THE SIMULTANEOUS NONLINEAR INEQUALITIES
PROBLEM AND APPLICATIONS
Simultaneous Nonlinear
Inequalities Problem
ZORAN D. MITROVIĆ
Zoran D. Mitrović
Faculty of Electrical Engineering
University of Banja Luka
78000 Banja Luka, Patre 5
Bosnia and Herzegovina
EMail: zmitrovic@etfbl.net
vol. 8, iss. 3, art. 84, 2007
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02 August, 2007
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Communicated by:
S.P. Singh
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2000 AMS Sub. Class.:
47J20, 41A50, 54H25.
Key words:
KKM map, Best approximations, Fixed points and coincidences, Variational inequalities, Saddle points.
Received:
22 June, 2007
Accepted:
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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Contents
1
Introduction and Preliminaries
3
2
Main Result
5
3
Applications
10
Simultaneous Nonlinear
Inequalities Problem
Zoran D. Mitrović
vol. 8, iss. 3, art. 84, 2007
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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 2X the family of all non-empty subsets of
X. If A is a subset of a vector space X, then coA denotes the convex hull of A in
X. Let K be a subset of a topological vector space X. Then a multivalued map
G : K → 2X is called a KKM-map if
co{x1 , . . . , xn } ⊆
n
[
G(xi )
Simultaneous Nonlinear
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vol. 8, iss. 3, art. 84, 2007
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i=1
for each finite subset {x1 , . . . , xn } of K.
Let K be a nonempty convex subset of a vector space X. For a map f : K → R,
the set
Ep(f ) = {(x, r) ∈ K × R : f (x) ≤ r}
is called the epigraph of f . Note that a map f is convex if and only if the set Ep(f )
is convex.
Let K be a nonempty set, n ∈ N and fi : K × K → R maps for all i ∈ [n], where
[n] = {1, . . . , n}. A simultaneous nonlinear inequalities problem is to find x0 ∈ K
such that it satisfies the following inequality
(1.1)
n
X
fi (x0 , y) ≥ 0
for all y ∈ K.
i=1
When n = 1 and f (x, x) = 0 for all x ∈ 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 all y ∈ 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]). Let X be a topological vector space, K a nonempty subset of X
and G : K → 2X T
be a KKM-map with closed values. If G(x) is compact for at least
one x ∈ K, then
G(x) 6= ∅.
Simultaneous Nonlinear
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vol. 8, iss. 3, art. 84, 2007
x∈K
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2.
Main Result
Now we will apply Theorem 1.1 to show the existence of a solution for our simultaneous nonlinear inequalities problem.
Theorem 2.1. Let K be a nonempty compact convex subset of a topological vector
space X and fi : K × K → R, i ∈ [n], continuous maps. If there exists λ ≥ 0, such
that
Zoran D. Mitrović
coEp(fi (x, ·)) ⊆ Ep(fi (x, ·) − λ)
(2.1)
Simultaneous Nonlinear
Inequalities Problem
vol. 8, iss. 3, art. 84, 2007
for all x ∈ K, i ∈ [n], then there exists x0 ∈ K such that
λn +
n
X
fi (x0 , y) ≥
i=1
n
X
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fi (x0 , x0 )
for all y ∈ K.
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i=1
Proof. Let us define the map G : K → 2K by
(
)
n
n
X
X
G(y) = x ∈ K : λn +
fi (x, y) ≥
fi (x, x) ,
i=1
for all y ∈ K.
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i=1
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We have that G(y) is nonempty for all y ∈ K, because y ∈ G(y) for all y ∈ K.
The fi , i ∈ [n] are continuous maps and we obtain that G(y) is closed for each
y ∈ K. Since K is a compact set, we have that G(y) is compact for each y ∈ K.
Now, we prove that G is a KKM-map. If G is not a KKM-map, then
Pmthere exists
a subset {y1 , . . . , ym } of K and there exists µj ≥ 0, j ∈ [m] with j=1 µj = 1,
such that
m
m
X
[
yµ =
µj y j ∈
/
G(yj ).
j=1
j=1
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So, we have
n
X
λn +
fi (yµ , yj ) <
n
X
i=1
fi (yµ , yµ ),
for all j ∈ [n].
i=1
On the other hand, since,
(yj , fi (yµ , yj )) ∈ Ep(fi (yµ , ·)),
for all i ∈ [n], j ∈ [m],
Zoran D. Mitrović
from condition (2.1) we obtain
m
X
yµ ,
Simultaneous Nonlinear
Inequalities Problem
vol. 8, iss. 3, art. 84, 2007
!
∈ Ep(fi (yµ , ·) − λ)
µj fi (yµ , yj )
for all i ∈ [n].
j=1
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Therefore, it follows that
fi (yµ , yµ ) − λ ≤
m
X
µj fi (yµ , yj )
for all i ∈ [n].
j=1
n
X
fi (yµ , yµ ) ≤ λn +
i=1
m
n X
X
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µj fi (yµ , yj ).
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Further, since
i=1 j=1
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This implies that
n X
m
X
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µj fi (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 ),
and
λn +
n
X
fi (yµ , yj ) <
n
X
i=1
we obtain
fi (yµ , yµ ),
for all j ∈ [m],
i=1
n
X
fi (yµ , yµ ) <
i=1
n
X
fi (yµ , yµ ).
i=1
This is a contradiction. Thus, G is a KKM-map.
By Theorem 1.1, there exists x0 ∈ K such that x0 ∈ G(y) for all y ∈ K, that is,
λn +
n
X
fi (x0 , y) ≥
n
X
i=1
fi (x0 , x0 )
for all y ∈ K.
i=1
Simultaneous Nonlinear
Inequalities Problem
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vol. 8, iss. 3, art. 84, 2007
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Corollary 2.2. Let K be a nonempty compact convex subset of a topological vector
space X and fi : K × K → R, i ∈ [n], continuous maps. If y 7→ fi (x, y) are convex
for all x ∈ K, i ∈ [n], then there exists x0 ∈ K such that
n
X
fi (x0 , y) ≥
i=1
n
X
fi (x0 , x0 )
for all y ∈ K.
i=1
Note that, if in Theorem 2.1 the maps x 7→ fi (x, y) are upper semicontinuous for
all y ∈ K and fi (x, x) ≥ 0 for all x ∈ K, i ∈ [n], we obtain the following result.
Corollary 2.3. Let K be a nonempty compact convex subset of a topological vector
space X and fi : K × K → R, i ∈ [n], maps such that
(i) fi (x, x) ≥ 0 for all x ∈ K,
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(ii) x 7→ fi (x, y) are upper semicontinuous for all y ∈ K,
(iii) y 7→ fi (x, y) are convex for all x ∈ K,
for all i ∈ [n]. Then there exists x0 ∈ K such that
n
X
fi (x0 , y) ≥ 0
for all y ∈ K.
Simultaneous Nonlinear
Inequalities Problem
i=1
From Theorem 2.1, we have the theorem on the existence of zeros of bifunctions.
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vol. 8, iss. 3, art. 84, 2007
Theorem 2.4. Let K be a nonempty compact convex subset of a topological vector
space X, fi : K × K → R, i ∈ [n] continuous maps and there exists λ ≥ 0 such
that the condition (2.1) is satisfied for all x ∈ K and i ∈ [n]. If for every x ∈ K,
with fi (x, x) 6= 0 for all i ∈ [n],
n
\
{y ∈ K : fi (x, x) − fi (x, y) > λ} =
6 ∅
i=1
then the set
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(
S=
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x ∈ K : λn +
n
X
i=1
fi (x, y) ≥
n
X
)
fi (x, x)
for all y ∈ K
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i=1
is nonempty and for each x ∈ S there exists i ∈ [n] such that fi (x, x) = 0.
Proof. By Theorem 2.1, there exists x0 ∈ S. We claim that such x0 is a zero of fi
for any i ∈ [n]. Suppose not, i.e., fi (x0 , x0 ) 6= 0 for all i ∈ [n]. Then we have the
existence of y0 ∈ K, such that
fi (x0 , x0 ) − fi (x0 , y0 ) > λ
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for all i ∈ [n].
Close
Consequently,
λn +
n
X
i=1
fi (x0 , y0 ) <
n
X
fi (x0 , x0 ),
i=1
so, x0 ∈
/ S and that is a contradiction. Therefore, fi (x0 , x0 ) = 0 for any i ∈ [n].
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vol. 8, iss. 3, art. 84, 2007
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3.
Applications
From Theorem 2.1, we have the following simultaneous approximations theorem for
metric spaces.
Theorem 3.1. Let K be a nonempty compact convex subset of a topological vector
space X with metric d and fi , gi : K → X, i ∈ [n] continuous maps. Suppose there
exists λ ≥ 0, such that fi , gi satisfy the condition
(3.1)
co{(y, r) : d(gi (y), fi (x)) ≤ r} ⊆ {(y, r) : d(gi (y), fi (x)) ≤ r + λ}
Simultaneous Nonlinear
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vol. 8, iss. 3, art. 84, 2007
for all x ∈ K, i ∈ [n]. Then there exists x0 ∈ K such that
λn +
n
X
i=1
d(gi (y), fi (x0 )) ≥
n
X
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d(gi (x0 ), fi (x0 ))
for all y ∈ K.
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i=1
Proof. Define
fi (x, y) = d(gi (y), fi (x)),
for x, y ∈ K, i ∈ [n].
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Now, the result follows by Theorem 2.1.
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Remark 1. Let X be a normed space and let gi : K → X be almost affine maps, see,
for example [6], [13], [15], [16], [17], [18], i. e.
||gi (αx1 + (1 − α)x2 ) − y|| ≤ α||gi (x1 ) − y|| + (1 − α)||gi (x2 ) − y||
for all x1 , 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. Let K be a nonempty compact convex subset of a normed space X,
fi , gi : K → X continuous maps and gi almost affine maps for all i ∈ [n]. Then
there exists x0 ∈ K such that
n
n
X
X
||gi (y) − fi (x0 )|| ≥
||gi (x0 ) − fi (x0 )|| for all y ∈ K.
i=1
i=1
Corollary 3.3. Let K be a nonempty compact convex subset of a normed space X
and fi : K → X, i ∈ [n], continuous maps. Then there exists x0 ∈ K such that
n
X
i=1
||y − fi (x0 )|| ≥
n
X
Simultaneous Nonlinear
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vol. 8, iss. 3, art. 84, 2007
||x0 − fi (x0 )||
for all y ∈ K.
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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, if X is a Hilbert space and n = 2, from Corollary 3.3 we obtain the
result of D. Delbosco [7].
As application of Theorem 2.4, we have the following coincidence point theorem
for metric spaces.
Theorem 3.4. Let K be a nonempty compact convex subset of a topological vector
space X with metric d and fi , gi : K → X, i ∈ [n], continuous maps. Suppose there
exists λ ≥ 0, such that fi , gi satisfy the condition (3.1) for all x ∈ K, i ∈ [n]. If for
every x ∈ K, with fi (x) 6= gi (x) for all i ∈ [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
d(gi (y), fi (x)) ≥
i=1
n
X
)
d(gi (x), fi (x))
for all y ∈ K
i=1
is nonempty and for each x ∈ S there exists i ∈ [n] such that fi (x) = gi (x).
Proof. Put
fi (x, y) = d(gi (y), fi (x)),
for x, y ∈ K, i ∈ [n].
Then fi , gi , i ∈ [n] satisfy all of the requirements of Theorem 2.4.
Corollary 3.5. Let K be a nonempty compact convex subset of a metric space X,
f, g : K → X continuous maps and
Simultaneous Nonlinear
Inequalities Problem
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vol. 8, iss. 3, art. 84, 2007
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d(g(λx1 + (1 − λ)x2 ), f (y)) ≤ λd(g(x1 ), f (y)) + (1 − λ)d(g(x2 ), f (y)),
for all x1 , x2 ∈ K, y ∈ X, λ ∈ [0, 1]. If for every x ∈ K, with f (x) 6= g(x) there
exists a y ∈ K such that
d(g(x), f (x)) > d(g(y), f (x)),
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then the set
S = {x ∈ K : d(g(y), f (x)) ≥ d(g(x), f (x))
for all y ∈ K}
is nonempty and f (x) = g(x) for each x ∈ S.
Corollary 3.6. Let K be a nonempty compact convex subset of a metric space X
and f : K → X a continuous map such that
x 7→ d(x, f (y)) is a convex map for all y ∈ X.
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If for every x ∈ K, with f (x) 6= x there exists a y ∈ K such that
d(x, f (x)) > d(y, f (x)),
then the set
S = {x ∈ K : d(y, f (x)) ≥ d(x, f (x))
for all y ∈ K}
Simultaneous Nonlinear
Inequalities Problem
is nonempty and f (x) = x for each x ∈ S.
Zoran D. Mitrović
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 Corollary 2.3.
Theorem 3.7. Let X be a reflexive Banach space with its dual X ? and K a compact
convex subset of X. Let Ti : K → X ? , i ∈ [n], be maps. If x 7→ hTi (x), y − xi are
upper semicontinuous for all y ∈ K, i ∈ [n], then there exists x0 ∈ K such that
n
X
hTi (x0 ), y − x0 i ≥ 0
for all y ∈ K.
i=1
i=1
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Proof. Let fi (x, y) = hTi (x), y − xi, for all x, y ∈ K, i ∈ [n]. By our assumptions,
the maps fi satisfy all the hypotheses of Corollary 2.3, and it follows that there exists
x0 ∈ K such that
n
X
vol. 8, iss. 3, art. 84, 2007
hTi (x0 ), y − x0 i ≥ 0
for all y ∈ K.
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Remark 3.
(i) If n = 1 then Theorem 3.7 reduces 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 find x0 ∈ K such that
hT (x0 ), µ(y, x0 )i ≥ 0
for all y ∈ K.
Zoran D. Mitrović
If in Corollary 2.3 a map
f1 (x, y) = hT (x), µ(y, x)i
Simultaneous Nonlinear
Inequalities Problem
vol. 8, iss. 3, art. 84, 2007
for all x, 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
Corollary 2.3 a map
f1 (x, y) = hT (x), µ(y, x)i − hA(x), µ(y, x)i
for all x, 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. Let K be a nonempty compact convex subset of a topological vector
space X and fi : K × K → R continuous maps and fi (x, x) = 0 for all x ∈ K,
i ∈ [n]. If there exists λ ≥ 0, such that
coEp(fi (x, ·)) ⊆ Ep(fi (x, ·) − λ)
for all x ∈ K
and
coEp(−fi (·, y)) ⊆ Ep(−fi (·, y) − λ)
for all y ∈ K,
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for all i ∈ [n], then
0 ≤ min max
x∈K y∈K
n
X
fi (x, y) − max min
y∈K x∈K
i=1
n
X
fi (x, y) ≤ 2λn.
i=1
Proof. Note that
0 ≤ min max
x∈K y∈K
n
X
fi (x, y) − max min
y∈K x∈K
i=1
n
X
fi (x, y)
i=1
holds in general. By our assumptions, fi , i ∈ [n] satisfy all the hypotheses of Theorem 2.1, and it follows that there exists x0 ∈ K such that
min
n
X
y∈K
vol. 8, iss. 3, art. 84, 2007
fi (x0 , y) ≥ −λn.
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i=1
max min
n
X
x∈K y∈K
fi (x, y) ≥ −λn.
n
X
y∈K
max
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gi (y0 , y) ≥ −λn,
i=1
so
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i=1
Let gi (x, y) = −fi (y, x) for all (x, y) ∈ K × K, i ∈ [n]. By Theorem 2.1, it follows
that there exists y0 ∈ K such that
min
Zoran D. Mitrović
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So, we obtain,
(3.2)
Simultaneous Nonlinear
Inequalities Problem
n
X
i=1
fi (x, y0 ) ≤ λn.
Therefore, we obtain,
min max
(3.3)
y∈K x∈K
n
X
fi (x, y) ≤ λn.
i=1
By combining (3.2) and (3.3), it follows that
min max
x∈K y∈K
n
X
fi (x, y) − max min
y∈K x∈K
i=1
n
X
fi (x, y) ≤ 2λn.
i=1
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vol. 8, iss. 3, art. 84, 2007
Corollary 3.9. Let K be a nonempty compact convex subset of a topological vector
space X. Suppose fi : K × K → R, i ∈ [n] are continuous maps such that
1. fi (x, x) = 0 for all x ∈ K,
2. y 7→ fi (x, y) is convex for all x ∈ K,
3. x 7→ fi (x, y) is concave for all y ∈ K,
y∈K x∈K
n
X
i=1
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for all i ∈ [n]. Then we have
max min
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fi (x, y) = min max
x∈K y∈K
n
X
i=1
fi (x, y).
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References
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[2] Q.H. ANSARI, A.H. SIDDIQI AND S.Y. WU, Existence and duality of generalized vector equilibrium problems, J. Math. Anal. Appl., 259 (2001), 115–126.
[3] Q.H. ANSARI AND J.-C. YAO, An existence result for the generalized vector
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Inequalities Problem
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vol. 8, iss. 3, art. 84, 2007
[4] A. BEHERA AND L. NAYAK, On nonlinear variational-type inequality problem, Indian J. Pure Appl. Math., 30(9) (1999), 911–923.
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[11] G.K. PANDA AND N. DASH, Strongly nonlinear variational-like inequalities,
Indian J. Pure Appl. Math., 31(7) (2000), 797–808.
[12] J. PARIDA, M. SAHOO AND A. KUMAR, A variational-like inequalitiy problem, Bull. Austral. Math. Soc., 39 (1989), 225–231.
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fixed points, Indian J. Pure Appl. Math., 25(5) (1994), 459–462.
Simultaneous Nonlinear
Inequalities Problem
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