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On the translations of quasimonotone maps and monotonicity

Abstract

We show that given a convex subset K of a topological vector space X and a multivalued map T:K X , if there exists a nonempty subset S of X with the surjective property on K and T+w is quasimonotone for each wS, then T is monotone. Our result is a new version of the result obtained by N. Hadjisavvas (Appl. Math. Lett. 19:913-915, 2006).

1 Introduction and some definitions

Throughout the paper, X and X denote a real topological vector space and the dual space of X, respectively. Suppose KX is a nonempty subset of X and T:K X is a multivalued map from K to X . Recall that T is said to be monotone if for all x T(x), y T(y) one has

x y , x y 0.

T is said to be pseudomonotone and quasimonotone, in the sense of Karamardian (see [1, 2]), respectively, if for any x T(x), y T(y) the following implications hold:

y , x y 0 x , x y 0

and

y , x y >0 x , x y 0.

It is clear that a monotone map is pseudomonotone, while a pseudomonotone map is quasimonotone. The converse is not true. If T is pseudomonotone (quasimonotone) and w X {0}, then T+w is not pseudomonotone (quasimonotone) in general. In the case of a single-valued linear map T defined on the whole space R n , it is known that if T+w is quasimonotone, then T is monotone [2]. Many authors (see, e.g., [4, 5]) extended this result for a nonlinear Gateaux differentiable map defined on a convex subset K (of a Hilbert space) with a nonempty interior.

Recently, Hadjisavvas [3] extended the above result to the multivalued maps defined on a convex subset of a real topological vector space with no assumption of differentiability or even continuity on the map T whose domain need not have a nonempty interior. In this paper, we first introduce the surjective property of a subset of X on a segment of K. By using this concept, we can extend the corresponding result obtained in [3]. Before stating the main result, we recall some definitions.

Definition 1 Let x, y be two elements of K. We say that S X has the surjective property on x and y whenever the following equality holds:

S,xy= { x , x y : x S } =R.

Remark that we can consider xy as a linear functional (denoted by x y ˆ ) on X which is defined by

x y ˆ ,f=f,xy.

Hence if S has the surjective property on x, y, then the image of S under the linear functional x y ˆ is all of the real numbers, and that is why we used the phrase surjective property.

Definition 2 Let KX be a nonempty set and S X . We say that S has the surjective property on K if for every xK there exists yK such that S has the surjective property on x and y.

Definition 3[3]

Let K be a convex subset of X. An element v of X is called perpendicular to K if v is constant on K, i.e.,

v,x=v,y,x,yK.

Also the straight line S={u+tv:tR}, where u,v X with v0, is said to be perpendicular to K if v is perpendicular to K.

Remark 1 If KX is a nonempty convex set and u,v X with v is not perpendicular to K, then the straight line S={u+tv:tR} has the surjective property on K. Indeed, let xK be an arbitrary member of K. Because v is not perpendicular to K, there exists yK such that c=v,xy0. For each aR, we put t= a u , x y c and so a=u+tv,xy. Hence S,xy=R. This means that S has the surjective property. Therefore, v being not perpendicular to K implies the surjective property while the simple example X= R 2 , S={(t,t)=(0,0)+(1,1)t:tR} and K={(x,x):xR} shows that the converse does not hold in general. In this example, one can see that S has the surjective property and v=(1,1) is perpendicular to K (note v=(1,1),(x,x)=v=(1,1),(y,y)=0). The notion v is not perpendicular to K, which plays a crucial rule in proving the main results in [3]; while in this note, the surjective property has an essential rule in the main result. Hence one can consider this paper as an improvement of [3] (slightly, of course).

We need the following lemma in the sequel.

Lemma 1 Let X be a real topological vector space, K a nonempty convex subset of X andT:K X a multivalued map. Supposex,yK, S X has the surjective property on x, y andT+wis quasimonotone on the line segment[x,y]={tx+(1t)y:t[0,1]}for allwS. Then T is monotone on[x,y].

Proof We can define an order on [x,y] as follows:

ab t 1 t 2 ,where a=x+ t 1 (yx),b=x+ t 2 (yx).

On the contrary, assume T is not monotone on [x,y]. So there exist a,b[x,y] and a T(a), b T(b) with ab and a b ,ab<0. Hence we have

a , y x > b , y x .

Since S is surjective on x, y, there exists wS such that

a , y x >w,xy> b , y x .

Therefore,

a + w , y x >0, b + w , y x <0,

which is a contradiction. This completes the proof. □

Now we are ready to present the main result.

Theorem 1 Let X be a real topological vector space, K a nonempty convex subset of X andT:X X a multivalued map. AssumeS X is connected and has the surjective property on K. IfT+wis quasimonotone for allwS, then T is monotone on K.

Proof Let x,yK, x T(x) and y T(y) be arbitrary elements. If S is surjective on x, y then, by Lemma 1, T is monotone on [x,y] and the proof is complete. Assume S does not have the surjective property on x, y. So S(xy)R. Since S has the surjective property on K, then there exists zK such that S is surjective on x, z; and since S is connected, then S,yz is a connected subset of the real numbers unbounded from above and below, and so it is equal to the real numbers. This means that S has the surjective property on y, z and also on x + y 2 , z. Therefore, it follows from Lemma 1 that T is monotone on [y,z] and [ x + y 2 ,z]. Similarly, T is monotone on the segments [x, z s ] and [y, z s ], for all s]0,1[, where z s =sz+(1s) x + y 2 . Therefore, for any z s T( z s ) and z T(z), we have

(1)
(2)
(3)

From (1) and (2), we deduce that

2s z s , z x + y 2 x , z s x + y , z s y .
(4)

Now from z z s =(1s)(z x + y 2 ) and (3), we obtain

z , z x + y 2 z s , z x + y 2 .
(5)

Combining (4) and (5), we have

2s z , z x + y 2 x , z s x + y , z s y .

So if in the previous inequality we tend s0, then z s x + y 2 , and hence we deduce

0 x , y x + y , y x .

This means T is monotone and the proof is now complete. □

Remark 1 shows that Theorem 1 is a new version of Theorem 1 in [3], although our proof is, in fact, completely similar to it.

References

  1. Karamardian S, Schaible S: Seven kinds of monotone maps. J. Optim. Theory Appl. 1990, 66: 37–46. 10.1007/BF00940531

    Article  MathSciNet  MATH  Google Scholar 

  2. Karamardian S, Schaible S, Crouuzeix JP: Characterizations of generalized monotone maps. J. Optim. Theory Appl. 1993, 76: 399–413. 10.1007/BF00939374

    Article  MathSciNet  MATH  Google Scholar 

  3. Hadjisavvas N: Translations of quasimonotone maps and monotonicity. Appl. Math. Lett. 2006, 19: 913–915. 10.1016/j.aml.2005.09.007

    Article  MathSciNet  MATH  Google Scholar 

  4. He Y: A relationship between pseudomonotone and monotone mappings. Appl. Math. Lett. 2004, 17: 459–461. 10.1016/S0893-9659(04)90089-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Isac G, Motreanu D: Pseudomonotonicity and quasimonotonicity by translations versus monotonicity in Hilbert spaces. Aust. J. Math. Anal. Appl. 2004, 1(1):1–8.

    MathSciNet  MATH  Google Scholar 

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Correspondence to S Plubtieng.

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Farajzadeh, A., Karamian, A. & Plubtieng, S. On the translations of quasimonotone maps and monotonicity. J Inequal Appl 2012, 192 (2012). https://doi.org/10.1186/1029-242X-2012-192

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