THE SIMULTANEOUS NONLINEAR INEQUALITIES PROBLEM AND APPLICATIONS

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Simultaneous Nonlinear Inequalities Problem

Zoran D. Mitrovi´c vol. 8, iss. 3, art. 84, 2007

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THE SIMULTANEOUS NONLINEAR INEQUALITIES PROBLEM AND APPLICATIONS

ZORAN D. MITROVI ´C

Faculty of Electrical Engineering University of Banja Luka 78000 Banja Luka, Patre 5 Bosnia and Herzegovina EMail:zmitrovic@etfbl.net

Received: 22 June, 2007

Accepted: 02 August, 2007

Communicated by: S.P. Singh

2000 AMS Sub. Class.: 47J20, 41A50, 54H25.

Key words: KKM map, Best approximations, Fixed points and coincidences, Variational inequalities, Saddle points.

Abstract: In this paper, we prove the existence of a solution to the simultaneous nonlinear inequality problem. As applications, we derive the results on the simultaneous approximations, variational inequalities and saddle points. The results of this paper generalize some known results in the literature.

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1. Introduction and Preliminaries

In this paper, using the methods of KKM-theory, see for example, Singh, Watson and Srivastava [17] and Yuan [20], we prove some results on simultaneous nonlinear inequalities. As corollaries, some results on the simultaneous approximations, variational inequalities and saddle points are obtained.

Let X be a set. We shall denote by 2^X the family of all non-empty subsets of X. IfAis a subset of a vector space X, then coAdenotes the convex hull ofA in X. Let K be a subset of a topological vector space X. Then a multivalued map G:K →2^X is called a KKM-map if

co{x₁, . . . , x_n} ⊆

n

[

i=1

G(x_i) for each finite subset{x₁, . . . , x_n}ofK.

LetKbe a nonempty convex subset of a vector spaceX.For a mapf :K →R, the set

Ep(f) ={(x, r)∈K×R:f(x)≤r}

is called the epigraph off. Note that a mapf is convex if and only if the setEp(f) is convex.

LetKbe a nonempty set,n∈Nandf_i :K×K →Rmaps for alli∈[n],where [n] = {1, . . . , n}.A simultaneous nonlinear inequalities problem is to findx₀ ∈K such that it satisfies the following inequality

(1.1)

n

X

i=1

f_i(x₀, y)≥0 for ally∈K.

When n = 1 and f(x, x) = 0 for allx ∈ K, (1.1) reduces to the scalar equilibrium problem considered by Blum and Oettli [5], that is, to find x0 ∈ K such

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that

f(x₀, y)≥0 for ally∈K.

This problem has been generalized and applied in various directions, see for example [1], [2], [3], [9], [10], [14].

The following result of Ky Fan [8] will be used to prove the main result of this paper.

Theorem 1.1 ([8]). LetXbe a topological vector space,Ka nonempty subset ofX andG:K →2^X be a KKM-map with closed values. IfG(x)is compact for at least onex∈K, then T

x∈K

G(x)6=∅.

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2. Main Result

Now we will apply Theorem1.1to show the existence of a solution for our simultaneous nonlinear inequalities problem.

Theorem 2.1. LetK be a nonempty compact convex subset of a topological vector spaceXandfi :K×K →R, i∈[n], continuous maps. If there existsλ≥0,such that

(2.1) coEp(f_i(x,·))⊆Ep(f_i(x,·)−λ) for allx∈K, i∈[n], then there existsx₀ ∈Ksuch that

λn+

n

X

i=1

fi(x0, y)≥

n

X

i=1

fi(x0, x0) for ally∈K.

Proof. Let us define the mapG:K →2^K by G(y) =

(

x∈K :λn+

n

X

i=1

f_i(x, y)≥

n

X

i=1

f_i(x, x) )

, for ally∈K.

We have thatG(y)is nonempty for ally∈K,becausey∈G(y)for ally∈K.

The f_i, i ∈ [n] are continuous maps and we obtain thatG(y) is closed for each y∈K. SinceKis a compact set, we have thatG(y)is compact for eachy ∈K.

Now, we prove thatGis a KKM-map. IfGis not a KKM-map, then there exists a subset {y₁, . . . , y_m} of K and there existsµ_j ≥ 0, j ∈ [m] withPm

j=1µ_j = 1, such that

y_µ=

m

X

j=1

µ_jy_j ∈/

m

[

j=1

G(y_j).

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So, we have

λn+

n

X

i=1

f_i(y_µ, y_j)<

n

X

i=1

f_i(y_µ, y_µ), for allj ∈[n].

On the other hand, since,

(y_j, f_i(y_µ, y_j))∈Ep(f_i(y_µ,·)), for alli∈[n], j ∈[m], from condition (2.1) we obtain

y_µ,

m

X

j=1

µ_jf_i(y_µ, y_j)

!

∈Ep(f_i(y_µ,·)−λ) for alli∈[n].

Therefore, it follows that

fi(yµ, yµ)−λ ≤

m

X

j=1

µjfi(yµ, yj) for alli∈[n].

This implies that

n

X

i=1

f_i(y_µ, y_µ)≤λn+

n

X

i=1 m

X

j=1

µ_jf_i(y_µ, y_j).

Further, since

n

X

i=1 m

X

j=1

µjfi(yµ, yj) =

m

X

j=1

µj n

X

i=1

fi(yµ, yj)≤ max

1≤j≤m n

X

i=1

fi(yµ, yj),

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and

λn+

n

X

i=1

f_i(y_µ, y_j)<

n

X

i=1

f_i(y_µ, y_µ), for allj ∈[m], we obtain

n

X

i=1

fi(yµ, yµ)<

n

X

i=1

fi(yµ, yµ).

This is a contradiction. Thus,Gis a KKM-map.

By Theorem1.1, there existsx₀ ∈Ksuch thatx₀ ∈G(y)for ally∈K,that is, λn+

n

X

i=1

f_i(x₀, y)≥

n

X

i=1

f_i(x₀, x₀) for ally∈K.

Corollary 2.2. LetK be a nonempty compact convex subset of a topological vector spaceXandf_i :K×K →R, i∈[n],continuous maps. Ify7→f_i(x, y)are convex for allx∈K, i∈[n], then there existsx₀ ∈K such that

n

X

i=1

f_i(x₀, y)≥

n

X

i=1

f_i(x₀, x₀) for ally∈K.

Note that, if in Theorem2.1the mapsx7→fi(x, y)are upper semicontinuous for ally∈K andfi(x, x)≥0for allx∈K, i∈[n], we obtain the following result.

Corollary 2.3. LetK be a nonempty compact convex subset of a topological vector spaceX andf_i :K×K →R, i∈[n],maps such that

(i) f_i(x, x)≥0for allx∈K,

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(ii) x7→f_i(x, y)are upper semicontinuous for ally∈K, (iii) y7→f_i(x, y)are convex for allx∈K,

for alli∈[n].Then there existsx₀ ∈Ksuch that

n

X

i=1

f_i(x₀, y)≥0 for ally∈K.

From Theorem2.1, we have the theorem on the existence of zeros of bifunctions.

Theorem 2.4. LetK be a nonempty compact convex subset of a topological vector spaceX, f_i : K ×K → R, i ∈ [n] continuous maps and there existsλ ≥ 0such that the condition (2.1) is satisfied for allx ∈ K andi ∈ [n]. If for everyx ∈ K, withf_i(x, x)6= 0for alli∈[n],

n

\

i=1

{y ∈K :f_i(x, x)−f_i(x, y)> λ} 6=∅

then the set

S = (

x∈K :λn+

n

X

i=1

f_i(x, y)≥

n

X

i=1

f_i(x, x) for ally∈K )

is nonempty and for eachx∈S there existsi∈[n]such thatf_i(x, x) = 0.

Proof. By Theorem2.1, there existsx₀ ∈ S. We claim that such x₀ is a zero off_i for anyi ∈ [n]. Suppose not, i.e.,f_i(x₀, x₀) 6= 0for alli ∈ [n]. Then we have the existence ofy₀ ∈K, such that

fi(x0, x0)−fi(x0, y0)> λ for alli∈[n].

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Consequently,

λn+

n

X

i=1

f_i(x₀, y₀)<

n

X

i=1

f_i(x₀, x₀),

so,x0 ∈/ Sand that is a contradiction. Therefore,fi(x0, x0) = 0for anyi∈[n].

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3. Applications

From Theorem2.1, we have the following simultaneous approximations theorem for metric spaces.

Theorem 3.1. LetK be a nonempty compact convex subset of a topological vector spaceX with metricdandfi, gi : K →X, i∈ [n]continuous maps. Suppose there existsλ≥0,such thatfi, gisatisfy the condition

(3.1) co{(y, r) :d(g_i(y), f_i(x))≤r} ⊆ {(y, r) :d(g_i(y), f_i(x))≤r+λ}

for allx∈K,i∈[n]. Then there existsx₀ ∈K such that

λn+

n

X

i=1

d(gi(y), fi(x0))≥

n

X

i=1

d(gi(x0), fi(x0)) for ally∈K.

Proof. Define

fi(x, y) = d(gi(y), fi(x)), forx, y ∈K, i∈[n].

Now, the result follows by Theorem2.1.

Remark 1. LetX be a normed space and letg_i :K →Xbe almost affine maps, see, for example [6], [13], [15], [16], [17], [18], i. e.

||g_i(αx₁+ (1−α)x₂)−y|| ≤α||g_i(x₁)−y||+ (1−α)||g_i(x₂)−y||

for allx₁, x₂ ∈ K, y ∈ X, α ∈ [0,1], i ∈ [n]. Then for λ = 0,assumption (3.1) is satisfied.

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Corollary 3.2. LetK be a nonempty compact convex subset of a normed spaceX, f_i, g_i : K → X continuous maps and g_i almost affine maps for all i ∈ [n]. Then there existsx₀ ∈K such that

n

X

i=1

||g_i(y)−f_i(x₀)|| ≥

n

X

i=1

||g_i(x₀)−f_i(x₀)|| for ally∈K.

Corollary 3.3. LetK be a nonempty compact convex subset of a normed spaceX andf_i :K →X, i∈[n],continuous maps. Then there existsx₀ ∈K such that

n

X

i=1

||y−f_i(x₀)|| ≥

n

X

i=1

||x₀−f_i(x₀)|| for ally∈K.

Remark 2.

(i) If n = 1 then Corollary 3.3 reduces to the well-known best approximations theorem of Ky Fan [8] and Corollary 3.2 reduces to the result of J.B. Prolla [15].

(ii) Note that, ifX is a Hilbert space andn = 2, from Corollary3.3we obtain the result of D. Delbosco [7].

As application of Theorem2.4, we have the following coincidence point theorem for metric spaces.

Theorem 3.4. LetK be a nonempty compact convex subset of a topological vector spaceXwith metricdandf_i, g_i :K →X, i∈[n],continuous maps. Suppose there existsλ ≥0,such thatf_i, g_isatisfy the condition (3.1) for allx∈ K,i ∈[n]. If for everyx∈K, withf_i(x)6=g_i(x)for alli∈[n],

n

\

i=1

{y∈K :d(gi(x), fi(x))> d(gi(y), fi(x)) +λ} 6=∅,

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then the set

S= (

x∈K :λn+

n

X

i=1

d(g_i(y), f_i(x))≥

n

X

i=1

d(g_i(x), f_i(x)) for ally∈K )

is nonempty and for eachx∈S there existsi∈[n]such thatf_i(x) =g_i(x).

Proof. Put

f_i(x, y) = d(g_i(y), f_i(x)), forx, y ∈K, i∈[n].

Thenfi,gi, i∈[n]satisfy all of the requirements of Theorem2.4.

Corollary 3.5. LetK be a nonempty compact convex subset of a metric spaceX, f, g:K →X continuous maps and

d(g(λx₁+ (1−λ)x₂), f(y))≤λd(g(x₁), f(y)) + (1−λ)d(g(x₂), f(y)), for allx₁, x₂ ∈ K, y ∈ X, λ ∈ [0,1]. If for everyx ∈ K, withf(x) 6= g(x)there exists ay∈K such that

d(g(x), f(x))> d(g(y), f(x)),

then the set

S={x∈K :d(g(y), f(x))≥d(g(x), f(x)) for ally∈K}

is nonempty andf(x) = g(x)for eachx∈S.

Corollary 3.6. Let K be a nonempty compact convex subset of a metric space X andf :K →Xa continuous map such that

x7→d(x, f(y))is a convex map for ally∈X.

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If for everyx∈K, withf(x)6=xthere exists ay∈Ksuch that d(x, f(x))> d(y, f(x)),

then the set

S ={x∈K :d(y, f(x))≥d(x, f(x)) for ally∈K}

is nonempty andf(x) = xfor eachx∈S.

We note that if f : K → K, then, from Corollary 3.6, we obtain the famous Schauder fixed point theorem.

Now, we establish an existence result for our simultaneous variational inequality problem by using Corollary2.3.

Theorem 3.7. LetXbe a reflexive Banach space with its dualX^?andK a compact convex subset ofX. LetT_i : K → X^?, i ∈[n], be maps. Ifx7→ hT_i(x), y−xiare upper semicontinuous for ally∈K, i∈[n], then there existsx₀ ∈K such that

n

X

i=1

hT_i(x₀), y−x₀i ≥0 for ally∈K.

Proof. Letf_i(x, y) =hT_i(x), y −xi, for allx, y ∈ K, i∈ [n]. By our assumptions, the mapsf_isatisfy all the hypotheses of Corollary2.3, and it follows that there exists x₀ ∈K such that

n

X

i=1

hT_i(x₀), y−x₀i ≥0 for ally∈K.

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Remark 3.

(i) Ifn = 1then Theorem3.7reduces to the classical result of F. E. Browder and W. Takahashi, see for example [17, Theorem 4. 33].

(ii) Given two maps T : K → X^? and µ : K ×K → X, the variational-like inequality problem, see for example [12], is to findx₀ ∈K such that

hT(x₀), µ(y, x₀)i ≥0 for ally∈K.

If in Corollary2.3a map

f₁(x, y) = hT(x), µ(y, x)i for allx, y ∈K

and n = 1, we obtain the result of X.Q. Yang and G.Y. Chen [19, Theorem 8], and the result of A. Behera and L. Nayak [4, Theorem 2.1]. Also, if in Corollary2.3a map

f₁(x, y) = hT(x), µ(y, x)i − hA(x), µ(y, x)i for allx, y ∈K,

where A : K → X^?, we obtain the result of G. K. Panda and N. Dash, [11, Theorem 2.1].

Finally, we give the following application to the existence for saddle points.

Theorem 3.8. LetK be a nonempty compact convex subset of a topological vector spaceX andf_i : K ×K → R continuous maps and f_i(x, x) = 0for all x ∈ K, i∈[n].If there existsλ ≥0,such that

coEp(f_i(x,·))⊆Ep(f_i(x,·)−λ) for allx∈K and

coEp(−fi(·, y))⊆Ep(−fi(·, y)−λ) for ally∈K,

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for alli∈[n],then

0≤min

x∈Kmax

y∈K n

X

i=1

f_i(x, y)−max

y∈K min

x∈K n

X

i=1

f_i(x, y)≤2λn.

Proof. Note that

0≤min

x∈Kmax

y∈K n

X

i=1

f_i(x, y)−max

y∈K min

x∈K n

X

i=1

f_i(x, y)

holds in general. By our assumptions,f_i, i ∈[n]satisfy all the hypotheses of Theo- rem2.1, and it follows that there existsx0 ∈K such that

miny∈K n

X

i=1

f_i(x₀, y)≥ −λn.

So, we obtain,

(3.2) max

x∈K min

y∈K n

X

i=1

f_i(x, y)≥ −λn.

Letg_i(x, y) = −f_i(y, x)for all(x, y)∈K×K, i∈[n].By Theorem2.1, it follows that there existsy₀ ∈K such that

miny∈K n

X

i=1

g_i(y₀, y)≥ −λn, so

maxx∈K n

X

i=1

fi(x, y0)≤λn.

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Therefore, we obtain,

(3.3) min

y∈Kmax

x∈K n

X

i=1

f_i(x, y)≤λn.

By combining (3.2) and (3.3), it follows that minx∈Kmax

y∈K n

X

i=1

fi(x, y)−max

y∈K min

x∈K n

X

i=1

fi(x, y)≤2λn.

Corollary 3.9. LetK be a nonempty compact convex subset of a topological vector spaceX. Supposef_i :K×K →R, i∈[n]are continuous maps such that

1. f_i(x, x) = 0for allx∈K,

2. y7→f_i(x, y)is convex for allx∈K, 3. x7→f_i(x, y)is concave for ally ∈K,

for alli∈[n].Then we have

maxy∈K min

x∈K n

X

i=1

f_i(x, y) = min

x∈Kmax

y∈K n

X

i=1

f_i(x, y).

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