Flattening of ellipses by growing nonlinearities

Nonlinear Analysis 35 (1999) 781 – 795

Flattening of ellipses by growing nonlinearities H. Gingold ∗ Department of Mathematics, West Virginia University, P.O. Box 6310, Morgantown, WV 26506-6310, USA Received 25 January 1996; accepted 14 May 1997

Keywords: Nonlinear ellipses; Nonlinear hyperbolas

1. Introduction Nonlinearities is a subject of great importance and interest. Unlike linearity which is well de ned and well understood much is yet to be discovered about nonlinearity. A growing literature on nonlinear problems indicates that our understanding of the subject is enhanced, when dierent methods are applied to dierent subclasses of nonlinear problems. The purpose of this paper is to contribute to the study of nonlinearities by focusing on specialized families of geometrical problems in which nonlinearities grow. This is a setting which could serve as a model for studies of other classes of nonlinear problems. Our current study lends support to the following principle. “Growing nonlinearities of bounded objects lead to perfect-like limiting con gurations.” It shows that “portions of nonlinear ellipses converge to faces of a perfect square as nonlinearities grow to in nity.” Since the convergence at vertices need not be uniform, an intriguing phenomenon takes place. Although the principle is not mathematically precise it conveys the gist of an interesting phenomenon. One may wonder what underlines the peculiar statements just annunciated. A simpli ed answer lies with a well-known universal limit. Namely, limn→∞ a1=n = 1; 0¡a¡∞. The interpretation of this limit is as follows. The positive roots of the equations x n − a = 0, as the nonlinearities n grow to in nity and tend to 1 for xed a. This could help us capture important features in more complicated settings. Other studies of families of nonlinear problems in the settings of function theory and dierential equations, which lend support to the above principle, will be discussed somewhere else. ∗

E-mail: [email protected].

0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 4 0 - 6

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Recently, a new pertubative method in problems of mathematical physics has been introduced. Algebraic equations with varying nonlinearities (as well as dierential equations with varying nonlinearities) play an important role in this method, to be called the “-method”, where  is a varying nonlinearity. See [2] for a review of the “-method.” This study could be bene cial to the “-method” in more ways than one. First it points out the possible existence of limits for  → ∞. Secondly, it warns of the possible existence of “kinks” ( jumps in tangent lines) and nonuniform convergence. Last but not the least, a rapid rate of convergence makes the conclusions valuable for moderate values of . Let us introduce some nomenclature. De nition 1.1. The set of points x = (x1 ; : : : ; xm ); xj ∈ R; with aj ∈ R+ ; nj ∈ N; j = 1; : : : ; m such that m X

2n

aj−2 xj j = 1

(1.1)

j=1

is called an ellipse. If one of the powers nj is not one, we call it a nonlinear ellipse. Pm 2n The set of points x such that j=1 aj−2 xj j ≤ 1 is called an ellipsoid. De nition 1.2. The set of points (x; y) = (x1 ; : : : ; xm ; y1 ; : : : ; yp ); m ≥ 1; p ≥ 1; with xj ∈R; aj ∈ R+ ; nj ∈ N; j = 1; : : : ; m; y ∈ R; b ∈ R+ ; l ∈ N;  = 1; : : : ; p such that m X

2nj

aj−2 xj

−

j=1

p X

2l b−2  y = 1

(1.2)

=1

is called a hyperbola. If a power nj or l is dierent than one we call it a nonlinear hyperbola. The set of points (x; y) satisfying m X j=1

2nj

aj−2 xj

−

p X

2l b−2  y ≥ 1

(1.3)

=1

is called a hyperboloid. De nition 1.3. Let (k); k = 1; 2; : : : be a sequence such that 0¡(k)¡1. We call (k) a gentle (1) sequence if [(k)]1=k → 1 as k → +∞. For example (k) = k  ;  ∈ R induces a gentle (1) sequence. We call (k) a gentle (0) sequence if [(k)]1=k → 0 as k → ∞. For example (k) = exp[− exp k] is gentle (0). Remark. The varying parameters nj ; l in Eqs. (1.1) – (1.3) are measures of the nonlinearities of the ellipses and hyperbolas, respectively. De nition 1.4. The set of points x on the ellipse (1.1) which satis es 1 − 2n

P

−2 j6=r aj

xj j ≥ dr ¿0 for a gentle (1) sequence dr and with xr = x+r or xr = x−r where x±r are

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given by  x±r = 1 −

X

1=(2nr ) 2n aj xj j 

;

x−r = −x+r

(1.4)

j6=r

will be called, respectively, an x+r and an x−r cap (on the ellipse). De nition 1.5. The set of points (x; y) on the hyperbola (1.2) such that  1=(2lr ) m X X 2n 2l b−2 − aj−2 xj j  ; y−r = −y+r ; y+r = 1 +  y

1+

X

2l b−2 −  y

(1.5)

j=1

6=r m X

aj−2 x2nj ≥ dr ¿0;

(1.6)

j=1

6=r

for a gentle (1) sequence dr is called, respectively, a y+r cap and a y−r cap. The set of points (x; y) on the hyperbola (1.2) such that  1=(2nr ) p X X 2n 2l b−2 − aj−2 xj j  ; x−r = −x+r ; (1.7) x+r = 1 +  y =1

subject to  1 +

p X =1

2l b−2 −  y

j6=r

X

 2n aj−2 xj j  ≥ dr ¿0

(1.8)

j6=r

and dr a gentle (1) sequence, is called an x+r cap and an x−r cap, respectively. Roughly speaking, the “most part” of these caps, are going to “ atten out.” The motivation for introducing the cap sets will become apparent later on. In the sequel an intriguing phenomenon of nonuniform convergence will come up. Although “major parts” of ellipses and hyperbolas are going to be attened out by increasing parameters of nonlinearities, certain thin sets on the nonlinear ellipses or hyperbolas cannot tend to a limit uniformly. The analysis that follows will identify these sets of nonuniform convergence. Rather than rst stating formal theorems, proving them and then showing examples, I will prefer on several occasions to follow an opposite order of exposition. By doing so I hope that the exposition motivation and purpose in this study will be more transparent. The main objective is to study the limiting geometrical con gurations which will result when one or several nonlinear parameters like nj ; j = 1; : : : ; m, in the ellipses (1.1) tend to in nity. The order of events in this paper is as follows. Section 2 is devoted to a detailed study of planar ellipses. Two types of results are presented. The one type lets just

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one parameter of nonlinearities grow to in nity and then the plus and minus caps of a planar ellipse are “ attened out.” The sets of nonuniform convergence are identi ed. The second type of result requires both parameters of nonlinearity to grow to in nity simultaneously. Then, the ellipse converges to a square. Moreover, subsets of points which converge to the vertices of this square, are identi ed. This is formalized in Theorem 2.1. In Section 3 we discuss m-dimensional families of ellipses and the k; k¡m, dimensional faces which could be obtained. Section 4 is devoted to a brief study of nonlinear hyperbolas. 2. Nonlinear families of planar ellipses We proceed now to analyze the limiting con guration of planar nonlinear ellipses a−2 x2m + b−2 y2n = 1 as n → ∞ or m → ∞ or both. Let (n) be a gentle (1) sequence. The inequality 1 ≥ b−2 y2n = 1 − a−2 x2m ≥ (n)¿0;

(2.1)

implies b2 (n) ≤ y2n ≤ b2 ;

b2=n 1=n (n) ≤ y2 ≤ b2=n :

(2.2)

Then uniformly for −xmn ≤ x ≤ xmn

(2.3)

xmn := a1=m [1 − (n)]1=2m

(2.4)

with we have, respectively, as n → ∞ (the caps satisfying) y+2 → 1; y−2 → −1. We remark that if supn (n) = ¡1, then by virtue of the monotonicity of the function w = 1=m ; (¿0), as a function of m, there exist arbitrarily small positive numbers  such that on the y±2 caps de ned in Eq. (2.1) with xmn = 1 −  we have y±2 tend to ±1, respectively, as n → ∞. However, the convergence cannot be uniform for −a1=m ≤ x ≤ a1=m . The reason being that at x = ±a1=m we have y±2 = 0, and conseˆ quently, a jump in the limits must occur. Moreover, for any gentle (0) sequence (n) we have for 1=2m ˆ −a1=m ≤ x ≤ −a1=m [1 − (n)]

(2.5)

and for 1=2m ˆ a1=m [1 − (n)] ≤ x ≤ a1=m ;

(2.6)

ˆ [1 − (n)] ≤ a−2 x2m ≤ 1;

(2.7)

that ˆ 0 ≤ 1 − a−2 x2m ≤ (n)

and ˆ y2n = b2 [1 − a−2 x2m ] ≤ b2 (n):

(2.8)

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Hence 1=n ˆ →0 0 ≤ y2 ≤ b2=n [(n)]

as n → ∞:

(2.9)

In the “boundary layers” −a1=m ≤ x ≤ − xˆ mn ;

xˆ mn ≤ x ≤ a1=m

(2.10)

with 1=2m ˆ xˆ mn = a1=m [1 − (n)]

(2.11)

we have y+2 ; y−2 tend to zero as n → ∞ for xed m. See Fig. 1a and b. When only one nonlinear parameter n or m tends to in nity, e.g., n, then a very simple argument utilizing the explicit representation of y as a function of x yields the desired conclusion. However, when n and m tend simultaneously to in nity, then a dierent argument is needed which will be given below. The basic idea is to enclose subsets of x and y caps, of a nonlinear ellipse, between certain parallel edges of certain rectangles. Then, it is shown that these certain parallel edges tend to the edges of a square with vertices at (±1; ±1). See Fig. 2a and b. Denote EL := {(x; y) | a−2 x2m + b−2 y2n ≤ 1}:

(2.12)

It can easily be veri ed that this ellipsoid is contained in the rectangle Ru := {(x; y) | 0 ≤ x2 ≤ a2=m ; 0 ≤ y2 ≤ b2=n };

(u for upper in Ru )

(2.13)

and actually contains every rectangle, Rl = {(x; y) | x2 ≤ (a2 2 )1=m ; y2 ≤ (b2 2 )1=n with 2 + 2 ≤ 1}; (l for lower in Rl ):

(2.14)

Evidently, Rl ⊆ EL ⊆ R n . We note that if

1 2

≤ ¡1 and 2 = 2 = (1 − ) then indeed

2 + 2 = 2(1 − ) ≤ 1:

(2.15)

Denote ynm = b1=n [1 − (m)]1=2n :

(2.16)

We will summarize various limiting behaviours of various sets of points given in rectangle Ru , as shown in Fig. 2, via the next theorem. Theorem 2.1. Let (k) be a gentle (1) sequence satisfying 1 2

≤ (k) ≤ sup (k) ≤ ¡1:

(2.17)

Then as n → ∞ and m → ∞ the caps y±2 = ±b1=n (1 − a−2 x2m )1=n de ned on 2 ; 0 ≤ x2 ≤ a2=m [1 − (n)]1=m = xmn

xmn ¿0;

(2.18)

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H. Gingold / Nonlinear Analysis 35 (1999) 781 – 795

Fig. 1. (a) The graph above pertains to a genuine ellipse, (1/68)x2 + (1/5)y2 = 1. (b) The second graph above shows the “ attened out y caps” on the ellipse (1/68)x2 +(1/5)y10 = 1. It turns out that even for small to moderate values of n, the y caps seem to “ atten out.” The set of points of nonuniform convergence can be detected on the graph.

and the caps x±1 = ±a1=m (1 − b−2 y2n )1=m de ned on 2 0 ≤ y2 ≤ b2=n [1 − (m)]1=n = ynm ;

ynm ¿0;

(2.19)

tend uniformly, respectively, to the “edges” of the square with vertices at (±1; ±1). Moreover, for each xed x; −xmn ≤ x ≤ xmn the set of points of the ellipse on the segment ynm ≤ y ≤ b1=n tends to +1 as n → ∞ uniformly in m and the set of points y; −b1=n ≤ y ≤ −ynm tends to −1 uniformly in m as n → ∞. In an analogous manner we have the following: For each xed y; −ynm ≤ y ≤ ynm ; the set of points of the ellipse on the segment xmn ≤ x ≤ a1=m tends to +1 as m → ∞; uniformly in n and the set of points x; −a1=m ≤ x ≤ −xmn tends to −1 as m → ∞; uniformly in n.

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Fig. 2. (a) Flattening of the nonlinear ellipses (1/68)x2m +(1/5)y2n = 1, when n and m tend simultaneously to in nity. The rst graph above is the case m = 5 and n = 5. (b) The second graph above is the case m = 12 and n = 10.

Proof. The theorem follows from previous observations and the fact that as m → ∞ and n → ∞; a1=m ; b1=n ; xmn , and ynm all tend to +1. We turn to the study of m-dimensional ellipses in the next section. 3. Nonlinear families of m-dimensional ellipses There are three types of results in this section. One is a simple extension of the rst part of Section 2.1 in which just one of the nonlinear parameters nj tends to in nity. The second requires all nonlinear parameters to tend simultaneously to in nity. This requires a construction analogous to the construction of Rl ; Ru in the previous section. Both results are given in Theorem 3.1. The third result, given in Theorem 3.2 relates to a k¡m dimensional face of an ellipse.

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We proceed to determine rectangular boxes Ru and Rl such that the ellipsoid EL de ned by   m  X  2n EL := x aj−2 xj j ≤ 1 (3.1)   j=1

will satisfy the relation Rl ⊆ EL ⊆ Ru :

(3.2)

Consider the rectangular box 1=(nj )

Ru = {x | −aj

1=(nj )

≤ xj ≤ aj

; j = 1; : : : ; m}:

(3.3)

2nk A point x outside Ru must have at least one coordinate xk such that a−2 k xk ¿1 and then m X

2nj

aj−2 xj

j=1

2nk ≥ a−2 k xk ¿1

(3.4)

implies that indeed x will be outside the ellipsoid EL. Consequently, EL ⊆ Ru . In order to construct a rectangular box Rl , choose any set of positive numbers j2 ; j = 1; : : : ; m Pm such that j=1 j2 ≤ 1. It can easily be veri ed that the set Rl de ned by Rl := {x | −(aj j )1=nj ≤ xj ≤ (aj j )1=(nj ) ; j = 1; : : : ; m}

(3.5)

is such that m X

2nj

aj−2 xj

≤

j=1

m X

j2 ≤ 1

(3.6)

j=1

and hence Rl ⊆ EL. Moreover, if j2 = (1 − ); j = 1; : : : ; m with 1 − m−1 ≤  ≤ 1; we still have Rl ⊆ EL. We are ready to formulate the next theorem. ˆ Theorem 3.1. Let (k) be a gentle (1) sequence and (k) be a gentle (0) sequence. Let r be xed. (a) Then; on the caps x±r de ned on X 2n aj−2 xj j ≥ (nr ); (3.7) 1− j6=r

we have limnr →∞ x±r = ± 1; respectively. Let x be such that X 2n ˆ r) ≥ 1 − (n aj−2 xj j ≥ 0: j6=r

Then x±r = ±(1 −

P

−2 2nj 1=(2nr ) j6=r aj xj )

converge uniformly in Eq. (3:8) to zero.

(3.8)

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(b) Let x ∈ Ru \Rl ( means the closure) be such that − [aj (1 − )]1=nj ≤ xj ≤ [aj (1 − )]1=nj ;

j 6= r; 1 − m−1 ≤ ¡1

(3.9)

either r) a1=(n ≥ xr ≥ [ar (1 − )]1=nr r

(3.10)

or − a1=(nr ) ≤ xr ≤ −[ar (1 − )]1=nr :

(3.11)

Then; uniformly in set (3:9) we have xr → 1 as nr → ∞ for Eq. (3:10) and xr → −1 as nr → ∞ for Eq. (3:11). (c) Let x ∈ Ru \Rl be such that 1=(n ) 1=(n ) [aj (1 − )]1=(nj ) ≤ xj ≤ aj j or −aj j ≤ xj ≤ [aj (1 − )]1=(nj ) ; j = 1; : : : ; m. Then; as nj → ∞; j = 1; : : : ; m; xj → +1 or xj → −1; respectively. Proof. Most details are omitted because they are simple. Statement (c) of course reminds us that there are 2m subsets of Ru \Rl which converge, respectively, to the 2m vertices (±1; : : : ; ±1) of the perfect rectangular box all of whose edges have length 2 and whose center is at the origin. It is interesting to note that this limiting con guration, obtained by a “varying nonlinearities process”, is a subset of the Hilbert cube. In the case that Eq. (1.1) is an ellipse with m¿2, the variety of limiting con gurations which could occur goes beyond the statements in Theorem 3.1. Therefore, we will conclude this section by studying the various ways that a “k-dimensional face” 1 ≤ k¡m, could be generated in the limiting con guration, which is a perfect cube in m-dimensional space. Let S ={ j1 ; j2 ; : : : ; jl } be a subset of the set of indices M = {1; 2; : : : ; m}. Let {xjl+1 ; : : : ; xjm } be the complementary set M \S. We can relabel the variable xj by denoting x˜  = xj ;  = 1; : : : ; m. Dropping the labels ∼ from the new variables x˜  we may assume then that without loss of generality we are given the subset S = {1; : : : ; l}. Put xˆ = (x1 ; : : : ; xl ); x∗ = (xl+1 ; : : : ; xm ). We will also use the convention x = (x1 ; : : : ; xl ; xl+1 ; : : : ; xm ) = (ˆx; x∗ ). Let nj ; j = 1; : : : ; l be xed and let nj → ∞ simultaneously for j = l + 1; : : : ; m with 1 ≤ l¡m. Let  be xed and 0¡¡1. Consider the set of points x on ellipse (1.1) such that 0¡2 ≤ w2 := 1 −

l X j=1

2nj

aj−2 xj

=

m X

2n

aj−2 xj j ;

1 ≤ l¡m:

(3.12)

j=l+1

Notice that w2 (ˆx) = w2 is bounded above and below, namely, 0¡ ≤ w2 ≤ 1. The set of points x satisfying Eqs. (1.1) and (3.12) lie on an “ellipse with variable coecients”. m X

2n

(aj w)−2 xj j = 1:

j=l+1

(3.13)

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We remark that each x∗ determined by Eq. (3.12) lies inside the l-dimensional ellipsoid   l   X 2n aj−2 xj j ≤ 1 − 2 ; 0¡2 ¡1 : (3.14) S = xˆ   j=1 De ne the function f(ˆx; x∗ ) by f(ˆx; x∗ ) :=

m X

2n

aj−2 w−2 (ˆx)xj j :

(3.15)

j=l+1

For each j; j = l + 1; : : : ; m de ne the sets 1=nj

B−j = {xj | −aj

≤ xj ≤ −[aj (1 − )1=2 ]1=nj }; 1=nj

B+j = {xj | [aj (1 − )1=2 ]1=nj ≤ xj ≤ aj

};

(3.16) (3.17)

with m−l−1 ≤ ¡1: m−l

(3.18)

Put j = +1 or −1 and consider a set PS which is the direct product PS = B(l+1) (l+1) × B(l+2) (l+2) × · · · × B(m) (m) :

(3.19)

There are 2m−l dierent sets of the form (3.19). The following proposition describes the limiting “k = m − l-dimensional face” which is obtained as nonlinearities grow without bound. Theorem 3.2. The set of points x = (ˆx; x∗ ); xˆ ∈ S and x∗ ∈ PS; which lie on the ellipse f(ˆx; x∗ ) = 1 is not empty. Uniformly for xˆ ∈ S where  and  are xed;  and  subject to Eqs. (3:14) and (3:18); we have for x∗ ∈ PS lim xj = j

as nj → ∞;

j = l + 1; : : : ; m:

(3.20)

Proof. Consider the rectangular box 1=nj

Ru = {x∗ = (xl+1 ; : : : ; xm ) | −aj

1=nj

≤ xj ≤ aj

; j = l + 1; : : : ; m}:

(3.21)

For each xed xˆ ∈ S, the set of points x∗ in the ellipsoid f(ˆx; x∗ ) ≤ 1;

(3.22)

must belong to Ru . This is so because if there exists an x∗ outside Ru which satis es 2n Eq. (3.22) then there exists an xj with j = l + 1; : : : ; m such that aj−2 xj j ¿1. However, 2nj

aj−2 w−2 (ˆx)xj

≥ aj−2 xj 2nj ¿1;

(3.23)

which is a contradiction. (Recall that by the de nition of w2 (ˆx) we have w−2 (ˆx) ≥ 1.)

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Consider the rectangular box Rl de ned by Rl = {x∗ = (xl+1 ; : : : ; xm ) | −[aj (1 − )1=2 ]1=nj ≤ xj ≤ [aj (1 − )1=2 ]1=nj ; j = l + 1; : : : ; m}:

(3.24)

Let us show that Rl is contained in the ellipsoid EL = {x∗ = (xl+1 ; : : : ; xm ) | f(ˆx; x∗ ) ≤ 1}:

(3.25)

We have f(ˆx; x∗ ) =

m X

2nj

aj−2 w−2 (ˆx)xj

j=l+1

≤

m X

2nj

aj−2 −2 xj

≤ (m − l)(1 − ) ≤ 1

(3.26)

j=l+1

by Eqs. (3.12), (3.24) and (3.18). Consider the continuous function f(ˆx; x∗ ). For each xed xˆ ; choose xa∗ = (xl+1 ; : : : ; xm ) ∈ PS such that xj = j [aj (1 − )1=2 ]1=nj . Evidently, f(ˆx; xa∗ ) =

m X

aj−2 w−2 (ˆx)2 aj2 j2 (1 − ) ≤ (m − l)(1 − ) ≤ 1:

(3.27)

j=l+1 1=nj

On the other hand, if we choose xb∗ with xj = j aj f(ˆx; xb∗ ) =

m X

aj−2 w−2 (ˆx)aj2 ≥ (m − l) ≥ 1:

we end up with (3.28)

j=l+1

By continuity there exist x∗ ∈ PS such that f(ˆx; x∗ ) = 1. The second part of our theorem follows from the inequalities (3.24). We turn to the next section. 4. Nonlinear hyperbolas The fact that hyperbolas are unbounded graphs, requires some modi cations in the previous arguments. Given the planar hyperbolas a−2 x2m − b−2 y2n = 1 +

(4.1)

x; y; ∈ R; a; b ∈ R and m; n ∈ N; we expect, for xed m; the positives parts of the branches of the hyperbolas to “ atten out” into portions of the straight line y = 1 as n → ∞ and the negative portions of the branches to “ atten out” into portions of y = −1. This is given in Figs. 3a and b. When n is xed and m → ∞ we expect each branch to “ atten out” into the straight lines x = ±1; respectively, as given in Fig. 4a and b. The observations above are roughly correct except that convergence of points on the hyperbolas is not necessarily uniform. The more accurate descriptions are given below.

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Fig. 3. (a) Flattening of the y caps on the nonlinear hyperbolas (0.8)x2m − (1=10)y2n = 1. The graph above pertains to the genuine hyperbola with n = 1 and m = 1. (b) The second graph above shows the “ attened out y caps” with m = 1 and n = 10. We expect two sets of nonuniform convergence on the graphs of the hyperbolas. The one for values of x near the vertex of the hyperbolas, the other for large x.

Theorem 4.1. Let (n) be a gentle (1) sequence. Denote by xnm = [a(1 + (n))1=2 ]1=m . Then; for m xed and L¿a; L xed and − L ≤ x ≤ −xnm

or

xnm ≤ x ≤ L;

(4.2)

we have; respectively; lim y− = −1

n→∞

and

lim y+ = +1;

n→∞

(4.3)

where y+ = b1=n [a−2 x2m − 1]1=2n

and

y− = −b1=n [a−2 x2m − 1]1=2n :

(4.4)

Let (n) be a gentle (0) sequence. Then for m xed and n → ∞ we have for a1=m ≤ x ≤ a1=m [1 + (n)]1=2m ;

y± := ± [b2 (a−2 x2m − 1)]1=2n ;

(4.5)

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Fig. 4. (a) Flattening of the x caps of the hyperbolas (1=1:44)x2m − (4)y2n = 1. Above is the graph of a genuine hyperbola with m = n = 1. This is in contrast with (b) where m is varied. (b) The nonlinear hyperbolas with m = 15 and n = 1.

lim x = a1=m

n→∞

and

lim y± = 0:

(4.6)

n→∞

Also; for m xed and n → ∞ we have for − a1=m [1 + (n)]1=2m ≤ x ≤ −a1=m ;

y± := ± [b2 (a−2 x2m − 1)1=2n ;

(4.7)

that lim x = −a1=m

n→∞

and

lim y± = 0:

n→∞

(4.8)

Let n be xed and denote x+ = [a2 (1 + b−2 y2n )]1=2m

and

x− = −[a2 (1 + b−2 y2n )]1=2m :

(4.9)

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H. Gingold / Nonlinear Analysis 35 (1999) 781 – 795

Then; for every compact portion of the hyperbola (4:1) where −K ≤ y ≤ K; K¿0 and K is xed we have uniformly for −K ≤ y ≤ K lim x+ = 1

and

m→∞

lim x− = −1:

m→∞

(4.10)

Proof. Relation (4.3) follows from the observation that 2 [b2 (a−2 L2m − 1)]1=n ≥ y± = [b2 (a−2 x2m − 1)]1=n ≥ b2 (n):

(4.11)

The statements (4.8) and (4.10) follow by noticing that Eq. (4.7) leads to 2n ≤ (n) 0 ≤ b−2 y±

(4.12)

from which the result follows. We turn to a study of multi-dimensional hyperbolas. Theorem 4.2. Let r be xed. Let (lr ) = dr be a gentle (1) sequence. Consider the points on the hyperbolas caps for which 0 ≤ y2 ≤ K2 ;  = 1; : : : ; p and  6= r; where K are xed positive numbers. Then; respectively, limnr →∞ y±r = ±1. Consider the caps (1:7) of the hyperbolas (1.2) subject to 0 ≤ y2 ≤ K2 ; 0 ≤ xj2 ≤ L2j ; j = 1; : : : ; m and j 6= r; where K ; Lj are xed positive numbers. Then; respectively; limnr →∞ x±r = ±1. Proof. The simple proofs are omitted. We notice that if all y are kept bounded in a rectangular box 0 ≤ y2 ≤ K2 ;  = 1; : : : ; p where K are positive numbers and l1 ; : : : ; lp are kept xed, then the points on the hyperbolas (1.2), can also be viewed as points on ellipses m X

2nj

aj−2 w−2 (y)xj

=1

(4.13)

j=1

with variable coecients where w2 (y) := 1 +

p X

2l b−2  y :

(4.14)

=1

Hence all conclusions of the previous section apply. It is then possible to characterize “k-dimensional faces” in the limiting con guration. Remark. An example in Ref. [1] p. 103 is now a special case of this analysis. It is demonstrated in here that “ attening” could occur, not only on bounded curves, but also on portions of unbounded curves like “nonlinear hyperbola”. Acknowledgements The graphs were derived by a graphing calculator (of an Apple compatible computer) authored by R. Avitzur, G. Robbins and S. Newman.

H. Gingold / Nonlinear Analysis 35 (1999) 781 – 795

795

References [1] E. Beckenbach, R. Bellman, An Introduction to Inequalities Random House, 1961. [2] C.M. Bender, K.A. Milton, S.S. Pinsky, A new perturbative approach to nonlinear problems, J. Math Phys. 30 (1989) 1447–1455. [3] A.V. Pogorelov, Topics in the Theory of Surfaces in Elliptic Space, Translated by Royer Roger, Inc. Gordon & Breach, New York, 1961. [4] F.G. Teixeira, Traite de Courbes Speciales Remarquables, Tome I, Chelsea Publishing Co. Bronx, NY, 1971.