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If, moreover,u satisfies the relation uK0, then inequality 2.3 implies The following definition, which is a modified version of one introduced in [3], vides a kind of the strict inequali

Volume 2007, Article ID 46041, 25 pages

doi:10.1155/2007/46041

Research Article

Equivalent Solutions of Nonlinear Equations in a Topological Vector Space with a Wedge

A Ront ´o and J ˇSremr

Received 31 December 2006; Revised 20 May 2007; Accepted 28 May 2007

Recommended by Simeon Reich

We obtain efficient conditions under which some or all solutions of a nonlinear equation

in a topological vector space preordered by a closed wedge are comparable with respect tothe corresponding preordering Conditions sufficient for the equivalence of comparablesolutions are also given The wedge under consideration is not assumed to be a cone, norany continuity conditions are imposed on the mappings considered

Copyright © 2007 A Ront ´o and J ˇSremr This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited

1 Introduction

The aim of this note is to establish certain order-theoretical properties of the set of tions of the equation

whereT : X → X is a (generally speaking, nonlinear and discontinuous) operator in a real

topological vector spaceX, λ is a real constant, and b is a given element of X.

The question on the set of all the admissible values ofx and λ in (1.1) is sometimesreferred to as a nonlinear eigenvalue problem [1] The present paper is motivated by someresults obtained in [2] and, recently, in [3–5] In the case where the wedge in question is anormal cone, we arrive at statements similar to the uniqueness results from [6] Note thatthe algebraic conditions used here, generally speaking, do not guarantee the solvability of(1.1) The study of the question on the existence of a solution, which is not treated in thispaper, depends upon further assumptions expressing a certain closer interplay betweenthe partial ordering and the topology inX For some efficient conditions sufficient for

the solvability of abstract second-kind equations of type (1.1) in a Banach space with anormal cone, we refer, for example, to [2].

2 Definitions, notation, and auxiliary statements

LetX be a topological vector space over the field R(of course,X = {0}) Throughoutthe paper, we assume that the spaceX is equipped with the preorderingK generated

by a certain wedgeK According to this preordering, elements x1andx2, by definition,satisfy the relationx1K x2 if and only ifx2− x1∈ K (this fact will also be expressed

alternatively asx2K x1in the sequel) Recall that, by a wedge [7], or a linear semigroup

[8] in a topological vector spaceX, a closed set K ⊂ X is meant such that α1x1+α2x2∈ K

for arbitrary{ α1,α2} ⊂[0, +∞) and{ x1,x2} ⊂ K It should be noted that the fulfilment

of both of the relationsx1K x2andx2K x1, generally speaking, does not imply thatx1

andx2should coincide with one another A wedgeK is said to be proper if K = {0}and

K = X.

The linear manifold

consisting of the elementsx that satisfy each of the relations xK0 andxK0 is called

the blade [7] of the wedgeK (here, as usual, αK : = { αx | x ∈ K }for all realα) The most extensively used class of wedges is constituted by the cones [8,9], that is, the wedges whoseblade is trivial

Definition 2.1 Two elements x1andx2fromX are said to be K-comparable if at least one

of the relationsx1K x2andx2K x1is satisfied

The property ofx1andx2beingK-comparable will be designated in the sequel by the

symbolK:x1K x2 It is clear thatx1K x2 means the same as the relationx2K x1.Elementsx1andx2such thatx2K x1will be referred to asK-incomparable.

In the general case, the relation x1K x2 is satisfied not for all pairs of elements(x1,x2)∈ X2

Definition 2.2 The relation x1K x2holds for two elementsx1andx2ofX if and only if

x1− x2∈ K 

Clearly,Kis an equivalence relation inX for an arbitrary choice of the wedge K Two

elementsx1andx2satisfying the relationx1K x2will be referred to asK-equivalent The

elementsx from X satisfying the relation xK0 (i.e., those belonging to the setK ) will

be calledK-negligible.

Example 2.3 If X = l ∞, the space of bounded real sequences with the usual topology, and

K is the wedge defined by the formula

K =x : N −→ R | x ∈ l ∞,x(k) ≥0∀ k ∈ S

(2.2)with some nonempty setS ⊆ N, then an elementx is K-negligible if and only if the equal-

ityx(k) =0 is satisfied for allk from S.

The lemma below states some simple properties of the symmetric two-sided ties that are often referred to in the sequel.

inequali-Lemma 2.4 Elements x and u from X satisfy the relation

that is,uK0 If, moreover,u satisfies the relation uK0, then inequality (2.3) implies

The following definition, which is a modified version of one introduced in [3], vides a kind of the strict inequality inX.

pro-Definition 2.5 Let H be a linear manifold in the space X Two elements f1andf2ofX are

In the case where the linear manifoldH coincides with the entire space X, the

sub-script “X” in the expression “ K;X” will be omitted Thus, the following definition isintroduced

Definition 2.6 One says that

if and only if, for an arbitraryx from X, relation (2.7) is true with someβ ∈[0, +∞)

In other words, the elements f1and f2satisfy relation (2.8) whenever (2.6) is true for

an arbitraryH.

Proposition 2.7 If some elements f1 and f2 from X satisfy relation ( 2.6 ) for a certain linear manifold H such that

then the relations

are necessarily satisfied In the case where condition ( 2.9 ) is not satisfied, an arbitrary pair of elements ( f1,f2)∈ X2possesses property ( 2.6 ).

Proof Indeed, according to Definition 2.5, relation (2.6) means that every elementx

fromH satisfies condition (2.7) with a certain constantβ ≥0 Amidst suchx, in view

of assumption (2.9), there are some that are notK-negligible, that is,

Forx satisfying (2.11), the constantβ in (2.7) cannot be equal to zero, and therefore

Lemma 2.4implies relations (2.10)

Condition (2.9) is violated if and only if every element fromH is K-negligible

There-fore, withβ =0, relation (2.7) is satisfied in this case for an arbitraryx from H and every

f1,f2fromX According toDefinition 2.5, this means that (2.6) is true independently of

natu-positive,” which circumstance makes the notion useless

Definition 2.8 [8] Two elements f1and f2are said to satisfy the relation f1 K f2if the

difference f1− f2is an interior element of the wedgeK.

It is well known [8] that if elements f1 and f2 satisfy the condition f1 K f2, thenrelation (2.8) is true The converse statement, generally speaking, is not true (seeExample2.9) Of course, the notion described byDefinition 2.8makes sense only ifK has non-

empty interior (i.e., is solid [8])

Example 2.9 In the Banach space L ∞[0, 1] of measurable and essentially bounded scalarfunctions on the interval [0, 1] with the coneK of functions that are nonnegative almost

everywhere on [0, 1], the corresponding relation f K0 is satisfied, for example, for thepositive-valued constant functions However, the interior of the abovementioned cone in

L ∞[0, 1] is empty

The proofs of the results of this paper rely upon properties of a certain nonlinear tional associated with the wedgeK and a certain suitably chosen element f from X Definition 2.10 Given some elements f and x, put

Thus, a mappingn K, f :X →[0, +∞] is associated with an arbitrary f from X Besides

the properties of this mapping stated inLemma 2.12below, we note the equality

For a suitable f , there is a close interplay between K  and the set of zeroes of themappingn K, f :X →[0, +∞]

Lemma 2.12 Let f be an element satisfying relation ( 2.12 ) with respect to a certain linear manifold H ⊆ X possessing property ( 2.9 ) Then,

(i)n K, f(x)= 0 if and only if xK 0.

(ii) For all x ∈ H ∪ K  , the relation

Proof Assertion (i) is established in the same manner as [3, Lemma 2.13] is in the case

of a Banach spaceX Indeed, if

Conversely, letx be an element from X such that equality (2.19) holds According to

Definition 2.10, there exists a sequence{ β m | m ∈ N} ⊂[0, +∞) such that

lim

for allm ∈ N In view of (2.20), limm →+∞ β m f =0 in the topology ofX Since (2.21) can

be rewritten in the form of the inclusion

with some sequence{ β m | m ∈ N} ⊂[0, +∞) such that (2.21) holds for allm ∈ N Passing

to the limit asm →+∞in (2.21) and using (2.24), we arrive at (2.16) 

In view ofProposition 2.7, there is no much sense to consider relations of type (2.12)with respect to the linear manifoldH for which condition (2.9) is not satisfied Thisfact explains the presence of assumption (2.9) inLemma 2.12and its absence from theformulations of the results of Sections3and4(seeRemark 3.2)

Remark 2.13 The fulfilment of assumption (2.12) inLemma 2.12implies, in particular,that the elementf satisfies the relations f K0 and f K0

In the statements established in Sections3and4, certain conditions generalizing theproperty of linearity of a mapping are used The corresponding notions are introduced

by Definitions2.14and2.16given below Note that other similar notions of subadditivity,superadditivity, convexity, and concavity for operators in various partially ordered spacesand their algebraic properties are treated in [9,13–16]

Definition 2.14 An operator A : X → X is said to be positively homogeneous on a set S ⊆ X

if the relation

is satisfied for arbitraryu ∈ S and α ∈[0, +∞)

Remark 2.15 It is clear that every mapping A : X → X which is continuous in a

neigh-bourhood of 0 and positively homogeneous on a nonempty set possesses the property

is true for allu1andu2fromS Similarly, an operator A : X → X will be called K-subadditive

u2 

(2.27)for allu1andu2fromS.

In the case where relation (2.26) (resp., (2.27)) is satisfied on the entire spaceX, one

will speak simply on theK-superadditivity (resp., K-subadditivity) of the operator A.

Every linear operator inX is of course positively homogeneous and both

K-superaddi-tive andK-subadditive with respect to an arbitrary wedge K ⊆ X A characteristic

exam-ple of a pair of nonlinear operators possessing the properties indicated is provided by thepositive and negative parts of a function

Example 2.17 Let X : = C([0,1],R) be the space of the continuous scalar-valued functions

on the interval [0, 1], letK : = C([0,1],R +) be the cone of nonnegative functions from

C([0,1],R), and letA :X → X, where  ∈ {−1, 1}, be the operator defined by the formula



A  x(t) := max

 x(t),0

Then,A isK-subadditive (resp., K-superadditive) on the entire space X if  =1 (resp.,

 = −1) In both cases, operator (2.28) is positively homogeneous

In some cases, theK-superadditivy and K-subadditivity conditions are satisfied

simul-taneously without implying the linearity of the mapping

Example 2.18 The operator A : C([0,1],R)→ C([0,1],R) given by formula

whereγ ∈(0, +∞),p(t, ·)∈ L1([0, 1],R) for allt ∈[0, 1], andp( ·,s) ∈ C([0,1],R) for a.e

s ∈[0, 1], is positively homogeneous and bothK-superadditive and K-subadditive on the

coneK : = C([0,1],R +) Note that operator (2.29) is nonlinear

3 Mutual comparability of solutions of ( 1.1 )

The aim of this section is to establish certain conditions under which each two solutions

of (1.1) lying in a certain linear manifold areK-comparable with one another.

3.1 Main theorems The theorem below claims that, under fairly general assumptions, a

certain two-sided condition imposed on the nonlinear mappingT guarantees the mutual

comparability of some or all solutions of (1.1) for| λ |large enough

Theorem 3.1 Assume that, for the mapping T : X → X, there exist a linear manifold Π ⊆ X and an operator A : X → X which is positively homogeneous and K-subadditive on the set Π and satisfies the condition

for arbitrary { y1,y2} ⊂ Π such that y1K y2and y1K y2 Let, moreover, the relation

K-In (3.3) and similar relations, the symbolT(M) stands for the image of a set M under

the mappingT Prior to the proof ofTheorem 3.1, we give some comments on the choice

of the linear manifoldH appearing in relation (3.3)

Remark 3.2 Let the mapping T : X → X satisfy relation (3.3) with some linear manifolds

H ⊆ X and Π ⊆ X If condition (2.9) does not hold, then for any nonzeroλ, all the

solu-tions of (1.1) that belong toΠ are K-equivalent to one another Indeed, any two solutions

{ x1,x2} ⊂Π of (1.1) obviously satisfy the relation

The consideration above shows that the assertions of the statements of Sections3and

4involving the linear manifoldH become trivial when (2.9) is violated, and we thus donot deal with this case in the proofs

Proof of Theorem 3.1 In view ofRemark 3.2, it will suffice to consider the case where thelinear manifoldH satisfies condition (2.9)

Letx1andx2be two distinct solutions of (1.1) lying in the setΠ Then (3.5) is true.Condition (3.3) and the linearity of the setH guarantee that

and hence relation (3.5) yieldsλ(x1− x2)∈ H In view of estimate (3.4),λ is nonzero, and

therefore, again by the linearity ofH, the last relation implies that

Relations (2.12), (3.7), property (2.9) of the linear manifoldH, and assertions (ii) and

(iii) ofLemma 2.12then guarantee that the valuen K, f(x1− x2) is a finite number, and theorder inequality

We need to prove the mutualK-comparability of the solutions x1andx2 Assume that,

on the contrary,x1andx2areK-incomparable, that is, the relation



x1+x2− u

(3.16)and using (3.10), we conclude thatx2K x1, which contradicts (3.9) Therefore, in addi-tion to (3.14), the relation

because relations (3.10) and (3.14) are satisfied.

In addition, both y1and y2 lie inΠ because, by assumption, the set mentioned is alinear manifold containing the elementf Therefore, in view of the assumption (3.1) andthe equalities



x1+x2− u

K A

12

in view of relations (3.11) and (3.17), we get relation (3.19) and the inclusion{ y1,y2} ⊂

Π By virtue of condition (3.1), we obtain



x1+x2− u

K A

12

2A(u),

(3.24)that is,

(3.26)and thus

The element f is assumed to satisfy condition (3.2), and therefore the last inequalityyields

By virtue of assertion (i) ofLemma 2.12, equality (3.35) yieldsx1K x2 However, sumption (3.9) implies, in particular, that

Proof It is easy to see that x is a solution of (1.1) if and only if the elementw : = − x is a

solution of the equation

SinceΠ is a linear manifold, together with z1andz2, it contains the vectorsy1:= − z2and

y2:= − z1 Note that

By assumption,T satisfies condition (3.1), and thus we have

Using (3.39) and (3.42), we can bring the last relation to form (3.41)

The operatorA is K-subadditive on the set Π Indeed, if u1,u2∈Π, then, by virtue oftheK-superadditivity of A on the set Π, we have

u2 

Moreover, it is clear that the operatorA is also positively homogeneous on Π.

SinceΠ is a linear manifold, we have−Π=Π, and hence (3.39) yields

Assumption (3.3) then implies thatT(Π) ⊆ H.

Finally, relation (3.37), in view of (3.42), can be rewritten as

Since every linear operator inX is of course positively homogeneous and both

K-superadditive andK-subadditive, Theorems3.1and3.3immediately yield

Corollary 3.4 Assume that, for the given mapping T : X → X, there exist a linear foldΠ⊆ X and a linear operator A : X → X such that the condition

Then, for an arbitrary real λ satisfying estimate ( 3.4 ) and an arbitrary element b from X, all the solutions of ( 1.1 ) that belong to the set Π are K-comparable to one another.

Remark 3.5 The assertion ofCorollary 3.4 can also be proved in the case whereA is

only assumed to be positively homogeneous andK-superadditive on the wedge K The

resulting theorem is somewhat strange due to the fact that theK-superadditive

opera-tors themselves are not typical representatives of the class of mappingsT satisfying the

symmetric conditions of form (3.49) We do not dwell on this here in more detail

Remark 3.6 We note that abstract Lipschitz-type conditions of form (3.49) in the casewhereX is a Banach space, K ∩(− K) = {0}, andΠ= X are used by some fixed point the-

orems (e.g., [6, Theorem 49.3] and [2, Theorem 2]) Certain assumptions on nonlinear

for arbitrary { y1,y2} ⊂ Π...

In addition, both y1and... (3.2), and therefore the last inequalityyields

By virtue of assertion (i) ofLemma 2.12, equality