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Some generalizations of common fixed point problems with applications

Abstract

In this paper, we establish a unified approach to study the existence of fixed points for common fixed point problems. Moreover, we introduce mixed systems of common fixed point problems, systems of common fixed point problems and common fixed point problems without convex assumptions. As applications, we establish a general type of some set-valued variational inequalities, and we obtain some existence theorems of solutions of set-valued variational inequalities. The results of this paper improve and generalize several known results on common fixed point problems.

MSC:47H10, 49J53, 49J40, 49J45.

1 Introduction

In three recent papers [13], by using some new concepts of generalized KKM mappings, the authors established common fixed point theorems for families of set-valued mappings in Hausdorff topological vector spaces. Recently, Agarwal et al. [4] established a common fixed point theorem for a family of self set-valued mappings on a compact and convex set in a locally convex topological vector space. As applications, an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities were obtained.

It is well known that the equilibrium problems are unified models of several problems, namely, optimization problems, saddle point problems, variational inequalities, fixed point problems, Nash equilibrium problems etc. Recently, Luc [5] introduced a more general model of equilibrium problems, which is called a variational relation problem (in short, VR). The stability of the solution set of variational relation problems was studied in [6, 7]. Some various types of variational relation problems or systems of variational relation problems have been investigated in many recent papers (see [812]). Recently, Agarwal et al. [13] presented a unified approach in studying the existence of solutions for two types of variational relation problems, and Balaj and Lin [14] established existence criteria for the solutions of two very general types of variational relation problems (see also [1522] for further studies of variational relation problems).

Motivated and inspired by research works mentioned above, in this paper, we establish a unified approach to study the existence of fixed points for common fixed point problems. As generalizations, mixed systems of common fixed point problems, systems of common fixed point problems, and common fixed point problems without convex assumptions are obtained.

2 Preliminary

In this short section, we recall some definitions and known results concerning set-valued mappings which will be needed throughout this paper.

Let X, Y, Z be three Hausdorff topological spaces. We adopt the following notations: for a set U and point x, α 1 (x,U) and, respectively, α 2 (x,U) means xU, and, respectively, xU. Denote α ˆ by α ˆ 1 = α 2 and α ˆ 2 = α 1 ; for two sets A, B, γ 1 (A,B) and, respectively, γ 2 (A,B) means AB, and, respectively, AB. Denote γ ˆ by γ ˆ 1 = γ 2 and γ ˆ 2 = γ 1 . A set-valued mapping F:XY is said to be: (1) upper semicontinuous at xX if, for any open subset O of Y with OF(x), there exists an open neighborhood U(x) of x such that OF( x ) for any x U(x); (2) upper semicontinuous on X, if it is upper semicontinuous at each xX; (3) lower semicontinuous at xX if, for any open subset O of Y with OF(x), there exists an open neighborhood U(x) of x such that OF( x ) for any x U(x); (4 lower semicontinuous on X, if it is lower semicontinuous in each xX; (5) closed if Graph(F)={(x,y)X×Y|yF(x)} is a closed subset of X×Y.

Lemma 2.1 (Corollary 17.55 (Kakutani-Fan-Glicksberg fixed point theorem) of [23])

Let X be a nonempty, convex and compact subset of a locally convex topological linear space, and F:XX be an upper semicontinuous set-valued mapping with nonempty convex compact values. Then there exists x X such that x F( x ).

Lemma 2.2 (Lemma 17.8, Theorem 17.10, Theorem 17.16, Theorem 17.19 of [23])

  1. (i)

    The image of a compact set under a compact-valued upper semicontinuous set-valued mapping is compact.

  2. (ii)

    If an upper semicontinuous set-valued mapping possess compact-valued, then it is closed.

  3. (iii)

    The correspondence φ is upper semicontinuous at x and φ(x) is compact, if and only if, for every net {( x α , y α )} in the graph of φ, that is, with y α φ( x α ) for each α, if x α x, then the net { y α } has a limit point in φ(x).

  4. (iv)

    If X and Y are topological spaces, a set-valued mapping F:XY is lower semicontinuous, if and only if, for any net { x α } in X, converging to xX, and each yF(x), there exists a net { y α } converging to y, with y α F( x α ) for all α.

3 Main results

Let S:XX, Q:XY, T:Y×ZZ and P:XZ be set-valued mappings with nonempty values. A common fixed point problem of type α (CFP-α) consists in finding x ˜ X such that x ˜ S( x ˜ ), for any yQ( x ˜ ),

α ( z , P ( x ˜ ) ) ,zT(y,z).

When α= α 1 , a common fixed point problem of type α 1 (CFP- α 1 ) consists in finding x ˜ X such that x ˜ S( x ˜ ), and

z y Q ( x ˜ ) T(y,z),zP( x ˜ ).

When α= α 2 , a common fixed point problem of type α 2 (CFP- α 2 ) consists in finding x ˜ X such that x ˜ S( x ˜ ) and, for any yQ( x ˜ ), there exists z ˜ P( x ˜ ) for which

z ˜ T(y, z ˜ ).

When X=Z and P(x)={x} for any xX, the problems (CFP- α 1 ) and (CFP- α 2 ) reduce the common fixed point problem (CFP): finding x ˜ X such that x ˜ S( x ˜ ), and

x ˜ y Q ( x ˜ ) T(y, x ˜ ).

Theorem 3.1 Assume that the data of problem (CFP-α) satisfy the following conditions:

  1. (i)

    X is a nonempty, convex and compact subset of a locally convex topological linear space, Y is Hausdorff linear topological space, and Z is a Hausdorff topological space;

  2. (ii)

    S is upper semicontinuous with nonempty convex compact values;

  3. (iii)

    U(y):={xX|yQ(x)and α ˆ (z,P(x)),zT(y,z)} is open in X.

Moreover, assume that there exists a set-valued mapping F:YX such that

  1. (iv)

    F(Q(x))S(x) for any xX;

  2. (v)

    F(y) is nonempty, convex and compact for any yY;

  3. (vi)

    F is convex, i.e., j = 1 n λ j F( y j )F( j = 1 n λ j y j ) for any y j Y and any λ j 0 with j = 1 n λ j =1;

  4. (vii)

    T is α-(F,P)-KKM, i.e., for any finite set { y 1 ,, y n } of Y and any xF(co{ y 1 ,, y n }), there is i 0 {1,,n} such that α(z,P(x)), zT( y i 0 ,z).

Then problem (CFP-α) has at least a solution.

Proof By way of contradiction suppose that, for any xX, xS(x), or, there is yQ(x) such that α ˆ (z,P(x)), zT(y,z). Denote U 0 ={xX|xS(x)}, which is open in X by (ii). Therefore,

X= U 0 y Y U(y).

By (iii), there is a finite subset { y 1 ,, y n } of Y such that

X= U 0 k = 1 n U( y k ).

Let { β k |k=0,1,2,,n} be the partition of unity subordinate to the open covering { U 0 ,U( y k )|k=1,2,,n} of X, i.e., { β k |k=0,1,2,,n} is a set of continuous functions with following properties: 0 β k (x)1, k = 0 n β k (x)=1, xX, k=0,1,2,,n; and if xU( y k ), for some k{1,,n}, then β k (x)=0, and if x U 0 , then β 0 (x)=0.

Now, we define the following set-valued mapping ϕ:XX:

ϕ(x)= β 0 (x)S(x)+ k = 1 n β k (x)F( y k ).

Clearly ϕ is upper semicontinuous on X. Moreover, since S(x) and F( y k ) are nonempty, convex and compact, ϕ(x) is nonempty, convex and compact in X for any xX. By Lemma 2.1, there exists x X such that x ϕ( x ). Let I( x )={k{1,,n}| β k ( x )>0}. Then, for any kI( x ), x U( y k ), it follows that y k Q( x ) for any kI( x ). By (iv), F( y k )F(Q( x ))S( x ) for any kI( x ). Therefore, x ϕ( x )S( x ), which implies that β 0 ( x )=0. It follows from the convexity of F that

x k = 1 n β k ( x ) F( y k )F ( k I ( x ) β k ( x ) y k ) .

By (vii), there is i 0 I( x ) such that α(z,P( x )), zT( y i 0 ,z), which implies that x U( y i 0 ), i.e., β i 0 ( x )=0. It contradicts the fact that i 0 I( x ), that is, β i 0 ( x )>0. This completes the proof. □

Remark 3.1 By Proposition 3.3 of [14], when α= α 1 , condition (iii) in Theorem 3.1 can be replaced by (a1) P:XZ is lower semicontinuous; (b1) {zZ|zT(y,z)} is closed in X for any yY; (c1) Q 1 (y) is open in X for any yY. Thus we have the following theorem.

Theorem 3.2 Assume that the data of problem (CFP- α 1 ) satisfy the conditions (i), (ii), (iv)-(vi) of Theorem  3.1 and

(a1) P:XZ is lower semicontinuous;

(b1) {zZ|zT(y,z)} is closed in X for any yY;

(c1) Q 1 (y) is open in X for any yY;

(d1) T is α 1 -(F,P)-KKM, i.e., for any finite set { y 1 ,, y n } of Y and any xF(co{ y 1 ,, y n }), there is i 0 {1,,n} such that zT( y i 0 ,z) for any zP(x).

Then problem (CFP- α 1 ) has at least a solution.

Remark 3.2 By Proposition 3.1 of [14], when α= α 2 , condition (iii) in Theorem 3.1 can be replaced by (a2) P:XZ is upper semicontinuous with nonempty compact values; (b2) {zZ|zT(y,z)} is closed in X for any yY; (c2) Q 1 (y) is open in X for any yY. Thus we have the following theorem.

Theorem 3.3 Assume that the data of problem (CFP- α 2 ) satisfy the conditions (i), (ii), (iv)-(vi) of Theorem  3.1 and

(a2) P:XZ is upper semicontinuous with nonempty compact values;

(b2) {zZ|zT(y,z)} is closed in X for any yY;

(c2) Q 1 (y) is open in X for any yY;

(d2) T is α 2 -(F,P)-KKM, i.e., for any finite set { y 1 ,, y n } of Y and any xF(co{ y 1 ,, y n }), there is i 0 {1,,n} such that there is zP(x) for which zT( y i 0 ,z).

Then problem (CFP- α 2 ) has at least a solution.

If X=Y and F(x)=x for all xX, we have problem (CFP-α).

Theorem 3.4 Let X be a nonempty, convex and compact subset of a locally convex topological linear space, Z be a Hausdorff topological space, and S,Q:XX, T:X×ZZ and P:XZ be set-valued mappings with nonempty values. Assume that

  1. (i)

    S is upper semicontinuous with nonempty convex compact values, and Q(x)S(x) for all xX;

  2. (ii)

    the set {xX|yQ(x)and α ˆ (z,P(x)),zT(y,z)} is open in X for any yX;

  3. (iii)

    T is α-P-KKM, i.e., for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that α(z,P(x)), zT( x i 0 ,z).

Then problem (CFP-α) has at least a solution.

When X=Y=Z, and F(x)=x, P(x)={x} for all xX, we obtain the following corollary.

Corollary 3.1 Let X be a nonempty, convex, and compact subset of a locally convex topological linear space, and S,Q:XX, T:X×XX be set-valued mappings with nonempty values. Assume that

  1. (i)

    S is upper semicontinuous with nonempty convex compact values;

  2. (ii)

    Q 1 (y) is open in X for any yX, and Q(x)S(x) for any xX;

  3. (iii)

    the set {xX|xT(y,x)} is closed in X for any yX;

  4. (iv)

    for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that xT( x i 0 ,x).

Then there exists x ˜ X such that x ˜ S( x ˜ ) and x ˜ y Q ( x ˜ ) T(y, x ˜ ).

Remark 3.3 Theorems 3.1-3.4 generalize the results of [14]. When X=Y and P(x)=F(x)=x for all xX, and S(x)=Q(x)=X for any xX, our Theorems 3.1-3.4 reduce to the results of [14].

Now, we introduce a new class of problems, called mixed systems of common fixed point problems (MSCFP). Let X, Y be nonempty sets in two Hausdorff topological vector spaces, Z be a Hausdorff topological space, S:X×YX, H:X×YY, Q:X×YX, M:X×YY, T:X×ZZ, F:Y×ZZ, P:X×YZ be set-valued mappings with nonempty values. A mixed system of common fixed point problems consists in finding ( x , y )X×Y such that x S( x , y ), y H( x , y ) and

u Q ( x , y ) , z P ( x , y ) , s.t.  z T ( u , z ) , v M ( x , y ) , z P ( x , y ) , s.t.  z F ( v , z ) .

Theorem 3.5 Assume that

  1. (i)

    X, Y, Z are three nonempty, compact and convex subsets of three Hausdorff linear topological spaces;

  2. (ii)

    C={(x,y)X×Y:xS(x,y)} and D={(x,y)X×Y:yH(x,y)} are nonempty and closed in X×Y;

  3. (iii)

    P is continuous with nonempty convex compact values;

  4. (iv)

    {zZ:zT(x,z)} and {zZ:zF(y,z)} are closed for any (x,y)X×Y;

  5. (v)

    Q(x,y), M(x,y), coQ(x,y)S(x,y), coM(x,y)H(x,y), and Q 1 (x), M 1 (y) are open for any (x,y)X×Y;

  6. (vi)

    for any fixed yY, any finite subset { u 1 ,, u n } of X and any xco{ u 1 ,, u n }, there is i{1,,n} such that, for any zP(x,y), zT( u i ,z);

  7. (vii)

    for any fixed xX, any finite subset { v 1 ,, v n } of Y and any yco{ v 1 ,, v n }, there is i{1,,n} such that there is zP(x,y) for which zF( v i ,z).

Then problem (MSCFP) has at least a solution.

Proof Define A:X×YX and B:X×YY as follows:

A ( x , y ) = { u X : z P ( x , y ) , z T ( u , z ) } , B ( x , y ) = { v Y : z F ( v , z ) , z P ( x , y ) } .

By (iii), (iv), and Remarks 3.1, 3.2, A 1 (u), B 1 (v) are open in X×Y for any (u,v)X×Y.

Suppose there exists (x,y)X×Y such that xcoA(x,y), then there is a finite subset { u 1 ,, u n } of A(x,y) such that xco{ u 1 ,, u n }. By (vi), there is i 0 {1,,n} such that zT( u i 0 ,z) for any zP(x,y), which contradicts the fact that u i A(x,y) for any i{1,,n}. Hence xcoA(x,y) for any (x,y)X×Y.

Suppose there exists (x,y)X×Y such that ycoB(x,y), then there is a finite subset { v 1 ,, v n } of B(x,y) such that yco{ v 1 ,, v n }. By (vii), there is i 0 {1,,n} such that there exists zP(x,y) for which zF( v i 0 ,z), which contradicts the fact that v i B(x,y) for any i{1,,n}. Hence ycoB(x,y) for any (x,y)X×Y.

Define the mappings A :X×YX and B :X×YY as follows:

A ( x , y ) = { A ( x , y ) Q ( x , y ) , if  ( x , y ) C , Q ( x , y ) , if  ( x , y ) C , B ( x , y ) = { B ( x , y ) M ( x , y ) , if  ( x , y ) D , M ( x , y ) , if  ( x , y ) D .

For any uX,

A 1 (u)= [ Q 1 ( u ) A 1 ( u ) ] [ ( ( X × Y ) C ) Q 1 ( u ) ]

is open in X×Y. Similarly, B 1 (v) is open for any vY. Hence, A 1 (u), B 1 (v) are open for any (u,v)X×Y, and xco A (x,y), yco B (x,y) for any (x,y)X×Y. By Theorem 3 of [24], there exists ( x , y )X×Y such that A ( x , y )= and B ( x , y )=, which implies that x S( x , y ), y H( x , y ) and

u Q ( x , y ) , z P ( x , y ) , s.t.  z T ( u , z ) , v M ( x , y ) , z P ( x , y ) , s.t.  z F ( v , z ) .

As a generalization, we introduce the following system of common fixed point problems. Let I be any index set. For any iI, let X i , Z i be Hausdorff topological spaces, and S i :X X i , Q i :X X i , T i : X i × Z i Z i and P i :X Z i be set-valued mappings with nonempty values. A system of common fixed point problems (SCFP) consists in finding x ˜ X such that, for each iI, x ˜ i S i ( x ˜ ) and, for any y i Q i ( x ˜ ),

α ( z i , P i ( x ˜ ) ) , z i T i ( y i , z i ).

 □

Theorem 3.6 Assume that, for each iI, the following conditions are satisfied:

  1. (i)

    X i is a nonempty, convex and compact subset of a locally convex topological linear space, and Z i is a Hausdorff topological space;

  2. (ii)

    the set Ω i :={xX: x i S(x)} is nonempty and closed in X;

  3. (iii)

    the set-valued mapping G i :X X i , defined by G i (x):={ y i X i | α ˆ ( z i , P i (x)), z i T i ( y i , z i )}, has open fibers;

  4. (iv)

    Q i 1 ( y i ) is open in X for any y i X i ;

  5. (v)

    for any finite set { y i 1 ,, y i n } of X i and any x i co{ y i 1 ,, y i n } with x= ( x i ) i I , there is j{1,,n} such that α( z i , P i (x)), z i T i ( y i j , z i ).

Then problem (SCFP) has at least a solution.

Proof For each iI, define the set

W i =[X Ω i ] { x X | G i ( x ) Q i ( x ) } .

By (ii) and (iii), W i is open in X for each iI. Since G i 1 ( y i ) is open in X for each iI, by a known continuous selection Theorem (see [[23], Theorem 17.63]), there is a continuous function f i :X X i such that f i (x)co G i (x). Thus, for each iI, we define the mapping F:XX as follows:

F i (x)={ f i ( x ) , if  x W i , X i , if  x W i , F(x)= i I F i (x).

By Lemma 2.1, there exists x X such that x F( x ), which implies that x i F i ( x ) for each iI. If x W i 0 for some i 0 I, x i 0 = f i 0 ( x )co G i 0 ( x ). Then there exists a finite set { y i 0 1 ,, y i 0 n } G i 0 ( x ) such that x i 0 co{ y i 0 1 ,, y i 0 n }. By (v), there is j 0 {1,,n} such that α( z i 0 , P i 0 ( x )), z i 0 T i 0 ( y i 0 j 0 , z i 0 ), which contradicts the fact that y i 0 j 0 G i 0 ( x ). Therefore, x W i for any iI, which implies that, for each iI, x i S i ( x ) and, for any y i Q i ( x ), α( z i , P i ( x )), z i T i ( y i , z i ). This completes the proof. □

As a generalization of Theorem 3.1, we derive the following existence result for the solution of problem (CFP-α) without convex assumptions.

Theorem 3.7 Assume that

  1. (i)

    X is a nonempty and compact subset of a Hausdorff topological vector space E, and has the fixed point property, and Z is a Hausdorff topological space;

  2. (ii)

    Ω:={xX:xS(x)} is closed;

  3. (iii)

    Q(x)S(x) for any xX;

  4. (iv)

    U(y):={xX| α ˆ (z,P(x)),zT(y,z)} and Q 1 (y) are open in X for any yX;

  5. (v)

    for any finite set { y 1 ,, y n } of X, there exists a continuous mapping ϕ n : Δ n X such that

(v1) for any λ=( λ 1 ,, λ n ) Δ n , there exists iJ(λ) such that α(z,P( ϕ n (λ))), zT( y i ,z);

(v2) if y i Q( ϕ n (λ)) for any iJ(λ), then ϕ n (λ)Q( ϕ n (λ)), where

Δ n := { ( λ 1 , , λ n ) R n | i = 1 n λ i = 1 , λ i 0 } ,J(λ):= { i { 1 , , n } | λ i > 0 } .

Then problem (CFP-α) has at least a solution, i.e., there exists x ˜ X such that x ˜ S( x ˜ ) and, for any yQ( x ˜ ),

α ( z , P ( x ˜ ) ) ,zT(y,z).

Proof Define the mapping G:XX as follows:

G(y)= [ X Q 1 ( y ) ] { x X | x S ( x )  and  α ( z , P ( x ) ) , z T ( y , z ) } .

By (ii) and (iv), G(y) is closed for any yX.

By (v), for any finite subset { y 1 ,, y n } of X, there exists a continuous mapping ϕ n : Δ n X such that, for any λ=( λ 1 ,, λ n ) Δ n , there exists iJ(λ) such that α(z,P( ϕ n (λ))), zT( y i ,z), then

  1. (1)

    if there exists i 0 J(λ) such that y i 0 Q( ϕ n (λ)), which implies that ϕ n (λ)X Q 1 ( y i 0 ). Thus ϕ n (λ)G( y i 0 );

  2. (2)

    if y i Q( ϕ n (λ)) for any iJ(λ), then ϕ n (λ)Q( ϕ n (λ)) by (v). By (iii), ϕ n (λ)S( ϕ n (λ)). Then ϕ n (λ)S( ϕ n (λ)) and, for any λ=( λ 1 ,, λ n ) Δ n , there exists iJ(λ) such that α(z,P( ϕ n (λ))), zT( y i ,z). Thus ϕ n (λ)G( y i ).

Hence, by Theorem 2.1 of [17],

y X G(y),

which implies that there is x ˜ X such that x ˜ S( x ˜ ) and, for any yQ( x ˜ ),

α ( z , P ( x ˜ ) ) ,zT(y,z).

 □

By Remarks 3.1 and 3.2, we have the following results.

Theorem 3.8 If (iv) of Theorem  3.7 is replaced by

(a3) P:XZ is lower semicontinuous;

(b3) {zZ|zT(y,z)} is closed in X for any yY, and Q 1 (y) is open in X for any yX.

Then problem (CFP- α 1 ) has at least a solution, i.e., there exists x ˜ X such that x ˜ S( x ˜ ) and

z y Q ( x ˜ ) T(y,z),zP( x ˜ ).

Theorem 3.9 If (iv) of Theorem  3.7 is replaced by

(a4) P:XZ is upper semicontinuous with nonempty compact values;

(b4) {zZ|zT(y,z)} is closed in X for any yY, and Q 1 (y) is open in X for any yX.

Then problem (CFP- α 2 ) has at least a solution, i.e., there exists x ˜ X such that x ˜ S( x ˜ ) and, for any yQ( x ˜ ), there is zP( x ˜ ) for which zT(y,z).

Remark 3.4 In [14], convex assumptions or the KKM property played an important role in the proofs of common fixed points. In Theorems 3.7-3.9, the existence of common fixed points does not depend on convex assumptions.

Remark 3.5 This paper extends the research on common fixed point problems. The classical common fixed point problem (see [14]) is a special case of problem (CFP-α). Moreover, we introduce the system of common fixed points, and common fixed point problems without convex assumptions are obtained.

4 Applications

4.1 Variational inclusions

In this section, we fix our attention on variational inclusions described below:

Let X be a nonempty, convex and compact subset of a locally convex topological linear space, Z be a Hausdorff topological linear space, and S,Q:XX, A,B:X×XZ be set-valued mapping with nonempty values.

A variational inclusion of type γ (VI-γ) consists in finding x ˜ X such that x ˜ S( x ˜ ) and γ(A( x ˜ ,y),B( x ˜ ,y)) holds for any yQ( x ˜ ).

A variational inclusion of type γ 1 (VI- γ 1 ) consists in finding x ˜ X such that x ˜ S( x ˜ ) and A( x ˜ ,y)B( x ˜ ,y) for any yQ( x ˜ ).

A variational inclusion of type γ 2 (VI- γ 2 ) consists in finding x ˜ X such that x ˜ S( x ˜ ) and A( x ˜ ,y)B( x ˜ ,y) holds for any yQ( x ˜ ).

Theorem 4.1 Let X be a nonempty, convex and compact subset of a locally convex topological linear space, Z be a Hausdorff topological linear space. Assume that

  1. (i)

    S is upper semicontinuous with nonempty convex compact values;

  2. (ii)

    Q 1 (y) is open in X for any yX, and Q(x)S(x) for any xX;

  3. (iii)

    the set {xX|γ(A(x,y),B(x,y))holds} is closed in X for any yX;

  4. (iv)

    for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that γ(A(x, x i 0 ),B(x, x i 0 )) holds.

Then problem (VI-γ) has at least a solution.

Proof Define the mapping T:X×XX as follows:

T(y,x)= { z X : γ ( A ( x , y ) , B ( z , y ) )  holds } .

By (iii), {xX:xT(y,x)} is closed in X for any yX. By Corollary 3.1, there exists x ˜ X such that x ˜ S( x ˜ ) and x ˜ y Q ( x ˜ ) T(y, x ˜ ), which implies x ˜ S( x ˜ ) and γ(A( x ˜ ,y),B( x ˜ ,y)) holds for any yQ( x ˜ ). This completes the proof. □

Theorem 4.2 Assume (ii) and (iii) of Theorem  4.1 are replaced by

(a5) for any yX, A(,y) is lower semicontinuous, and B(,y) is closed;

(b5) for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that A(x, x i 0 )B(x, x i 0 ).

Then problem (VI- γ 1 ) has at least a solution.

Proof As soon as we show that the set {xX:A(x,y)B(x,y)} is closed for any yX. Let { x α } be a net in X converging to x, such that A( x α ,y)B( x α ,y) for any α. For any zA(x,y), since A(,y) is lower semicontinuous, by Lemma 2.2, there is z α A( x α ,y)B( x α ,y) such that z α z. It follows from the closeness of B(,y) that zB(x,y). Then A(x,y)B(x,y). Thus, the set {xX:A(x,y)B(x,y)} is closed for any yX. □

Theorem 4.3 Assume (ii) and (iii) of Theorem  4.1 are replaced by

(a6) for any yX, A(,y) is upper semicontinuous with nonempty compact values, and B(,y) is closed;

(b6) for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that A(x, x i 0 )B(x, x i 0 ).

Then problem (VI- γ 2 ) has at least a solution.

Proof As soon as we show that the set {xX:A(x,y)B(x,y)} is closed for any yX. Let { x α } be a net in X converging to x, such that A( x α ,y)B( x α ,y) for any α. Then there exists z α Z such that z α A( x α ,y)B( x α ,y) for any α. Since A(,y) is upper semicontinuous with nonempty compact values, by (iii) of Lemma 2.2, there is a subnet { z α β } of { z α } converging to some zA(x). Since z α β B( x α β ,y), and B(,y) is closed, zB(x,y). Thus zA(x,y)B(x,y). Hence, the set {xX:A(x,y)B(x,y)} is closed for any yX. □

4.2 Generalized multiplied minimax inequality of Ky Fan type

Let X be a nonempty, convex and compact subset of a locally convex topological linear space, and S,Q:XX, f:X×X×XR be a real-valued function. A generalized multiplied minimax inequality of Ky Fan type consists in finding x ˜ X such that x ˜ S( x ˜ ) and f(y, x ˜ , x ˜ )0 for any yQ( x ˜ ).

Theorem 4.4 Let X be a nonempty, convex and compact subset of a locally convex topological linear space. Assume that

  1. (i)

    S is upper semicontinuous with nonempty convex compact values;

  2. (ii)

    Q 1 (y) is open in X for any yX, and Q(x)S(x) for any xX;

  3. (iii)

    f(y,,) is lower semicontinuous on X×X for any yX;

  4. (iv)

    for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that f( x i 0 ,x,x)0.

Then the generalized multiplied minimax inequality of Ky Fan type has at least a solution.

Proof Define the mapping T:X×XX as follows:

T(y,x)= { z X : f ( y , x , z ) 0 } .

By (iii), {xX:xT(y,x)} is closed in X for any yX. By Corollary 3.1, there exists x ˜ X such that x ˜ S( x ˜ ) and x ˜ y Q ( x ˜ ) T(y, x ˜ ), which implies x ˜ S( x ˜ ) and f(y, x ˜ , x ˜ )0 for any yQ( x ˜ ). This completes the proof. □

Remark 4.1 Theorem 4.4 is different from Theorems 4.2, 4.3 of [4]. (1) Our Theorem 4.4 with constraining mappings S, Q is a more general problem than Theorems 4.2, 4.3 of [4]. (2) The existence conditions are different between Theorem 4.4 and Theorems 4.2, 4.3 of [4].

From Theorem 4.4, when S(x)=Q(x)=X for any xX, we obtain a multiplied minimax inequality of Ky Fan type.

Theorem 4.5 Let X be a nonempty, convex and compact subset of a locally convex topological linear space. Assume that

  1. (i)

    f(y,,) is lower semicontinuous on X×X for any yX;

  2. (ii)

    for any finite set { x 1 ,, x n } of X and any xco{ x 1 ,, x n }, there is i 0 {1,,n} such that f( x i 0 ,x,x)0.

Then the multiplied minimax inequality of Ky Fan type has at least a solution, i.e., f(y, x ˜ , x ˜ )0 for any yX.

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Acknowledgements

This research is supported by the Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (no. 13CG35), and open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (no. 201309KF02).

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Yang, Z. Some generalizations of common fixed point problems with applications. Fixed Point Theory Appl 2014, 189 (2014). https://doi.org/10.1186/1687-1812-2014-189

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Keywords

  • common fixed point
  • general types
  • set-valued variational inequality