View of Compatibility of some systems of inequalities

Rev. Anal. Num´er. Th´eor. Approx., vol. 32 (2003) no. 1, pp. 3–9 ictp.acad.ro/jnaat

COMPATIBILITY OF SOME SYSTEMS OF INEQUALITIES

MIRCEA BALAJ^∗

Abstract. In this paper, necessary and sufficient conditions for the compati- bility of some systems of quasi-convex, or convex inequalities are established.

Finally a new proof for a theorem of Shioji and Takahashi (1988) is given.

MSC 2000. 52A41, 52A07.

Keywords. Quasi-convex inequalities, concave-convex-like function, minimax inequality.

1. INTRODUCTION

Ky Fan studied in [4] the existence of solutions for some systems of convex inequalities involving lower semicontinuous functions defined on a compact convex set in a topological vector space (all the topological vector spaces con- sidered in this paper are real and Hausdorff). Particularly, he proved the following theorem.

Theorem A. Let C be a nonempty compact convex subset of a topological vector space and let F be a family of real-valued lower semicontinuous convex functions defined on C. Then the following assertions are equivalent:

(i) The system of convex inequalities

(1) f(x)≤0, f ∈ F,

is compatible onC, i.e., there exists x∈C satisfying (1).

(ii) For any n nonnegative numbers αi with ^Pn_i=1αi = 1 and for any f₁, f₂, . . . , f_n∈ F, there existsx∈C such that

n

X

i=1

α_if_i(x)≤0.

For closed results and extensions of Fan’s theorem see [5], [7], [8], [10] and [11]. In Section 2 we study the compatibility of some systems of inequalities (1) in the case when all functions f ∈ F are quasi-convex (Theorems 3 and 5), respectively convex (Theorems 2 and 6).

Shioji and Takahashi in [10, Th. 1] have established a Fan type theorem in the case when some function of two variables associated to the system of inequalities (1) is convex-like in one of the variables. This theorem receives a new proof in Section 3.

∗Department of Mathematics, University of Oradea, 3700, Oradea, Romania, e-mail:

[email protected].

For any positive integernwe denote by Sn the set

Sn=ⁿα= (α1, α2, . . . , αn)∈Rⁿ, α1 ≥0, α2 ≥0, . . . , αn≥0,

n

P

i=1

αi= 1^o The standard abbreviations convA, clA, cardAare used to define the convex hull, closure and cardinality of a setA, respectively.

2. SYSTEMS OF QUASI-CONVEX INEQUALITIES

We recall that a real-valued function f defined on a convex set C is said to be quasi-convex if for every real number α, the set {x ∈C : f(x) ≤α} is convex.

In proving Theorem 2 we shall need the following lemma which is an anal- ogous result of a classical Fan’s section theorem.

Lemma 1. [6, Th. 2.2]. Let C be a nonempty compact convex subset of a locally convex topological vector space X and K a nonempty closed convex subset of a topological vector space Y. Let A be a subset of C×K having the following properties:

(a) A is closed;

(b) for anyy∈K, {x∈C : (x, y)∈A} is nonempty and convex;

(c) for anyx∈C, {y ∈K : (x, y)∈/ A} is convex (possibly empty).

Then there exists x0∈C such that {x₀} ×K ⊂A.

Theorem2. LetCbe a nonempty compact convex subset of a locally convex topological vector space X and let F be a family of continuous quasi-convex functions f :C→R,satisfying the condition

(2) any convex combination of functions inF is quasi-convex.

Then the following assertions are equivalent:

(i) The system of inequalities(1) is compatible on C.

(ii) For each integern,1≤n≤cardF, for each(α1, α2, . . . , αn)∈Snand anyf₁, f₂, . . . , fn∈ F there existsx∈C such that ^Pn_i=1αifi(x)≤0.

Proof. It is clear that (i) implies (ii). In order to prove the reverse im- plication we shall apply Lemma 1 taking in the posture of Y, the vector space of all continuous functions f :C→R, endowed with the uniform norm kfk= max{|f(x)|:x ∈C}. Also we take K = cl(convF) (the closure being taken with respect to the uniform topology),A={(x, f)∈C×K :f(x)≤0}

and we show that conditions (a), (b), (c) in Lemma 1 are satisfied.

(a) Let ((xi, fi))_i∈I be a net in A converging to (x, f). It follows that f_i(x)≤0 for each i∈I and x_i →^X x, f_i →^Y f. Let us take an arbitrary >0.

By the continuity of the functionf, there is i1 ∈I such thatkf−fik< ₂,for all i∈I,i > i₁, and byf_i →^Y f there is i₂ ∈I such that kf −f_ik < ₂,for all

i∈I,i > i2. Then for every i∈I satisfying i > i1,i > i2 we have f(x) = (f(x)−f(xi)) + (f(xi)−fi(xi)) +fi(xi)

≤ f(x)−f(xi) +kf−fik

< ₂+ ₂ =.

(b) Let f ∈ K = cl(convF) and (fn)_n∈N a sequence in convF uniformly converging to f (such a sequence there exists since Y is a normed space).

By (ii) it follows that for eachn∈Nthere existsxn∈C such thatfn(xn)≤0.

Since C is compact the sequence (xn)_n∈N has a subsequence (xn_k)_k∈N con- verging to an x ∈ C. Therefore ((x_nk, f_nk))_k∈N is a sequence in A conver- gent to (x, f) and A being closed, it follows that f(x) ≤ 0. Hence the set {x∈C: (x, f)∈A}is nonempty.

According to (2), all the functions in convF are quasi-convex. It is easily checked that the quasi-convexity is conserved by the pointwise convergence, hence by the uniform convergence too.

(c) For everyx∈C, the set {f ∈K;f(x)>0} is obviously convex.

So all the conditions of Lemma 1 are satisfied. Therefore there existsx₀ ∈C such that {x₀} ×cl(convF) ⊂ A. Particularly, for every f ∈ F we have

(x₀, f)∈A, that is,f(x₀)≤0.

In [10, Th. 2], Shioji and Takahashi extend Fan’s theorem to families of lower semicontinuous convex functions with values in (−∞,∞]. More exactly they have established

Theorem 3. Let C be a nonempty compact convex subset of a topological vector space X and let F be a family of lower semicontinuous convex func- tions f : C → (−∞,∞]. Then the assertions (i) and (ii) in Theorem 2 are equivalent.

It should be mentioned that in the case when the topological vector space X is locally convex, Theorem 3 can be derived from Theorem 2. Indeed let C be a nonempty compact convex subset of a locally convex space and letF be a family of lower semicontinuous convex functions f :C→(−∞,∞]. It is clear that (i) implies (ii). In order to prove the reverse implication, for each f ∈ F let A_f be the set of all continuous affine functionsg:C →Rsatisfying g(x)≤f(x), for allx∈C, and denote byG =∪{A_f :f ∈ F }.

It is known (see [2, p. 99] or [9, p. 30]) that for a semicontinuous convex functionf :C→(−∞,∞] the following equality holds

(3) f(x) = supg(x) :g∈ A_f .

We show that the system

g(x)≤0, g∈ G,

is compatible on C. Obviously, the family G satisfies condition (2) in Theo- rem 2. Letnbe a positive integer, (α₁, α₂, . . . , α_n)∈S_nandg₁, g₂, . . . , g_n∈ G.

If g1, g2, . . . , g_k ∈ A_f1,g_k+1, g_k+2, . . . , g_l ∈ A_f2,. . ., gr+1, gr+2, . . . , gn ∈ A_fm, then for every x∈C we have

α1g1(x) +α2g2(x) +· · ·+αngn(x)≤

≤(α1+α2+· · ·+α_k)f1(x) + (α_k+1+α_k+2+· · ·+α_l)f2(x) +. . . + (αr+1+αr+2+· · ·+αn)fm(x).

The sum in the right-hand side of the above inequality is a convex combi- nation of the functions f1, f2, . . . , fm hence, according to (ii), it is ≤0 for at least one x ∈ C. This shows that G satisfies (ii). Theorem 2 applied to the family of functions G puts into evidence an x₀ ∈ C such that g(x₀) ≤ 0 for each g∈ G. By (3) it follows immediately that f(x0)≤0, for eachf ∈ F.

Remark 1. Observe that if the family of functions F is finite, having cardF =n, the condition (ii) in each of Theorems A, 2, 3 can be replaced by (ii⁰) For each(α1, α2, . . . , αn) ∈Sn and any f1, f2, . . . , fn∈ F there exists

x∈C such that^Pn_i=1α_if_i(x)≤0.

In Theorem 5 we shall give a set of sufficient conditions for the compatibility of systems of convex inequalities. The proof will be based on Theorem 3 and on the following intersectional result for convex sets (see [1] or [3]).

Lemma 4. Let C be a compact convex subset of a topological vector space, A a family of closed convex subsets of C, and k, l two positive integers with k≤l+ 1≤cardA. Suppose that

(i) ∪A⁰=C,for any subfamily A⁰ of Awith cardA⁰=k;

(ii) ∩A⁰6=∅,for any subfamily A⁰ of A withcardA⁰ =l.

Then ∩A 6=∅.

Theorem 5. Let C be a nonempty compact convex subset of a topological vector space,F be a family of lower semicontinuous convex functions f :C→ (−∞,∞],and k, l two positive integers withk≤l+ 1≤cardF. Suppose that (a) for each kfunctions, pairwise distinct, f₁, f₂, . . . , f_k∈ F and any x∈

C there exists (α1, α2, . . . , αk)∈Sk such that

k

X

j=1

αjfj(x)≤0;

(b) for each l functions f₁, f₂, . . . , f_l ∈ F and any (α₁, α₂, . . . , α_l) ∈ S_l there existsx∈C such that

f(x)≤0, for all f ∈ F.

Proof. Denote byA the family of all sets Ai ={x ∈C :fi(x) ≤0},where f_i ∈ F. Since the functions f_i ∈ F are lower semicontinuous and convex, the corresponding sets Ai are closed in C and convex. The proof of Theorem 5 will be achieved whenever we verify the conditions (i) and (ii) in Lemma 4 for the familyA.

If A does not satisfy the condition (i) then there exists k functions, pair- wise distinct, f₁, f₂, . . . , f_k in F and x in C such that f_j(x) > 0, for all j ∈ {1,2, . . . , k}. But in this case for any (α1, α2, . . . , α_k) ∈ S_k we have Pk

j=1α_jf_j(x)>0, which contradicts condition (a).

Now given a subfamily{A₁, A2, . . . , Al} ofl members of A, i.e.,Aj ={x∈ C : f_j(x) ≤ 0}, f_j ∈ F, then condition (b) together with Theorem 3, via Remark 1, yield an x ∈C such that fj(x) ≤0, for all j ∈ {1,2, . . . , k}, that

is,A₁∩A₂∩ · · · ∩A_l6=∅.

The following result can be proved by applying the same argument as in the previous proof, using Theorem 2 instead of Theorem 3.

Theorem6. LetCbe a nonempty compact convex subset of a locally convex topological vector space, F a family of continuous quasi-convex functions f : C → R satisfying condition (2) in Theorem 2, and k, l two positive integers with k ≤ l+ 1 ≤cardF. If conditions (a) and (b) in Theorem 5 hold, then there exists x∈C such that

f(x)≤0, for all f ∈ F.

3. THE SHIOJI-TAKAHASHI THEOREM

In [10, Th. 1] Shioji and Takahashi have extended Fan’s theorem to functions more general than the convex ones. The goal of this section is to give a new proof of this result, using a minimax theorem.

Before going to this result, we first recollect the following definitions (see [2, p. 161]).

LetA, B be arbitrary sets. A functionF :A×B →(−∞,∞] is said to be:

(i) concave-like in its first variable, if for any x1, x2 ∈A and 0 < α <1, there existsx₀ ∈A such that

αF(x₁, y) + (1−α)F(x₂, y)≤F(x₀, y), for all y∈B;

(ii) convex-like in its second variable, if for anyy₁, y₂∈B and 0< α <1, there existsy0∈B such that

F(x, y0)≤αF(x, y1) + (1−α)F(x, y2), for allx∈A;

(iii) concave-convex-like, if it is concave-like in its first variable and convex- like in its second variable.

Remark2. It is clear from condition (i) that the following property results (i⁰) for every x1, x2, . . . , xn ∈ A and (α1, α2, . . . , αn) ∈ Sn, there exists

x₀ ∈A such that

n

X

i=1

α_iF(x_i, y)≤F(x₀, y), for ally ∈B.

A similar statement for condition (ii) holds.

Lemma7. LetAandB be compact topological spaces and letF :A×B →R be an upper-lower semicontinuous concave-convex-like function. Then

maxx∈A min

y∈BF(x, y) = min

y∈B max

x∈A F(x, y).

The above lemma has been formulated in [2, Th. 3.5], in the case when A and B are compact convex sets, each in a topological vector space, but the proof given there holds too in the conditions imposed by us.

The following theorem was obtained by Shioji and Takahashi in [10, Th. 1].

We present another proof relied on Lemma 7.

Theorem 8. Let C be a nonempty compact space (not necessarily Haus- dorff). Let F be a family of lower semicontinuous functions f :C →R such that the functionF :F ×C→Rdefined byF(f, x) =f(x),for eachf ∈ F and x ∈ C, is convex-like in its second variable. Then the assertions (i) and (ii) in Theorem 2 are equivalent.

Proof. We have only to prove the implication (ii) ⇒ (i). The set C be- ing compact and the functions f ∈ F being lower semicontinuous, it follows immediately that an infinite system (1) is compatible if and only if every fi- nite subsystem is compatible. So, we may assume that the familyF is finite, namely F ={f₁, f₂, . . . , f_n}. Define the function L:S_n×C→Rby

L(α, x) =

n

X

i=1

α_if_i(x), for each (α₁, α₂, . . . , α_n)∈S_n and x∈C.

Clearly L is linear, hence continuous in its first variable. On the other side, from hypothesis it follows that L is lower semicontinuous convex-like in its second variable.

Our assumption (ii) can be written as a minimax inequality, namely

(4) max

α∈Sn

minx∈CL(α, x)≤0.

The existence of anx∈C satisfying allninequalitiesf_i(x)≤0,1≤i≤n, is equivalent to the truth of the relation

minx∈C max

1≤i≤nf_i(x)≤0 or, which is the same,

minx∈C max

α∈Sn

n

X

i=1

αifi(x) = min

x∈C max

α∈Sn

L(α, x)≤0.

This relation can be obtained by Lemma 7 and relation (4) as follows minx∈C max

α∈Sn

L(α, x) = max

α∈Sn

minx∈CL(α, x)≤0.

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Received by the editors: April 13, 1998.