{pxn, xm:m > n}0. Letαn 1/εϕfx1fxn−r,n∈N.
Form, n∈Nwithm > n, sinceϕis nondecreasing, we have
pxn, xm≤ m−1
jn
pxj, xj1
≤αn. 4.3
Then sup{pxn, xm : m > n} ≤ αn for each n ∈ N. Since limn→ ∞fxn r, we
Cauchy sequence inW, and hence we have limn→ ∞dxn, xn1 0. So all the conditions of
Theorem 3.7are satisfied. ApplyingTheorem 3.7, there existsv∈ Wsuch that
Fiv, v 1, ∀i∈I, 4.4
Γv {v}. 4.5
Sincev∈ W, we have the conclusiona. From4.5,εpv, x> ϕfvfv−fx
for allx∈ Wwithx /v. For anyx∈X\ W, since
εpu, v pv, x≥εpu, x
> ϕfufu−fx
≥εpu, v ϕfvfv−fx,
4.6
it follows thatεpv, x> ϕfvfv−fxfor allx∈X\ W. So the conclusionbholds. Moreover, assume that conditionHholds. On the contrary, iffv>infx∈Xfx, then
there existsw∈Xwithw /vsuch thatεpv, w≤ϕfvfv−fw. But, byb, we have
εpv, w> ϕfvfv−fw≥εpv, w, 4.7
a contradiction. Thereforefv infz∈Xfz. The proof is completed.
By usingTheorem 4.1, we can immediately obtain the followingτ-function version of generalized Ekeland’s variational principle, generalized Takahashi’s nonconvex minimiza-tion theorem, and generalized Caristi’s common fixed point theorem for multivalued maps.
Theorem 4.2. LetX, d,f,ϕ, andpbe the same as inTheorem 4.1. LetIbe any index set. For each
i∈I, letTi:X → 2X be a multivalued map with nonempty values such that, for eachi, x∈I×X
and anyε >0, there existsyi,x∈Tixsuch thatεpx, yi,x≤ϕfxfx−fyi,x. Then for
eachε >0andu∈Xwithfu<∞andpu, u 0, there existsv∈Xsuch that
aεpu, v≤ϕfufu−fv,
bεpv, x> ϕfvfv−fxfor allx∈Xwithx /v,
cvis a common fixed point for the family{Ti}i∈I.
Moreover, if one further assumes that
Hfor eachε >0and anyx∈Xwithfx>infz∈Xfz, there existsy∈Xwithy /xsuch
thatεpx, y≤ϕfxfx−fy,
thenfv infz∈Xfz.
Remark 4.3. Theorem 4.2extends some results in2,8,14,15,19,22and references therein.
Theorem 4.4. LetX, dbe a complete metric space,Ea nonempty set withα∈E, andL:X×X →
2Ea multivalued map. LetIbe any index set. For eachi∈I, letF
ibe a fuzzy map onX. Assume that
D1for eachx∈X, the set{y∈X:xyorα∈Lx, y}is a closed subset ofX,
D2x, y, z∈Xwithα∈Lx, yandα∈Ly, zimplies thatα∈Lx, z,
D3if a sequence{xn}n∈NinXsatisfiesα∈Lxn, xn1for eachn∈N, thendxn, xn1 → 0
asn → ∞,
D4for anyi, x∈I×X, there existsyi,x∈Xsuch thatα∈Lx, yi,xandFix, yi,x 1.
Then there existsv∈Xsuch that
aFiv, v 1for alli∈I,
bα /∈Lv, xfor allx∈X\ {v}.
Proof. Takeg≡idas an identity map. DefineΓ:X → 2Xby
Γx y∈X:xyorα∈Lx, y. 4.8
Clearly,h1S,h3, andh4inTheorem 3.7hold. To seeh2, letx,y ∈X withy ∈
Γx. We need to consider the following two possible cases:
Case 1. Ifxy, thenΓy Γx⊆Γxis obvious.
Case 2. Ifx /y, thenα ∈Lx, y. For anyz∈ Γy, ifz y, one hasz ∈Γx. Otherwise, if
α∈Ly, z, then it follows fromα∈Lx, yandD2thatα∈Lx, z. Soz∈Γx. Therefore
Γy⊆Γx.
By Cases1and2, we prove thath2holds. ApplyingTheorem 3.7, there existsv∈X
such that
1Fiv, v 1 for alli∈I,
2 Γv {v}.
From2, we obtainα /∈Lv, xfor allx∈X\ {v}.
Remark 4.5. Theorem 4.4generalizes17, Theorems 3.1which is one of the main results of
Lin and Chuang17.
Here, we give an example illustratingTheorem 4.4.
Example 4.6. Let X 0,3 with the metric dx, y |x−y|for x,y ∈ X. Then X, d is
a complete metric space. Let E −1000,−500∪ {−360} ∪ {−210} ∪−100,∞ and letL :
X×X → 2Ebe defined by
LetC0{0}andCn n−1, n, for everyn∈ {1,2,3}, and define a fuzzy mapF:X×X →
0,1by
Fx, y
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0, ifx, y∈C0×C0,
1 4, if
x, y∈X×C1,
1 2, if
x, y∈X×C2,
1, ifx, y∈X×C3.
4.10
Clearly, 5∈Lx, xfor eachx∈X. Note that, for eachx∈X,{y∈X:xyor 5∈Lx, y}
{y ∈ X : 5 ∈Lx, y} x,3is nonempty and closed inX. SoD1andD4hold. To see
D2, letx, y, z∈X with 5∈ Lx, yand 5∈Ly, z. It is easy to see that 5 ∈Lx, zholds. Finally, let{xn}n∈Nbe a sequence inX satisfing 5 ∈Lxn, xn1for eachn ∈N. So{xn}is a
nondecreasing sequence andxn ≤ 3 for each n ∈ N. Thus{xn}converges in X and hence
dxn, xn1 → 0 asn → ∞. SoD3also holds. ByTheorem 4.4, there existsv∈X in fact,
we takev3such thatFv, v 1 and 5/∈Lv, yfor ally∈X\ {v}.
The following conclusion is immediate fromTheorem 4.4.
Theorem 4.7. LetX, d,E,α,L,D1,D2, andD3be the same as inTheorem 4.4. LetIbe any
index set. For eachi∈I, letTi:X → 2Xbe a multivalued map with nonempty values. Suppose that
for eachi, x∈I×X, there existsyi,x∈Tixsuch thatα∈Lx, yi,x.
Then there existsv∈Xsuch that
avis a common fixed point for the family{Ti}i∈I,
bα /∈Lv, xfor allx∈X\ {v}.
Following a similar argument as inTheorem 4.4, we can easily obtain the following existence theorem of nonconvex version of variational inclusion problem in metric spaces.
Theorem 4.8. InTheorem 4.4, if conditionsD1andD2are replaced by the conditionD1and
D2, where
D1for eachx∈X, the set{y∈X:xyorα /∈Lx, y}is a closed subset ofX,
D2x, y, z∈Xwithα /∈Lx, yandα /∈Ly, zimplies thatα /∈Lx, z,
then there existsv∈Xsuch that
aFiv, v 1for alli∈I,
bα∈Lv, xfor allx∈X\ {v}.
Theorem 4.9. LetX, d,E,α,L,D1,D2, andD3be the same as inTheorem 4.8. LetI be
any index set. For eachi∈I, letTi:X → 2Xbe a multivalued map with nonempty values. Suppose
Then there existsv∈Xsuch that
avis a common fixed point for the family{Ti}i∈I,
bα∈Lv, xfor allx∈X\ {v}.
The following existence theorem of nonconvex version of variational inclusion and disclusion problem in the Ekeland’s sense is immediate fromTheorem 4.4.
Theorem 4.10. LetX, dbe a complete metric space,paτ-function onXwithpx,·being l.s.c. for
eachx ∈X,V a topological vector space with originθ,G :X×X → 2V a multivalued maps, and
k0∈V \ {θ}. LetIbe any index set. For eachi∈I, letFibe a fuzzy map onX. Assume that
S1for eachx∈X, the set{y∈X:xyorθ∈Gx, y px, yk0}is closed inX, S2x, y, z ∈ X with θ ∈ Gx, y px, yk0 and θ ∈ Gy, z py, zk0 implies that
θ∈Gx, z px, zk0,
S3if a sequence{xn}n∈NinXsatisfiesθ∈Gxn, xn1 pxn, xn1k0for eachn∈N, then
dxn, xn1 → 0asn → ∞,
S4for anyi, x∈I×X, there existsyi,x∈Xsuch thatθ∈Gx, yi,x px, yi,xk0and
Fix, yi,x 1.
Then for eachu∈Xwithθ∈Gu, uandpu, u 0, there existsv∈Xsuch that
iθ∈Gu, v pu, vk0,
iiθ /∈Gv, x pv, xk0for allX\ {v}, iiiFiv, v 1for alli∈I.
Proof. Letu ∈ X be given andL : X×X → 2V defined byLx, y : Gx, y px, yk
0
forx, y ∈X×X. PutM :{x∈ X : θ ∈ Lu, x}. Sinceu∈ M,M/∅. ByS1,M, dbe a complete metric space. It is not hard to see that all conditions inTheorem 4.4are satisfied fromS1–S4. ApplyingTheorem 4.4, there existsv∈ Msuch thatFiv, v 1 for alli∈I
andθ /∈Lv, xfor allx∈ M \ {v}or, equivalently,
aθ∈Gu, v pu, vk0,
bθ /∈Gv, x pv, xk0for allM \ {v}.
For anyx∈X\ M, ifθ∈Gv, x pv, xk0, then, byS2anda, we havex∈ M, which is
a contradiction. Thereforeθ /∈Gv, x pv, xk0for allX\ {v}.
Remark 4.11. Theorem 3.2 in17is a special case ofTheorem 4.10.
Acknowledgments
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