**PROBLEMS WITH AN INCREASING COST** **FUNCTION AND POROSITY**

ALEXANDER J. ZASLAVSKI
*Received 18 July 2002*

We consider the minimization problem *f*(x)*→*min,*x**∈**K, whereK*is a closed
subset of an ordered Banach space*X*and*f* belongs to a space of increasing lower
semicontinuous functions on*K. In our previous work, we showed that the com-*
plement of the set of all functions*f*, for which the corresponding minimization
problem has a solution, is of the first category. In the present paper we show that
this complement is also a*σ*-porous set.

**1. Introduction**

The study of a generic existence of solutions in optimization has recently been a rapidly growing area of research (see [1,2,3,4,5,6,8,9,10,12,13,14,15]

and the references mentioned there). Instead of considering the existence of so- lutions for a single cost function, we study it for a space of all such cost functions equipped with an appropriate complete uniformity and show that a solution ex- ists for most of these functions. Namely, we show that in the space of functions, there exists a subset which is a countable intersection of open everywhere dense sets such that for each cost function in this subset, the corresponding minimiza- tion problem has a unique solution. This approach allows us to establish the ex- istence of solutions of minimization problems without restrictive assumptions on the functions and on their domains.

Let*K*be a nonempty closed subset of a Banach ordered space (X,* · **,**≥*). A
function *f* :*K**→*R^{1}*∪ {*+*∞}*is called increasing if

*f*(x)*≤**f*(y) *∀**x, y**∈**K*such that*x**≤**y.* (1.1)
Increasing functions are considered in many models of mathematical econom-
ics. As a rule, both utility and production functions are increasing with respect
to natural order relations.

Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:11 (2003) 651–670 2000 Mathematics Subject Classification: 49J27, 90C30, 90C48 URL:http://dx.doi.org/10.1155/S1085337503212094

In this paper, we study the existence of a solution of the minimization prob- lem

*f*(x)*−→*min, *x**∈**K,* (1.2)

where *f* :*K**→*R^{1}*∪ {*+*∞}*is an increasing lower semicontinuous function. In
[10,12], it was established the generic existence of solutions of problem (1.2)
for certain classes of increasing lower semicontinuous functions*f*. Note that the
perturbations which are usually used to obtain a generic existence result are not
suitable for these classes since they break the monotonicity. In [10], we proposed
the new kind of perturbations which allowed us to establish the generic existence
of solutions for certain classes of increasing lower semicontinuous functions. In
the present paper, we show that the complement of the set of all functions *f*, for
which the corresponding minimization problem has a solution, is not only of
the first category but also*σ-porous.*

Before we continue, we briefly recall the concept of porosity [2,4]. As a matter of fact, several diﬀerent notions of porosity have been used in the literature. In the present paper, we will use porosity with respect to a pair of metrics, a concept which was introduced in [15].

When (Y, d) is a metric space, we denote by*B** _{d}*(y, r) the closed ball of center

*y*

*∈*

*Y*and radius

*r >*0. Assume that

*Y*is a nonempty set and

*d*1

*, d*2:

*Y*

*×*

*Y*

*→*[0,

*∞*) are two metrics which satisfy

*d*1(x, y)

*≤*

*d*2(x, y) for all

*x, y*

*∈*

*Y*. A subset

*E*

*⊂*

*Y*is called porous in

*Y*with respect to the pair (d1

*, d*2) (or just porous in

*Y*if the pair of metrics is understood) if there exist

*α*

*∈*(0,1) and

*r*0

*>*0 such that for each

*r*

*∈*(0, r0] and each

*y*

*∈*

*Y*, there exists

*z*

*∈*

*Y*for which

*d*2(z, y)

*≤*

*r*and

*B*

*d*1(z, αr)

*∩*

*E*

*= ∅*. A subset of the space

*Y*is called

*σ*-porous in

*Y*with respect to (d1

*, d*2) (or just

*σ*-porous in

*Y*if the pair of metrics is understood) if it is a countable union of porous (with respect to (d1

*, d*2)) subsets of

*Y*. Note that if

*d*1

*=*

*d*2, then by [15, Proposition 1.1] our definitions reduce to those in [2,4].

We use porosity with respect to a pair of metrics because in applications a space
is usually endowed with a pair of metrics and one of them is weaker than the
other. Note that porosity of a set with respect to one of these two metrics does
not imply its porosity with respect to the other metric. However, it is shown
in [15, Proposition 1.2] that if a subset*E**⊂**Y* is porous in*Y* with respect to
(d1*, d*2), then*E*is porous in*Y* with respect to any metric which is weaker than
*d*2and stronger than*d*1.

We obtain our main results as a realization of a general variational principle which is established inSection 3.

**2. Well-posedness of optimization problems with increasing cost functions**
In this paper, we use the following notations and definitions. Let (X,* · **,**≥*) be a
Banach ordered space and*X*+*= {**x**∈**X*:*x**≥*0*}*the cone of its positive elements.

Assume that*X*+is a closed convex cone such that*x** ≤ **y*for each*x, y**∈**X*+

satisfying*x**≤**y. We assume that the coneX*+has the following property:

(A) if*{**x**i**}*^{∞}_{i}* _{=}*1

*⊂*

*X,x*

*i+1*

*≤*

*x*

*i*, for all integers

*i*

*≥*1 and sup

*{*

*x*

*i*:

*i*

*=*1,2, . . .

*}*

*<*

*∞*, then the sequence*{**x**i**}*^{∞}_{i}* _{=}*1converges.

The property (A) is well known in the theory of ordered Banach spaces (see,
e.g., [7,10,11]). Recall that the cone*X*+has the property (A) if the space*X*is re-
flexive. The property (A) also holds for the cone of nonnegative functions (with
respect to usual order relation) in the space *L*1 of all integrable on a measure
space functions.

Assume that*K*is a closed subset of*X. For each function* *f* :*Y**→*[*−∞**,+**∞*],
where*Y* is a nonempty set, we define

dom(*f*)*=*

*y**∈**Y*:^{}*f*(y)^{}*<**∞*

*,* inf(*f*)*=*inf^{}*f*(y) :*y**∈**Y*^{}*.* (2.1)
We use the convention that*∞ − ∞ =*0,*∞**/**∞ =*1, and ln(*∞*)*= ∞*.

Assume thatᏭis a nonempty set and*d**w**, d**s*:Ꮽ*×*Ꮽ*→*[0,*∞*) are two metrics
which satisfy*d** _{w}*(a, b)

*≤*

*d*

*(a, b) for all*

_{s}*a, b*

*∈*Ꮽ. We assume that the metric space (Ꮽ

*, d*

*s*) is complete. The topology induced inᏭby the metric

*d*

*s*is called the strong topology and the topology induced inᏭby the metric

*d*

*w*is called the weak topology.

We assume that with every*a**∈*Ꮽa lower semicontinuous function *f**a*:*K**→*
[*−∞**,+**∞*] is associated and *f**a*is not identically*∞*for all*a**∈*Ꮽ.

Let*a**∈*Ꮽ. We say that the minimization problem for *f** _{a}*on

*K*is strongly well posed with respect to (Ꮽ

*, d*

*w*) if the following assertions hold:

(1) the infimum inf(*f**a*) is finite and attained at a point*x*^{(a)}*∈**K* such that
for each*x**∈**K*satisfying *f** _{a}*(x)

*=*inf(

*f*

*), the inequality*

_{a}*x*

*≤*

*x*

^{(a)}holds;

(2) for any*>*0, there exist*δ >*0 and a neighborhood*U*of*a*inᏭwith the
weak topology such that for each*b**∈**U, inf(f**b*) is finite; and if*x**∈**K*
satisfies*f** _{b}*(x)

*≤*inf(

*f*

*) +*

_{b}*δ, then*

*|*

*f*

*(x*

_{a}^{(a)})

*−*

*f*

*(x)*

_{b}*|*

*<*and there is

*u*

*∈*

*X*such that

*u*

*<*and

*x*

*≤*

*x*

^{(a)}+

*u.*

Note that if*X*+*= {*0*}*, then our definition reduces to those in [6,13].

For each integer *n**≥*1, denote by Ꮽ*n* the set of all*a**∈*Ꮽ which have the
following property:

(P1) there exist*x**∈**K*and positive numbers*r,η, andc*such that

*−∞**< f**a*(x)*<*inf^{}*f**a*
+1

*n,* (2.2)

and for each *b**∈*Ꮽsatisfying*d** _{w}*(a, b)

*< r, inf(f*

*) is finite; and if*

_{b}*z*

*∈*

*K*satisfies

*f*

*b*(z)

*≤*inf(

*f*

*b*) +

*η, then*

*z*

*≤*

*c,*

*|*

*f*

*b*(z)

*−*

*f*

*a*(x)

*| ≤*1/n, and there is

*u*

*∈*

*X*such that

*u*

*≤*1/nand

*z*

*≤*

*x*+

*u.*

Proposition2.1. *Assume thata**∈ ∩*^{∞}*n**=*1Ꮽ*n**. Then the minimization problem for*
*f*_{a}*onKis strongly well posed with respect to*(Ꮽ*, d** _{w}*).

*Proof.* By (P1) for each integer*n**≥*1, there exist*x**n**∈**K*,*r**n**>*0,*η**n**>*0, and*c**n**>*0
such that

*−∞**< f**a*
*x**n*

*<*inf^{}*f**a*

+ 2^{−}* ^{n}* (2.3)

and the following property holds:

(P2) for each*b**∈*Ꮽsatisfying*d** _{w}*(a, b)

*< r*

*, inf(*

_{n}*f*

*) is finite; and if*

_{b}*z*

*∈*

*K*sat- isfies

*f*

*b*(z)

*≤*inf(

*f*

*b*) +

*η*

*n*, then there exists

*u*

*∈*

*X*such that

*u** ≤*2^{−}^{n}*,* *z**≤**x** _{n}*+

*u,*

*z*

*≤*

*c*

_{n}*,*

^{}

*f*

*(z)*

_{b}*−*

*f*

_{a}^{}

*x*

_{n}^{}

*≤*2

^{−}

^{n}*.*(2.4) We may assume without loss of generality that for all integers

*n*

*≥*1,

*η*_{n}*, r*_{n}*<*4^{−}^{n}^{−}^{1}*,* *η*_{n}*< η*1*.* (2.5)
There exists a strictly increasing sequence of natural numbers*{**k**n**}*^{∞}*n**=*1such that

4*·*2^{−}^{k}^{n+1}*< η*^{}*k**n*

for all integers*n**≥*1. (2.6)
Let*n**≥*1 be an integer. Inequality (2.3) implies that

*−∞**< f**a*

*x**k**n+1*

*<*inf^{}*f**a*

+ 2^{−}^{k}^{n+1}*<*inf(*f**a*) +*η*^{}*k**n*

*.* (2.7)

By (2.7), (2.5), and the definition of*c*1,

*x**k**n+1**≤**c*1*.* (2.8)

It follows from (2.7), (P2) (see (2.4)), and the definitions of*x**k**n*and*η**k**n*that there
exists*u**n**∈**X*such that

*u**n**≤*2^{−}^{k}^{n}*,* *x**k**n+1**≤**x**k**n*+*u**n**,* (2.9)
*f**a*

*x**k**n+1*

*−**f**a*

*x**k**n**≤*2^{−}^{k}^{n}*.* (2.10)

Set

*y**n**=**x**k**n*+
*∞*
*i**=**n*

*u**i**.* (2.11)

Clearly, the sequence*{**y**n**}*^{∞}*n**=*1is well defined. By (2.11) and (2.9), for each integer
*n**≥*1,

*y**n+1**−**y**n**=**x**k**n+1*+
*∞*
*i**=**n+1*

*u**i**−*

*x**k**n*+
*∞*
*i**=**n*

*u**i* *=**x**k**n+1**−**x**k**n**−**u**n**≤*0. (2.12)

Equation (2.11) and inequalities (2.9) and (2.8) imply that

sup^{}*y**n*:*n**=*1,2, . . .^{}*<**∞**.* (2.13)
It follows from (2.13), (2.12), and the property (A) that there is*x*^{(a)}*=*lim*n**→∞**y**n*.
Combined with (2.11) and (2.9), this equality implies that

*x*^{(a)}*=*lim

*n**→∞**x**k**n**.* (2.14)

By (2.14), (2.3), and the lower semicontinuity of*f** _{a}*,

*f*_{a}^{}*x*^{(a)}^{}*=*inf^{}*f*_{a}^{}*.* (2.15)
Assume now that*x*_{∈}*K* and *f** _{a}*(x)

*inf(*

_{=}*f*

*). By the definition of*

_{a}*x*

_{k}*,*

_{n}*n*

*1,2, . . .(see the property (P2)), for each integer*

_{=}*n*

*≥*1, there is

*v*

_{n}*∈*

*X*such that

*v*_{n}^{}*≤*2^{−}^{k}^{n}*,* *x**≤**x*_{k}* _{n}*+

*v*

_{n}*.*(2.16)

These inequalities and (2.14) imply that*x**≤**x*^{(a)}.
By (2.15) and the property (P2), for all integers*n**≥*1,

*f**a*

*x*^{(a)}^{}*−**f**a*

*x**n**≤*2^{−}^{n}*.* (2.17)

Let*>*0. Choose a natural number*m*for which

*x*^{(a)}*−**x**k**m**<*4^{−}^{1}*,* 2^{−}^{k}^{m}*<*4^{−}^{1}*.* (2.18)

Assume that*b**∈**B**w*(a, r*k**m**/2),x**∈**K, and*
*f**b*(x)*≤*inf^{}*f**b*

+*η**k**m**.* (2.19)

By (2.19) and the definitions of*η**k**m*,*r**k**m*, and*x**k**m*(see the property (P2)),
*f**b*(x)*−**f**a*

*x**k**m**≤*2^{−}^{k}* ^{m}* (2.20)

and there is*v**∈**X*such that

*v**<*2^{−}^{k}^{m}*,* *x**≤**x**k**m*+*v.* (2.21)

It follows from (2.21) and (2.18) that

*x**≤**x**k**m*+*v**=**x*^{(a)}+^{}*x**k**m**−**x*^{(a)}+*v*^{}*,*
*x**k**m**−**x*^{(a)}+*v*^{}*≤**x*^{(a)}*−**x**k**m*+*v**<*^{}

2*.* (2.22)

Inequalities (2.20), (2.17), and (2.18) imply that
*f**a*

*x*^{(a)}^{}*−**f**b*(x)^{}*≤**f**a*

*x*^{(a)}^{}*−**f**a*

*x**k**m*+^{}*f**a*

*x**k**m*

*−**f**b*(x)^{}

*≤*2^{−}^{k}* ^{m}*+ 2

^{−}

^{k}

^{m}*<*

^{}2*.* (2.23)

This completes the proof ofProposition 2.1.

An element*x**∈**K* is called minimal if for each *y**∈**K* satisfying *y**≤**x, the*
equality*x**=**y*is true. Denote by*K*minthe set of all minimal elements of*K*.

For each integer*n**≥*1, denote by ˜Ꮽ*n*the set of all*a**∈*Ꮽwhich has the prop-
erty (P1) with*x**∈**K*min.

Analogously to the proof ofProposition 2.1, we can prove the following re- sult.

Proposition2.2. *Assume that the setK*min*is a closed subset of the Banach space*
*X* *anda**∈ ∩*^{∞}_{n}* _{=}*1Ꮽ˜

*n*

*. Then the minimization problem for*

*f*

_{a}*onK*

*is strongly well*

*posed with respect to*(Ꮽ, d

*w*)

*and*inf(

*f*

*a*)

*is attained at a unique point.*

In the proof ofProposition 2.2, we choose*x**n**∈**K*min,*n**=*1,2, . . . .This implies
that inf(*f**a*) is attained at the unique point*x*^{(a)}*∈**K*min(see (2.13)).

*Remark 2.3.* Note that assertion (1) in the definition of a strongly well-posed
minimization problem for *f** _{a}*can be represented in the following way: inf(

*f*

*) is finite and the set*

_{a}argmin

*x**∈**K*

*f*_{a}*=*

*x**∈**K*:*f** _{a}*(x)

*=*inf

^{}

*f*

_{a}^{}(2.24) has the largest element.

We construct an example of an increasing function *h* for which the set
argmin(h) is not a singleton and has the largest element. Define a continuous
increasing function*ψ*: [0,*∞*)*→*R^{1}by

*ψ(t)**=*0, *t**∈*

0,1 2

*,* *ψ(t)**=*2t*−*1, *t**∈*
1

2*,**∞*

*.* (2.25)

Let*n*be a natural number and consider the Euclidean spaceR* ^{n}*. Let

*K*

*= {*

*x*

*=*(x1

*, . . . , x*

*n*)

*∈*R

*:*

^{n}*x*

*i*

*≥*0,

*i*

*=*1, . . . , n

*}*. Define a function

*h*:

*K*

*→*R

^{1}by

*h(x)**=**ψ*^{}max^{}*x**i*:*i**=*1, . . . , n^{}*,* *x**∈**K.* (2.26)

It is easy to see that*h*is a continuous increasing function:

inf^{}*h(x) :x**∈**K*^{}*=*0 (2.27)

and the set

*x**∈**K*:*h(x)**=*0^{}*=*

*x**=*

*x*1*, . . . , x*_{n}^{}*∈*R* ^{n}*:

*x*

_{i}*∈*

0,1 2

*, i**=*1, . . . , n

(2.28) is not a singleton and has the largest element (1/2, . . . ,1/2).

*Remark 2.4.* The following example shows that in some cases the setsᏭ*n* can
be empty. LetᏭ*=**K**=*R^{1}. For each*a**∈*R^{1}, consider the function *f** _{a}*:

*K*

*→*R

^{1}, where

*f*

*a*

*=*0 for any

*x*

*≤*

*a*and

*f*

*a*(x)

*>*0 for any

*x > a. It is easy to see that the*setᏭ

*n*is empty for any natural number

*n.*

**3. Variational principles**

We use the notations and definitions introduced inSection 2. The following are the basic hypotheses about the functions:

(H1) for each*a**∈*Ꮽ, inf(*f**a*) is finite;

(H2) for each*>*0 and each integer*m**≥*1, there exist numbers*δ >*0 and
*r*0*>*0 such that the following property holds:

(P3) for each*a**∈*Ꮽsatisfying inf(*f**a*)*≤**m*and each*r**∈*(0, r0], there exist

¯

*a**∈*Ꮽ, ¯*x**∈**K*, and ¯*d >*0 such that
*d** _{s}*(a,

*a)*¯

*≤*

*r,*inf

^{}

*f*

*a*¯

*≤**m*+ 1, *f**a*¯( ¯*x)**≤*inf^{}*f**a*¯

+*,* (3.1)

and if*x**∈**K*satisfies

*f**a*¯(x)*≤*inf^{}*f**a*¯

+*δr,* (3.2)

then*x** ≤**d*¯and there exists*u**∈**X*for which*u** ≤*and*x**≤**x*¯+*u;*

(H3) for each integer *m**≥*1, there exist *α**∈*(0,1) and*r*0*>*0 such that for
each*r**∈*(0, r0], each*a*1*, a*2*∈*Ꮽsatisfying*d**w*(a1*, a*2)*≤**αr, and eachx**∈*
*K*satisfying min*{**f*_{a}_{1}(x), f_{a}_{2}(x)*} ≤**m, the inequality**|**f*_{a}_{1}(x)*−**f*_{a}_{2}(x)*| ≤**r*
is valid.

Theorem3.1. *Assume that (H1), (H2), and (H3) hold. Then there exists a set*
Ᏺ*⊂*Ꮽ*such that the complement*Ꮽ*\*Ᏺ*isσ-porous in*Ꮽ*with respect to*(d*w**, d**s*)
*and for eacha**∈*Ᏺ, the minimization problem for *f*_{a}*onK* *is strongly well posed*
*with respect to*(Ꮽ*, d**w*).

*Proof.* Recall that for each integer*n**≥*1,Ꮽ*n*is the set of all*a**∈*Ꮽwhich has the
property (P1). ByProposition 2.1, in order to prove the theorem, it is suﬃcient
to show that the setᏭ*\*Ꮽ*n*is*σ*-porous inᏭwith respect to (d_{w}*, d** _{s}*) for any
integer

*n*

*≥*1. Then the theorem is true withᏲ

*= ∩*

^{∞}

_{n}*1Ꮽ*

_{=}*n*.

Let*n**≥*1 be an integer. We will show that the setᏭ*\*Ꮽ*n*is*σ*-porous inᏭ
with respect to (d*w**, d**s*). To meet this goal, it is suﬃcient to show that for each
integer*m**≥*1, the set

Ω*nm*:*=*

*a**∈*Ꮽ*\*Ꮽ*n*: inf^{}*f**a*

*≤**m*^{} (3.3)

is porous inᏭwith respect to (d_{w}*, d** _{s}*).

Let*m**≥*1 be an integer. By (H3), there exist
*α*1*∈*(0,1), *r*1*∈*

0,1 2

(3.4)
such that for each*r**∈*(0, r1], each*a*1*, a*2*∈*Ꮽsatisfying*d** _{w}*(a1

*, a*2)

*≤*

*α*1

*r, and each*

*x*

*∈*

*K*satisfying

min^{}*f**a*1(x), f*a*2(x)^{}*≤**m*+ 4, (3.5)
the inequality*|**f**a*1(x)*−**f**a*2(x)*| ≤**r*holds.

By (H2), there exist*α*2*, r*2*∈*(0,1) such that the following property holds:

(P4) for each*a**∈*Ꮽsatisfying inf(*f** _{a}*)

*≤*

*m*+ 2 and each

*r*

*∈*(0, r2], there exist

¯

*a**∈*Ꮽ, ¯*x**∈**K, and ¯d >*0 such that
*d** _{s}*(a,

*a)*¯

*≤*

*r,*inf

^{}

*f*

*a*¯

*≤**m*+ 3, *f**a*¯( ¯*x)**≤*inf^{}*f**a*¯

+ (2n)^{−}^{1}*,* (3.6)
and if*x**∈**K*satisfies *f**a*¯(x)*≤*inf(*f**a*¯) + 4rα2, then*x** ≤**d*¯and there ex-
ists*u**∈**X*for which*u** ≤*(2n)^{−}^{1}and*x**≤**x*¯+*u.*

Choose

¯
*α**∈*

0,*α*1*α*2

16

*,* *r*¯*∈*

0,*r*1*r*2*α*¯
*n*

*.* (3.7)

Let*a**∈*Ꮽand*r**∈*(0,*r]. There are two cases*¯
*B*_{d}_{s}

*a,r*

4

*∩*

*ξ**∈*Ꮽ: inf^{}*f*_{ξ}^{}*≤**m*+ 2^{}*= ∅**,* (3.8)
*B**d**s*

*a,r*

4

*∩*

*ξ**∈*Ꮽ: inf^{}*f**ξ*

*≤**m*+ 2^{}*= ∅**.* (3.9)

Assume that (3.8) holds. We will show that for each*ξ**∈**B*_{d}* _{w}*(a,

*r), the inequality*¯ inf(

*f*

*ξ*)

*> m*is valid. Assume the contrary. Then there exists

*ξ*

*∈*Ꮽsuch that

*d** _{w}*(ξ, a)

*≤*

*r,*¯ inf

^{}

*f*

_{ξ}^{}

*≤*

*m.*(3.10) There exists

*y*

*∈*

*K*such that

*f** _{ξ}*(y)

*≤*

*m*+1

2*.* (3.11)

It follows from the definitions of*α*1,*r*1(see (3.4), (3.5)), (3.11), (3.10), and (3.7)
that

*f**a*(y)*−**f**ξ*(y)^{}*≤**α** ^{−}*1

^{1}

*r*¯

*≤*1

4*.* (3.12)

This inequality and (3.11) imply that
inf^{}*f**a*

*≤**f**a*(y)*≤**f**ξ*(y) +1

4^{≤}*m*+ 1, (3.13)

a contradiction (see (3.8)). Therefore
*B**d**w*(a,*r)*¯ *⊂*

*ξ**∈*Ꮽ: inf^{}*f*_{ξ}^{}*> m*^{} (3.14)
and by (3.3),

*B**d**w*(a,*r)*¯ *∩*Ω*nm**= ∅**.* (3.15)
Thus, we have shown that (3.8) implies (3.15).

Assume that (3.9) holds. Then there exists*a*1*∈*Ꮽsuch that
*d**s*

*a, a*1

*≤**r*

4*,* inf^{}*f**a*1

*≤**m*+ 2. (3.16)

By the definitions of*α*2,*r*2, the property (P4), (3.16), and (3.7), there exist ¯*a**∈*Ꮽ,

¯

*x**∈**K, ¯d >*0 such that
*d**s*

*a*1*,a*¯^{}*≤**r*

4*,* inf^{}*f**a*¯

*≤**m*+ 3, *f**a*¯( ¯*x)**≤*inf^{}*f**a*¯

+ (2n)^{−}^{1} (3.17)
and that the following property holds:

(P5) if*x**∈**K*satisfies*f**a*¯(x)*≤*inf(*f**a*¯) +*rα*2, then*x** ≤**d*¯and there exists*u**∈*
*X*for which*u** _{ ≤}*(2n)

^{−}^{1}and

*x*

_{≤}*x*¯+

*u.*

Inequalities (3.17) and (3.16) imply that
*d**s*(a,*a)*¯ *≤**r*

2*.* (3.18)

Assume that

*ξ**∈**B*_{d}* _{w}*( ¯

*a,αr).*¯ (3.19)

By (3.17),

inf^{}*f**a*¯

*=*inf

*f**a*¯(z) :*z**∈**K, f**a*¯(z)*≤**m*+7
2

*.* (3.20)

Let*x**∈**K*satisfy

*f**a*¯(x)*≤**m*+7

2*.* (3.21)

It follows from (3.19), (3.21), (3.7), and the definitions of*α*1,*r*1(see (3.4), (3.5))
that

*f**a*¯(x)*−**f** _{ξ}*(x)

^{}

*≤*

*αrα*¯

^{−}_{1}

^{1}

*≤*

*α*2

*r*

16*.* (3.22)

Since these inequalities hold for any*x**∈**K*satisfying (3.21), the relation (3.20)
implies that

inf^{}*f**ξ*

*≤*inf

*f**ξ*(x) :*x**∈**K, f**a*¯(x)*≤**m*+7
2

*≤*inf

*f**a*¯(x) +*α*2*r*

16 :*x**∈**K, f**a*¯(x)*≤**m*+7
2

*=**α*2*r*

16 + inf^{}*f**a*¯
*.*

(3.23)

Moreover, since (3.21) holds with*x**=**x*¯ (see (3.17)), we obtain that*|**f**a*¯( ¯*x)**−*
*f**ξ*( ¯*x)**| ≤**α*2*r/16. Thus*

inf^{}*f*_{ξ}^{}* _{≤}*inf

^{}

*f*

*a*¯

+*α*2*r*

16*,* ^{}*f**a*¯( ¯*x)*_{−}*f** _{ξ}*( ¯

*x)*

^{}

_{≤}*α*2

*r*

16*.* (3.24)

Let*x**∈**K*satisfy

*f**ξ*(x)*≤*inf^{}*f**ξ*

+1

4*.* (3.25)

Inequalities (3.25), (3.24), (3.17) and (3.7) imply that*f**ξ*(x)*≤**m*+ 7/2. It follows
from this inequality, (3.19), (3.7), and the definitions of*α*1,*r*1(see (3.4), (3.5))
that

*f**a*¯(x)*−**f** _{ξ}*(x)

^{}

*≤*

*αrα*¯

^{−}_{1}

^{1}

*≤*

*α*2

*r*

16*.* (3.26)

Thus, the following property holds:

(P6) if*x**∈**K*satisfies (3.25), then*|**f**a*¯(x)*−**f**ξ*(x)*| ≤**α*2*r/16.*

The property (P6) implies that
inf^{}*f**a*¯

*≤*inf

*f**a*¯(x) :*x**∈**K, f**ξ*(x)*≤*inf^{}*f**ξ*

+1 4

*≤*inf

*f** _{ξ}*(x) +

*α*2

*r*

16 :*x**∈**K, f** _{ξ}*(x)

*≤*inf

^{}

*f*

_{ξ}^{}+1 4

*=**α*2*r*

16 + inf^{}*f**ξ*

*.*

(3.27)

Therefore

inf^{}*f**a*¯

*≤*inf^{}*f*_{ξ}^{}+*α*2*r*

16*.* (3.28)

Combined with (3.24) and (3.17), this inequality implies that
inf^{}*f**a*¯

*−*inf^{}*f**ξ**≤**α*2*r*

16*,* *f**ξ*( ¯*x)**≤*inf^{}*f**ξ*
+*α*2*r*

8 + (2n)^{−}^{1}*.* (3.29)
Assume that*x**∈**K*and

*f**s*(x)*≤*inf^{}*f**ξ*
+*α*2*r*

16*.* (3.30)

By (P6),

*f**a*¯(x)*−**f**ξ*(x)^{}*≤**α*2*r*

16*.* (3.31)

Inequalities (3.30), (3.29), (3.17) and (3.7) imply that
*f**ξ*(x)*−**f**a*¯( ¯*x)*^{}*≤**f**ξ*(x)*−*inf^{}*f**ξ*+^{}inf^{}*f**ξ*

*−*inf^{}*f**a*¯
+^{}inf^{}*f**a*¯

*−**f**a*¯( ¯*x)*^{}

*≤**α*2*r*
16 +*α*2*r*

16 + (2n)^{−}^{1}*< n*^{−}^{1}*,*

(3.32)
*f**ξ*(x)*−**f**a*¯( ¯*x)*^{}*< n*^{−}^{1}*.* (3.33)
It follows from (3.31), (3.30), and (3.29) that

*f**a*¯(x)*≤**f**ξ*(x) +*rα*2

16 * ^{≤}*inf

^{}

*f*

*ξ*

+*α*2*r*

8 * ^{≤}*inf

^{}

*f*

*a*¯

+3α2*r*

16 *,* (3.34)

*f**a*¯(x)*≤*inf^{}*f**a*¯

+3α2*r*

16 *.* (3.35)

It follows from (3.35) and the property (P5) that*x** ≤**d*¯and there is*u**∈**X*such
that*u** ≤*(2n)^{−}^{1}and*x**≤**u*+ ¯*x. Therefore, ifx**∈**K* satisfies (3.30), then (3.33)
is valid,*x** ≤**d, and there is*¯ *u**∈**X*for which*u** ≤*(2n)^{−}^{1}and*x**≤**u*+ ¯*x.*

Thus, we have shown that for each*ξ**∈**B**d**w*( ¯*a,αr), the inequalities (3.29) are*¯
true and if*x**∈**K*satisfies (3.30), then (3.33) is valid,*x** ≤**d, and there is*¯ *u**∈**X*
for which*u** ≤*(2n)^{−}^{1}and*x**≤**u*+ ¯*x.*

By the definition ofᏭ*n*and (3.3),
*B**d**w*

*a,*¯ *αr*¯

2

*⊂*Ꮽ*n**⊂*Ꮽ*\*Ω*nm**.* (3.36)

Since (3.8) implies (3.15), we obtain that, in both cases,*B**d**w*( ¯*a,αr/2)*¯ *∩*Ω*nm**= ∅*
with ¯*a**∈*Ꮽ satisfying (3.18). (Note that if (3.8) is valid, then ¯*a**=**a.) Hence,*
the setΩ*nm* is porous in Ꮽwith respect to (d*w**, d**s*). This implies thatᏭ*\*Ꮽ*n*

is*σ*-porous inᏭwith respect to (d*w**, d**s*) for all integers*n**≥*1. Therefore,Ꮽ*\*
(*∩*^{∞}_{n}* _{=}*1Ꮽ

*n*) is

*σ-porous in*Ꮽwith respect to (d

_{w}*, d*

*). This completes the proof of*

_{s}Theorem 3.1.

We also use the following hypotheses about the functions:

(H4) for each*>*0 and each integer*m**≥*1, there exist numbers*δ >*0 and
*r*0*>*0 such that the following property holds:

(P7) for each*a**∈*Ꮽsatisfying inf(*f** _{a}*)

*≤*

*m*and each

*r*

*∈*(0, r0], there exist

¯

*a**∈*Ꮽ, ¯*x**∈**K*min, and ¯*d >*0 such that (3.1) is true; and if*x**∈**K*satisfies
(3.2), then *x** ≤**d*¯and there exists*u**∈**X* for which*u** ≤*,*x**≤*

¯
*x*+*u.*

Theorem3.2. *Assume that (H1), (H3), and (H4) hold andK*min*is a closed subset*
*of the Banach spaceX. Then there exists a set*Ᏺ*⊂*Ꮽ*such that the complement*
Ꮽ*\*Ᏺ*isσ-porous in*Ꮽ*with respect to*(d*w**, d**s*)*and that for eacha**∈*Ᏺ*the following*
*assertions hold:*

(1)*the minimization problem for* *f*_{a}*onK* *is strongly well posed with respect to*
(Ꮽ*, d**w*),

(2)*the infimum*inf(*f**a*)*is attained at a unique point.*

We can proveTheorem 3.2analogously to the proof ofTheorem 3.1. Recall
that for each integer*n**≥*1, ˜*A** _{n}*is the set of all

*a*

*∈*Ꮽwhich have the property (P1) with

*x*

*∈*

*K*min. SetᏲ

*= ∩*

^{∞}

_{n}*1*

_{=}*A*˜

*n*. ByProposition 2.2for each

*a*

*∈*Ᏺ, assertions (1) and (2) hold. Therefore, in order to proveTheorem 3.2, it is suﬃcient to show that for each integer

*n*

*≥*1, the setᏭ

*\*Ꮽ˜

*n*is

*σ*-porous inᏭwith respect to (d

*w*

*, d*

*s*). We can show this fact analogously to the proof ofTheorem 3.1.

**4. Spaces of increasing functions**

In the sequel, we use the functional*λ*:*X** _{→}*R

^{1}defined by

*λ(x)**=*inf^{}*y*:*y**≥**x*^{}*,* *x**∈**X.* (4.1)
The function*λ*has the following properties (see [10, Proposition 6.1]):

(i) the function*λ*is sublinear. Namely,

*λ(αx)**=**αλ(x)* *∀**α**≥*0 and all*x**∈**X,*
*λ*^{}*x*1+*x*2

*≤**λ*^{}*x*1

+*λ*^{}*x*2

*∀**x*1*, x*2*∈**X,* (4.2)
(ii)*λ(x)**=*0 if*x**≤*0,

(iii) if*x*1*, x*2*∈**X*and*x*1*≤**x*2, then*λ(x*1)*≤**λ(x*2),
(iv) 0*≤**λ(x)**≤ **x*for all*x**∈**X.*

Clearly,*|**λ(x)**−**λ(y)**| ≤ **x**−**y*for each*x, y**∈**X.*

Denote byᏹthe set of all increasing lower semicontinuos bounded-from-
below functions *f* :*K* *→*R^{1}*∪ {*+*∞}* which are not identically +*∞*. For each

*f , g**∈*ᏹ, set

*d*˜*s*(*f , g)**=*sup^{}*f*(x)*−**g(x)*^{}:*x**∈**X*^{}*,*

*d**s*(*f , g)**=**d*˜*s*(*f , g)*^{}1 + ˜*d**s*(*f , g)*^{}^{−}^{1}*.* (4.3)

It is not diﬃcult to see that the metric space (ᏹ*, d**s*) is complete. Denote byᏹ*v*

the set of all finite-valued functions*f* *∈*ᏹand byᏹ*c*the set of all finite-valued
continuous functions *f* *∈*ᏹ. Clearly,ᏹ*v*andᏹ*c*are closed subsets of the metric
space (ᏹ*, d** _{s}*).

We say that the set*K* has property (C) if*K*minis a closed subset of*K*and for
each*x*_{∈}*K, there isy*_{∈}*K*minsuch that*y*_{≤}*x.*

Denote byᏹ*g*the set of all *f* *∈*ᏹsuch that *f*(x)*→ ∞*as*x** → ∞*. Clearly,
ᏹ*g*is a closed subset of the metric space (ᏹ, d*s*). Setᏹ*gc**=*ᏹ*g**∩*ᏹ*c*andᏹ*gv**=*
ᏹ*g**∩*ᏹ*v*.

It is easy to see that

ᏹ*c**⊂*ᏹ*v**⊂*ᏹ*,* ᏹ*gc**⊂*ᏹ*gv**⊂*ᏹ*g**⊂*ᏹ*.* (4.4)
*Remark 4.1.* Let*K**=**X*+and define

*f*1(x)*= **x**,* *x**∈**K,* *f*2(x)*= **x**,* *x**∈**K**\ {*0*}**,* *f*2(0)*= −*1,
*f*3(x)*= **x* if*x**∈**K,* *x** ≤*1, *f*3(x)*=*+*∞* if*x**∈**K,* *x**>*1. (4.5)
Clearly,

*f*1*∈*ᏹ*gc**,* *f*2*∈*ᏹ*gv**\*ᏹ*gc**,* *f*3*∈*ᏹ*g**\*ᏹ*gv**.* (4.6)
Theorem4.2. *Assume that*Ꮽ*is either*ᏹ*g**,*ᏹ*gv**, or*ᏹ*gc* *and that* *f**a**=**afor all*
*a**∈*Ꮽ. Then there exists a setᏲ*⊂*Ꮽ*such that the complement*Ꮽ*\*Ᏺ*isσ-porous*
*in*Ꮽ*with respect to*(d_{s}*, d** _{s}*)

*and that for each*

*f*

*∈*Ᏺ

*the minimization problem for*

*f*

*onKis strongly well posed with respect to*(Ꮽ, d

*s*). If

*Khas the property (C), then*

*for each*

*f*

*∈*Ᏺ,inf(

*f*)

*is attained at a unique point.*

*Proof.* By Theorems3.1and3.2, we need to show that (H1), (H2), and (H3) hold
and that the property (C) implies (H4). Clearly, (H1) holds. For each *f , g**∈*ᏹ,
we have that

*d*˜*s*(*f , g)**=**d**s*(*f , g)*^{}1*−**d**s*(*f , g)*^{}^{−}^{1} (4.7)
and that if *d**s*(*f , g)**≤*1/2, then ˜*d**s*(*f , g)**≤*2d*s*(*f , g). Combined with (4.7), this*
property implies (H3).

We will show that (H2) holds and that the property (C) implies (H4).

Let *f* *∈*Ꮽ,*∈*(0,1), and*r**∈*(0,1]. Choose ¯*x**∈**K*such that
*f*( ¯*x)**≤*inf(*f*) +^{}*r*

8 *.* (4.8)

If*K* has the property (C), then we assume that ¯*x* is a minimal element of*K.*

Define

*f*¯(x)*=* *f*(x) + 2^{−}^{1}*r*min^{}1, λ(x*−**x)*¯ ^{} *∀**x**∈**K.* (4.9)

Evidently, ¯*f* *∈*Ꮽ,*d**s*(*f ,f*¯)*≤**d*˜*s*(*f ,f*¯)*≤**r/2, and*

inf^{}*f*¯^{}*≤**f*¯( ¯*x)**=* *f*( ¯*x)**≤*inf(*f*) +^{}*r*

8*.* (4.10)

Let*x**∈**K*and ¯*f*(x)*≤*inf( ¯*f*) +*r/8. Then by (4.9) and (4.8),*
*f*(x) + 2^{−}^{1}*r*min^{}1, λ(x*−**x)*¯ ^{}*=* *f*¯(x)*≤*inf^{}*f*¯^{}+^{}*r*

8 ^{≤}*f*¯( ¯*x) +*^{}*r*
8

*=* *f*( ¯*x) +*^{}*r*

8 ^{≤}*f*(x) +^{}*r*
4*,*
min^{}1, λ(x*−**x)*¯ ^{}*≤*

2*,* *λ(x**−**x)*¯ *≤*
2*.*

(4.11)

By (4.1), there exists*u**∈**X* such that *x**≤**x*¯+*u* and*u**<*. Since ¯*f*(y)*→ ∞*
as*y** → ∞*, we obtain that*x** ≤**d, where ¯*¯ *d >*0 is a constant which depends
only on ¯*f*. Thus, (H2) is true and if*K* has the property (C), then (H4) holds.

Theorem 4.2is proved.

Theorem4.3. *Assume that there existsz*¯*∈**Xsuch thatz*¯*≤**xfor allx**∈**K, that a*
*space*Ꮽ*is either*ᏹ,ᏹ*v**, or*ᏹ*c**, and thatf*_{a}*=**afor alla**∈*Ꮽ. Then there exists a set
Ᏺ*⊂*Ꮽ*such that*Ꮽ*\*Ᏺ*isσ-porous in*Ꮽ*with respect to*(d*s**, d**s*)*and that for each*
*f* *∈*Ᏺ, the minimization problem for *f* *onK* *is strongly well posed with respect*
*to*(Ꮽ*, d** _{s}*). If

*K*

*has the property (C), then for each*

*f*

*∈*Ᏺ,inf(

*f*)

*is attained at a*

*unique point.*

*Proof.* We can proveTheorem 4.3analogously to the proof ofTheorem 4.2. The
existence of a constant ¯*d*is obtained in the following manner. Let*x**∈**K*,*u**∈**X,*
*x*_{≤}*x*¯+*u, and*_{}*u*_{}*<** _{}*. Then

*x** ≤ **x**−**z*¯+*z*¯* ≤ **z*¯+*x*¯+*u**−**z*¯* ≤*2*|**z*¯+*x*¯+*,*

*x** ≤**d,*¯ (4.12)

where ¯*d**=*2*z*¯+*x*¯+.

Denote byᏹ^{+}the set of all *f* *∈*ᏹsuch that *f*(x)*≥*0 for all*x**∈**K. Clearly,*
ᏹ^{+}is a closed subset of the metric space (ᏹ*, d** _{s}*). Define

ᏹ^{+}*v**=*ᏹ^{+}*∩*ᏹ*v**,* ᏹ^{+}*c* *=*ᏹ^{+}*∩*ᏹ*c**,* ᏹ^{+}*g**=*ᏹ^{+}*∩*ᏹ*g**,*

ᏹ^{+}_{gv}*=*ᏹ^{+}*∩*ᏹ*gv**,* ᏹ^{+}_{gc}*=*ᏹ^{+}*∩*ᏹ*gc**.* (4.13)
For each *f , g**∈*ᏹ^{+}, set

*d*˜*w*(*f , g)**=*sup^{}ln^{}*f*(z) + 1^{}*−*ln^{}*g*(z) + 1^{}:*z**∈**K*^{}*,* (4.14)
*d**w*(*f , g)**=**d*˜*w*(*f , g)*^{}1 + ˜*d**w*(*f , g)*^{}^{−}^{1}*.* (4.15)
It is not diﬃcult to see that the metrtic space (ᏹ^{+}*, d**w*) is complete and thatᏹ^{+}* _{v}*,
ᏹ

^{+}

*c*,ᏹ

^{+}

*g*,ᏹ

^{+}

*gv*, andᏹ

^{+}

*gc*are closed subsets of (ᏹ

^{+}

*, d*

*). Clearly,*

_{w}*d*

*(*

_{w}*f , g)*

*≤*

*d*

*(*

_{s}*f , g)*for all

*f , g*

*∈*ᏹ

^{+}.