In the paper, a strengthening of the core-accessibility theorem by the author is proposed. It is shown that for any imputation outside of the nonempty core of TU cooperative game a strongly monotonic trajectory originating from this imputation exists, which converges to some element of the core. Here, strong monotonicity means that each imputation from the trajectory dominates several preceding elements and, besides, the number of these dominated imputations tends to in nity. To show that transferable utility assumption is relevant for strong accessibility of the core, we give an example of NTU cooperative game with a \black hole" being a nonempty closed subset of dominated imputations that contains all the sequential improvement trajectories originating from its points.
I(v) = fx 2 RN j ∑ xi = v(N ); xi v(i); i 2 N g : Further, we apply one more shortening: for any x = (x1; : : : ; xn) 2 RN and S 2 2N we put i2N x(S) = ∑ xi : i2S Copyright ⃝c by the paper's authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org In order to introduce classic domination relation v on I(v); some additional terms are required. First, recall that imputation x is said to be dominated by imputation y via a coalition S; if xi < yi; i 2 S; and y(S) v(S): Further, we say, for short, that x 2 I(v) is dominated by y 2 I(v); if there exists a coalition S such that x is dominated by y via S: Finally, we say that x 2 I(v) is dominated, if there exists an imputation y that dominates x. Respectively, an imputation x is said to be un-dominated, if there is no imputation that dominates it.
To summarize, we get formal descriptions of the classic domination relation v on
I(v); given by the formula
x
v y , 9S 2 2N n f∅g [(xi < yi; i 2 S) & (y(S)
v(S))] ;
and the core C( v) of v being the set of un-dominated imputations of v :
v y ]g :
Recall (see, e.g., [
In the paper, we consider one of the problem relating to the asymptotic behavior of monotonic sequences, so-called core-accessibility problem. Namely, we are interesting if there exists, for any imputation outside the core, at least one monotonic sequence starting at that imputation and converging to some un-dominated imputation. And if so, what additional improvements of this sequence can be made. To isolate the cores satisfying accessibility property mentioned, we present one of the key de nitions of the paper.
De nition 2 (Vasil'ev, 1987). The core C( v) of TU cooperative game v is called accessible, if C( v) ̸= ∅; and for any imputation x 2= C( v) there exists a convergent v-monotonic sequence fxrgr1=0 of imputations such that x0 = x and limr!1 xr belongs to the core C( v) of the game v.
It turned out that the core-accessibility takes place for any TU cooperative game with non-empty core.
Theorem 1 (Accessibility Theorem). The core C( v) of any TU cooperative game
v is accessible whenever it is nonempty.
This theorem was established in [
A strengthening of Accessibility Theorem, proposed in the paper, is motivated by the lack of transitivity of domination v: Sequential improvement process fxrgr1=0; based on non-complete and non-transitive binary relation v; may contain several fragments of type xr 1 v xr v xr+1 with xr+1 that doesn't dominate xr 1: Even more, we may get a cycle xr 1 v xr v xr+1 v xr 1 or more lengthy cycle xr 1 v xr v : : : v xm v xr 1 with m > r + 1: Unfortunately, we have no idea relating to the exclusion of this type of cycling in sequential improvement process. What can we do is just providing each imputation xr+1 in the process fxrgr1=0 to be able dominate a wider collection of previous imputations than a singleton fxrg: Certainly, it would be very nice to organize an improvement process in such a way that any imputation xr+1 dominates all the previous xm; m = 0; 1; : : : ; r: But already very simple 3-person games demonstrate impossibility of such a total domination phenomenon. Nevertheless, it turned out that we can essentially re ne improvement process provided that instead of v-monotonicity we apply its strengthening, given in the following de nition.
De nition 3. A sequence fxrgr1=0 of imputations of TU cooperative game v is said to be strongly v-monotonic (strongly monotonic, for short), if there exists a sequence of natural numbers kr ! 1 such that kr r; r 1; and xr m v xr for any r 1 and m 2 [1; kr]:
Now, we are in position to present main de nition of the paper.
De nition 4. Let the core C( v) be non-empty. Then C( v) is said to be strongly
accessible, if for any x 2 I(v)nC( v) there exists a convergent and strongly v-monotonic
sequence of imputations fxrgr1=0 such that x0 = x and lim xr belongs to the core C( v):
In the paper, we propose a strengthening of above mentioned Accessibility Theorem:
for any TU cooperative game v the core C( v) is strongly accessible whenever it is
nonempty. Thus, in case of TU cooperative games we can raise the quality of a
monotonic improvement process to a rather high level. Note that another variant of raising
quality of improvement process is given in [
The paper is organized as follows. Besides Introduction, it contains two parts and
appendix. First part is devoted to the proof of the strong accessibility theorem, second
part contains an example of NTU cooperative game that has nonempty core, which is
not accessible. Appendix, for the sake of completeness, is devoted to brief outline of
the proof of Accessibility Theorem. In contrast to [
Proof of the Strong Accessibility Result In this section we give a proof of strong accessibility of the core of TU cooperative game. We show that any sequential improvement process that exists according to Theorem 1 admits a considerable improvement of its quality, suffering due to the non-transitivity of classic domination relation. This lack of transitivity in the process of sequential improvement of dominated alternatives may be inadmissible in many applications of game theory to economical, sociological and political problems. By reconstructing a monotonic process in appropriate way, we can get a strongly monotonic improvement process with each imputation (except the rst and second ones) dominating several preceding alternatives. Even more, the number of this dominated alternatives tends to in nity. Passing on to the formal presentation of the strong accessibility result, we stress once more that its proof heavily relies upon Accessibility Theorem and some elementary topological properties of the domination relation v: Theorem 2. Let v be an n-person TU cooperative game with nonempty core C( v). Then for any imputation x 2 I(v) n C( v) there exist a sequence of natural numbers kr ! 1 with kr r; r 1; and convergent sequence of imputations fxrg01 such that x0 = x; limr!1 xr belongs to the core C( v); and, besides, xr m v xr for all natural numbers r 1 and m 2 [1; kr]: Proof. Let v be an n-person TU cooperative game with nonempty core C( v); and x be some dominated imputation. Without loss of generality we may assume that v satis es requirement: v(fig) = 0 for each i 2 N: Due to Accessibility Theorem there exists an v-monotonic convergent sequence of imputations fxrgr1=0 such that x0 = x and x = limr!1 xr 2 C( v): By De nition 1, for any r 1 there exists a coalition Sr such that imputation xr = (xr;1; : : : ; xr;n) dominates imputation xr 1 = (xr 1;1; : : : ; xr 1;n) via Sr : xr 1;i < xr;i; i 2 Sr; xr(Sr) v(Sr); r = 1; 2; : : : : (1) Below, we exploit max-metric 1(x; y) = ∥x y∥1 = max fjxi yij i 2 N g (respectively, all the distances and neighborhoods we use in this section are given in this metric). By applying max-metric we de ne vicinities Ur of imputations xr in order to modify the sequence fxrgr1=0 in a way required by the de nition of strong accessibility. Namely, for any r > 1 we put "r = min fxr;i
xr 1;i i 2 Srg=2 and de ne r = min f"r; "r+1g; r > 1: Further, for any r > 1 denote by Ur the r-neighborhood of the imputation xr :
Ur = fy 2 I(v) 1(xr; y) < rg; r = 2; 3; : : : : (2) Fix some r 1: Since xr 1 is dominated by xr via the coalition Sr; directly from (1) and 0-normalization condition v(fig) = 0; i 2 N; it follows that xr(Sr) v(Sr) and
xr;i > 0; i 2 Sr : Hence, we may construct a \bundle" fxrmgrm=11 dition with respect to the coalition Sr : Ur satisfying v-monotonicity con1 2 r 1 < x r;i ; i 2 Sr ; xr;i < xr;i < : : : < xr;i (3) where xrm;i is an i-th component of the imputation xrm = (xrm;1; : : : ; xrm;n) of the game under consideration. One can rather easily check that xr1; : : : ; xr r 1 may be given by the formulae: xrm = xr + (m r)c with ci = { r(n sr)=nr ; i 2 Sr ; rsr=nr ; i 2 N n Sr ; where sr = jSrj; r 1: In particular, inclusions xr +(m follow directly from the elementary relations1 r)c 2 Ur; m 2 [1; r 1]; r 1; (r (r m)ci = r m)ci = r r r ∑ ci = r sr(n
sr) nr
= i2Sr m n r r n
sr < r; i 2 Sr; m 2 [1; r m sr < r; i 2 N n Sr; m 2 [1; r n ∑ ci;
r i2NnSr possess some \transitivity features": each element of the collection under consideration dominates all the preceding elements. As to xr; it dominates (together with xr 1) each imputation xrm; m = 1; : : : ; r 1: Note also, that according to the choice of the vicinities Ur we get: the rst imputation xr1 (together with the others xrm; m r 1) dominates xr 1 and xrm 1; m = 1; : : : ; r 2; via the coalition Sr:
Passing on to the formation of a sequence fysgs1=0, originating from the above mentioned imputation x 2= C( v) and satisfying all the requirements of strong accessibility, we introduce rst the numbers er = r(r + 1)=2; r = 1; : : : : Further, by exploiting above mentioned elements xr and xrm we de ne the sought for sequence fysgs1=0 by the formula ys = 8 x0 = x ; if s = 0; <
xr ; if s = er; : xrm+1 ; if s = er + m and m 2 [1; r] : (4) 1 One may proceed by investigating these or some other appropriate concrete bundles. Below, we apply more general consideration, which admits extensions to a more abstract settings. We start with the proof that sequence fysgs1=0 is convergent, and its limit is equal to x = limr!1 xr: Fix an arbitrary " > 0 and show that there exists a number s0 such that 1(ys; x ) < " for any s > s0: To this end let us mention that due the convergency xr ! x there exists r0 such that 1(xr; x ) < "=2 for any r > r0: Further, from xr ! x it follows also that limr!1 r = 0: In fact, by de nition of the numbers r we get r 1(xr; xr 1); r 1: But the right-hand sides of these inequalities converge to zero, and, consequently, we get desired: r ! 0: Therefore, there exists r1 such that r < "=2 for any r > r1: Put s0 = er = r(r + 1)=2; where r = max fr0; r1g + 1: To estimate the distance 1(ys; x ) we consider two different cases: 1) s = er for some r r; and 2) s = er + m for some r r and m 2 [1; r]: In the rst case, due to the inequality s = er(s) > er we get r(s) > r0: Hence, by equality yer(s) = xr(s) following from the de nition of ys we get ∥ys x ∥1 = ∥xr(s) x ∥1 < "=2 : As to the second case, when s = er(s) + m > s0 with m belonging to the interval [1; r(s)]; according to the formula (4) we have ys = xrm(s(s)+)1: Since xrm(s(s)+)1 belongs to the vicinity Ur(s)+1 of the imputation xr(s)+1 we get (due to the de nition of Ur(s)+1 and obvious inequality r(s) + 1 > r1) ∥ys x ∥1 ∥xrm(s(s)+)1 xr(s)+1∥1 + ∥xr(s)+1 x ∥1 < r(s)+1 + "=2 < ": Thus, in both cases we get estimations desired, which complete the proof of convergency ys ! x 2 C( v):
To nalize the proof of Theorem 2 we have to show that for any s > 1 imputation ys dominates together with the preceding element ys 1 some more distant previous imputations ys m; m 2 [2; ks] with ks ! 1: Put ks = r for any s 2 [er; er+1); r 1: It is clear that ks chosen tends to in nity. To prove that for any s 2 [er; er+1) imputation ys dominates at least r nearest preceding imputations of the sequence fysgs1=0 we consider two possible cases: 1)s = er; and 2) s = er + m with m 2 [1; r]: In the rst case according to the formulae (3), (4), and v-monotonicity of the sequence fxrgr1=0 we get: yer 1 = xr 1 v xr = yer = ys and yer 1+p = xrp v xr = yer = ys for all p 2 [1; r 1]: In the second case without loss of generality we consider situation m = 1 only. Taking account our choice of vicinities Ur (see formula (2)), we obtain 1 (below, domination is realized via coalition Sr+1): ys 1 = xr v xr+1 = ys and, besides, ycr 1+p = xrp v xr1+1 = ys for all p 2 [1; r 1]:
Thus, the sequence fysgs1=0 meets all the requirements of De nition 3, and this fact completes the proof of Theorem 2. ⊔⊓ 3
To demonstrate that transferable utility assumption (namely, assumption that games
under consideration are TU games) is essential even if we deal with the weakest form
of accessibility, we give an example of NTU cooperative game with \black hole" being
a nonempty closed subset of dominated imputations, which contains all the monotonic
trajectories originating from its points2.
2 A bit more complicated counterexample can be found, also, in [
Recall (see, e.g. [
We say that x 2 G(N ) is an efficient payoff of G if there are no y 2 G(N ) such that yi > xi; i 2 N: To introduce an analog of the set I(v) we deal with in TU case, we propose the set I(G) de ned below. Put gi = max fxi 2 Rfig xi 2 G(fig)g; i 2 N: An element x = (x1; : : : ; xn) 2 G(N ) is said to be individually rational payoff of a game G if xi gi; i 2 N: We are now in position to de ne the following analog of imputation set in case of of NTU game G
I(G) = fx 2 G(N ) x is efficient and individually rationalg: Elements of I(G) are said to be imputations of NTU game G; and I(G) is said to be imputation set of G: For the next step we introduce domination relation G; an analog of binary relation v x
G y , 9S 2 2N n f∅g[8i 2 S(xi < yi)&(yS 2 G(S))]; x; y 2 I(G) ; where yS 2 RS is ordinary restriction of y = (yi)i2N to S : (yS )i = yi; i 2 S.
We introduce G-core C( G) (the core C( G), for short) of NTU game G as the set of un-dominated (maximal with respect to the binary relation G) elements of I(G) C( G) = fx 2 I(G) ̸ 9y 2 I(G)[ x
G y ]g Finally, we present an analog of accessibility of the core in NTU case. De nition 5. The core C( G) of NTU cooperative game G is called accessible if C( G) ̸= ∅; and for any x 2 I(G) n C( G) there exists a convergent sequence of imputations fxrgr1=0 of this game such that limr!1 xr belongs to the core C( G); x0 = x; and xr G xr+1 for any r 0:
Passing on to the example of NTU cooperative game with the above-mentioned \black hole" we consider a three-person NTU cooperative game G with imputation set I(G) containing some closed subset B that doesn't intersect the core C( G) and meets the saturation condition (x 2 B)&(x G y) ) y 2 B: So, let N = f1; 2; 3g and function G is de ned by the formulae
G(f1; 2; 3g) = fx 2 Rf1;2;3g
G(f1; 2g) = fx 2 Rf1;2g x1 + x2 G(f1; 3g) = fx 2 Rf1;3g x1 + x3 G(f2; 3g) = fx 2 Rf2;3g x2 + x3
G(fig) = fxi 2 Rfig xi First of all we show that the core C( G) of this game is a singleton. In fact, we prove the formula
C( G) = fug ; where u = (3; 4; 0): It is clear that imputation u is un-dominated. To prove that any imputation x ̸= u is dominated we note rst that imputation set I(G) of the game G under consideration is given by the formula
I(G) = fx 2 Rf+1;2;3g j ∑ xi = 7g: i2N Further, we consider separately two possible cases: 1) x3 = 0; and 2) x3 > 0: For the rst case let us check two subcases 1a) x1 < 3; and 1b) x1 > 3 (note, that x1 = 3 implies x = u). It is clear that in case 1a) imputation x is dominated by y = (3; 2; 2) via the coalition S = f1; 3g: As to the case 1b) we have x2 < 4: Hence, imputation y = (1; 4; 2) dominates x via coalition S = f2; 3g: Further, in case 2) consider 3 subcases: 2a) x3 2 (0; 2]; x2 < 4; 2b) x3 2 (0; 2]; x2 4; and 2c) x3 > 2: In the rst subcase imputation x is, obviously, dominated by y = (1; 4 ; 2 + ) with > 0 and 4 > x2 via coalition S = f2; 3g: In the second subcase, due to the inequality x3 > 0; we have x1 < 3 and, since x1 + x3 < 5 we get: imputation (x1 + ; x2 2 ; x3 + ) dominates x via coalition S = f1; 3g provided that > 0 and min f2; 3 x1g: Finally, in the third subcase we obtain: x is dominated by (x1 + ; x2 + ; 2) via S = f1; 2g; where = [5 (x1 + x2)]=2: Thus, summarizing all the cases considered we con rm the equality C( G) = f(3; 4; 0)g:
To de ne so-called \black hole" put
B = fx 2 I(G) x1 1; x2 2; x3 2g: It is clear that B \ C( G) = ∅: As B is closed, the only thing we need to prove non-accessibility of the core C( G) is the following implication
(x G y)&(x 2 B) ) y 2 B: Fix an arbitrary x 2 B and some y 2 I(G) such that x G y. To prove y 2 B let us look through three possible situations, which correspond to the two-person dominating coalitions S = fi; jg : (yi; yj) 2 G(fi; jg); xi < yi; xj < yj.
1: S = f1; 2g: By de nition of domination x by y via coalition S = f1; 2g it holds: y1 > x1 1; y2 > x2 2: Further, due to relations y1 + y2 5 and y1 + y2 + y3 = 7 we get y3 2; which completes the proof of inclusion y 2 B:
2: S = f1; 3g: Since directly by de nition of domination via coalition S = f1; 3g we get y1 > x1 1; y3 > x3 2; the only thing we have to prove is inequality y2 2: But by de nition of the set G(f1; 3g) we get y1 + y3 5; and, consequently, y2 = 7 (y1 + y3) 2:
3: S = f2; 3g: By de nition of domination via S = f2; 3g we have that y2 > x2 2; y3 > x3 2: Further, since y2 + y3 6; we get y1 = 7 (y2 + y3) 1; which completes the proof of inclusion y 2 B:
Thus, in all possible situations x G y and x 2 B imply y 2 B. Taking account that B is closed and B \ C( G) = ∅, we complete the proof of non-accessibility of the core C( G) of 3-player NTU cooperative game under consideration.
Acknowledgements The research was supported in part by Russian Fund for
Basic Research (grant 16 - 06 - 00101). The author thanks Prof. P.Sudholter for helpful
comments and remarks. Also many thanks to anonymous referees for useful
recommendations and suggestions.
Appendix: Outline of the Proof of Accessibility Theorem
For the sake of completeness, we propose a brief outline of the straightforward proof of
Accessibility Theorem, mentioned in the Introduction (for more details, see [
x 2 I(v) ;
v(x) = fy 2 I(v) x
v yg;
x 2 I(v):
3 Here and below we apply terms from [
For correspondence : X ! 2X let gr = f(x; y) 2 X X y 2 (x)g; (x) = ∪m1=1 m(x) with 1(x) = (x) and m+1(x) = ∪y2 m(x) (y): De nition 6. We say that a d.s. φ admits a generalized Lyapunov function if there exists a bounded function l : gr φ ! R such that (L1) l(x; y) (L2) l(x; z) d(x; y) for all x 2 X; y 2 φ(x) ; l(x; y) + l(y; z) for all x 2 X; y 2 φ (x); z 2 φ (y) : It is worth to note that the existence of a generalized Lyapunov function doesn't yet ensure the convergence of any trajectory of a d.s. φ to some element of its set of endpoints
Eφ = fx 2 X
φ(x) = fxgg : At the same time, there exists, for lower semi-continuous d.s. φ; a standard way of formation of subsystems (gr gr φ) possessing the property mentioned and satisfying equality E = Eφ Proposition 1. Let φ be a lower semi-continuous d.s. on X admitting a generalized Lyapunov function. Let
φ(x) = supfd(x; y) j y 2 φ(x)g; x 2 X : Then for any 2 (0; 1) all trajectories of the dynamic system φ (x) = { fxg fy 2 φ(x) d(x; y) >
; x 2 Eφ ; φ(x)g ; x 2 X n Eφ ; converge to elements of Eφ: So, for the proof of accessibility of the core C( v) on the basis of Proposition 1 it is enough to construct a lower semi-continuous subsystem of d.s.φv that admits a generalized Lyapunov function and satis es the relation E = C( v): Considering as the desired subsystem of d.s. φv a dynamic system l of the form
l(x) = fy 2 v(x) l(x; y) > d(x; y)g [ fxg; x 2 I(v) ;
with l to be some real-valued function on I(v)
sufficient condition of accessibility (see, e.g., [
I(v); one can obtain the following Theorem 3. Suppose there exists a bounded lower semi-continuous with respect to the rst argument function l : I(v) I(v) ! R+ such that l(x; z) l(x; y) + l(y; z);
x 2 I(v); y 2 φv(x); z 2 φv(y) : If, in addition, function l meets the requirement
8 x 2 I(v) n C( v) 9 y 2 v(x) [ l(x; y) > d(x; y)] ; then the core C( v) is accessible.
Below, we apply the following useful corollary of Theorem 3.
Corollary 1. If there exists a continuous function u : I(v) ! R+ such that 8 x 2 I(v) n C( v) 9 y 2 v(x) [ u(x) u(y) > d(x; y)] ; then the core C( v) of TU cooperative game v is accessible.
In order to apply Corollary 1 we put u = uv; where uv(x) = 4nd(x; C( v)); with d(x; C( v)) to be the distance from x to the core C( v) in Euclidean norm jjxjj2 = ( )1=2 ∑x2
i
N which plays a crucial role in justi cation of Accessibility Theorem.
. To complete an outline we will only sketch the proof of the main lemma, Lemma 1. Function uv satis es condition 8x 2 I(v) n C( v)9y 2 v(x)[uv(x) Sketch of the proof of Lemma 1. First of all we may assume w.l.g. that v(fig) = 0 for all i 2 N; and 0 v(S) v(N ) for all S N: It follows directly from the de nition of C( v) that condition v(S) v(N ); S N; implies that the core of 0-normalized TU cooperative game v is given by the formula
N C( v) = fx 2 R+ x(N ) = v(N ); x(S) v(S);
S
N g : In order to prove (5) let us x an arbitrary imputation x 2 I(v) n C( v). We will show that there exists z 2 I(v) satisfying inequality and the following weak domination requirement where uv(x) uv(z) > jjx
zjj2 x ~v z ; x ~v z , 9S [(x(S) < z(S) v(S))&(xi If x ~v z, then every neighborhood of z contains an element y 2 I(v) , which dominates x w.r.t. v, and the inequality (6) , by continuity of uv, holds in some neighborhood of z. Therefore the existence of z, satisfying (6) and (7) , proves the lemma .
In order to construct z, mentioned above, consider the imputation u 2 C( v), closest to x w.r.t. the norm jj jj2. It is not very hard to verify that there exists a coalition S N such that x(S) < u(S) = v(S). Put
S1 = fi 2 S xi < uig ; S2 = fi 2 N n S xi > uig ; T1 = fi 2 S xi T2 = fi 2 N n S xi uig ; uig : If T1+ = fi 2 T1j xi > uig = ∅ , then u may be used as the sought for imputation. If T1+ ̸= ∅, then setting we de ne z 2 RN by the formula e(x; S) = v(S)
x(S) ; qj = e(x; S)= u(Sj)
x(Sj) ; j = 1; 2 ; zi = <8 xi + q1(ui
xi + q2(ui : xi xi) ; i 2 S1 ; xi) ; i 2 S2 ; ; i 2 T1 [ T2 : (8) (9) we have where Since q1; q2 2 (0; 1], vector z clearly belongs to I(v). Taking account that S1 ̸= ∅ and z(S) = x(S) + e(x; S) = v(S); xi = zi; i 2 T1 ; x(S) < z(S) v(S); xi zi; i 2 S : By de nition of the weak domination ~v, these inequalities imply that imputation x is ~v-dominated by z via coalition S = S1 [ T1.
In order to check the inequality (6), note that and the lower bound of the difference jjx ujj2 jjz
ujj2 has the following form jjx zjj2
p2 e(x; S) ; jjx ujj2 jjz ujj2 (Q1 + Q2)=2jjx
ujj2 ; Qj = (2qj qj2) By Cauchy-Schwartz inequality , from (10) we obtain At last , by de nition of Qj, we have ui)2 (u(Sj) jSjj ∑ (xi i2Sj (10) (11) Estimating denominator in the right hand side of (9) we have for the rst step jjx ujj2 : So, applying de nitions of q1; q2, and combining inequalities (8),(9) and (11), by lengthy, but elementary calculations we obtain the sought for relationship which completes the sketch of the proof of Lemma 1. ⊓⊔
To conclude the outline of the proof of Theorem 1 we have to exploit some properties
of the dynamic system
φv(x) = { fxg
fy 2 φv(x) uv(x)
uv(y) > jjx
where is an arbitrary number from the interval (0; 1) and
v(x) = supfjjx
yjj2 y 2 φv(x); uv(x)
uv(y) > jjx
yjj2g; x 2= C( v) :
Since by Lemma 1 the dynamic system φv admits Lyapunov function, and φv is lower
semi-continuous dynamic system, it is not very hard to verify (like in [