A Nonlinear Eigenvlaue Problem^{∗}^{†}

Yu-Tai Huang Liberal Arts Center, Da-Yeh Univeristy Chung-Hwa, Taiwan, R. O. C.

Abstract The nonlinear eigenvalue problem cos x = λx motivates our study of a more general problem f(x) = λx, where f captures the essential properties of the cosine function. It is shown that each real λ is an eigenvalue and the exact number of corresponding eigensolutions are found.

A familiar linear eigenvalue problem takes the form Ax = λx, where A is a real n by n matrix. Nonlinear eigenvalue problems are also interesting but more difficult (see [1] for examples of nontrivial nonlinear eigenvalue problems). In this note, we consider one relatively simple case of the form

f(x) = λx,

which is motivated by the eigenvalue problem

cos x = λx.

Before investigating such a problem, let us consider the case where A is a real number a. Then the eigenvalue problem Ax = λx has exactly one eigenvalue λ = a and the corresponding eigensolutions are nontrivial real numbers. In contrast, we will show for the eigenvalue problem cos x = λx, each real number is an eigenvalue, and the number of corresponding eigensolutions is finite.

We will assume that f captures the essential features of the cosine function. More specifically, let f : R → R be a real continuous function which satisfies one or more of the following properties:

∗Mathematics Subject Classifications: 65H17

†Key words and phrases: nonlinear eigenvalue problems, convexity, cosine function

(H1) f is symmetric so that f(x) = f(−x) for x ∈ R.

(H2) lim_{x→0^{+}} f(x)/x = +∞, and |f(x)/x| < C for all large x.

(H3) The set of all extremal points of f over (0, ∞) is of the form {m_{1}, M_{1}, m_{2}, M_{2}, ...} such that

0 < m_{1} < M_{1} < m_{2} < M_{2} < ...,
f(m_{i}) < 0andisarelativeminimum,i = 1, 2, ...,

and

f(M_{i}) > 0andisarelativemaximum,i = 1, 2, ... .

Note that in view of (H2) and (H3), there is a sequence {r_{0}^{+}, r_{1}^{−}, r_{1}^{+}, r_{2}^{−}, r_{2}^{+}, ...} of roots of f in (0, ∞) such that

0 < r_{0}^{+} < m_{1} < r_{1}^{−} < M_{1} < r_{1}^{+} < m_{2} < r_{2}^{−} < M_{2} < r_{2}^{+} < ... .

Note that if a function h is continuous on [α, β], h(α) is a relative minimum, h(β) is a relative maximum and h(x) is not a relative extremum for any x ∈ (α, β), then f is strictly increasing on [α, β]. Therefore, f is strictly increasing on [m_{i}, M_{i}] for i = 1, 2, ..., and decreasing on (0, m_{1}) and [M_{i}, m_{i+1}] for each i ≥ 1.

We will make an additional assumption as follows:

(H4) There is a sequence {p_{0}^{+}, p_{1}^{−}, p_{1}^{+}, p_{2}^{−}, p_{2}^{+}, ...} of points in (0, ∞) such that

0 < p_{0}^{+} < m_{1} < p_{1}^{−} < M_{1} < p_{1}^{+} < m_{2} < p_{2}^{−} < M_{2} < p_{2}^{+} < ...,
fisstrictlyconvexover (p_{i}^{+}, p_{i+1}^{−}) , i = 0, 1, 2, ...,

and

fisstrictlyconcaveover (p_{i}^{−}, p_{i}^{+}) , i = 1, 2, ... .

Recall that a real function h is said to be convex over an interval I if P = P (x_{−1}, y_{−1}), Q = Q(x_{0}, y_{0}) and R = R(x_{1}, y_{1}) are any three points on the graph of h over I such that x_{−1} < x_{0} < x_{1}, then Q is below the chord \overline{P R} joining the points P and R. In terms of slopes, this is equivalent to

slope of \overline{P Q} ≤ slope of \overline{P R} ≤ slope of \overline{QR},  (1)

with strict inequalities when h is strictly convex over I.

LEMMA 1. Suppose f : R → R satisfies (H4). Suppose there is an interval I with positive length in (0, ∞) such that f(x) ≤ ax for x ∈ I and f(x_{0}) = ax_{0}, where a > 0, and x_{0} ∈ I. Then f is strictly concave on a neighborhood of x_{0}, or else x_{0} is an inflection point of f.

PROOF. Assume that x_{0} is not an inflection point of f. Then by (H4), either f is strictly convex on a neighborhood of x_{0}, or f is strictly concave on a neighborhood of x_{0}. Suppose to the contrary that f is strictly convex on a neighborhood B(x_{0}) of x_{0}. Let x_{−1}, x_{1} ∈ B(x_{0}) ⋂ I such that x_{−1} < x_{0} < x_{1}, and let P = P (x_{−1}, f(x_{−1})), Q = Q(x_{0}, f(x_{0})) as well as R = R(x_{1}, f(x_{1})). We see from (1) and the condition f(x) ≤ ax for x ∈ I that

a ≤ slope of \overline{P Q} < slope of \overline{P R} < slope of \overline{QR} ≤ a.

This is the desired contradiction.

We remark that a symmetric statement can also be made: If f : R → R satisfies (H4), and if there is an open interval I in (0, ∞) such that f(x) ≥ ax for x ∈ I and f(x_{0}) = ax_{0}, where x_{0} ∈ I, then f is strictly convex on a neighborhood of x_{0}, or else x_{0} is an inflection point of f.

Next, we consider the function g : (0, ∞) → R defined by

g(x) = \frac{f(x)}{x}, x ∈ (0, ∞).  (2)

It is clear that when f satisfies (H3), then the set of roots of g is {r_{0}^{+}, r_{1}^{−}, r_{1}^{+}, r_{2}^{−}, r_{2}^{+}, ...}, which is just the set of roots of f in (0, ∞). We assert that when f satisfies (H2) and (H3), g will have similar properties.

LEMMA 2. Suppose f satisfies (H2) and (H3). Then the function g defined by (2) is increasing on [m_{i}, r_{i}^{−}] for each i = 1, 2, ..., and decreasing on (0, r_{0}^{+}) as well as (M_{i}, r_{i}^{+}] for each i = 1, 2, ... .

PROOF. Let m_{i} ≤ x_{1} < x_{2} ≤ r_{i}^{−}, where i ≥ 1. Since x_{1}, x_{2} > 0 and f(x_{1}) < f(x_{2}) ≤ 0, we see that

\frac{f(x_{1})}{x_{1}} < \frac{f(x_{1})}{x_{2}} < \frac{f(x_{2})}{x_{2}}.

The other two assertions are similarly proved.

LEMMA 3. Suppose f satisfies (H2), (H3) and (H4). Then the function g defined by (2) has only one (positive) local maximum in (r_{i}^{−}, M_{i}] and only one (negative) local minimum in (r_{i−1}^{+}, m_{i}] for each i = 1, 2, ... . Furthermore, it has no local minimum in (r_{i}^{−}, M_{i}), nor local maximum in (r_{i−1}^{+}, m_{i}) for each i = 1, 2, ... .

PROOF. Since g is continuous on [r_{i}^{−}, M_{i}], g(r_{i}^{−}) = 0 and g(x) > 0 for x ∈ (r_{i}^{−}, M_{i}], g has a positive local maximum in (r_{i}^{−}, M_{i}]. Suppose g(M_{i}^{∗}) and g(M_{i}^{∗∗}) are neighboring local maxima in (r_{i}^{−}, M_{i}] where M_{i}^{∗} < M_{i}^{∗∗}, then

\frac{f(x)}{x} ≤ max \{\frac{f(M_{i}^{∗})}{M_{i}^{∗}}, \frac{f(M_{i}^{∗∗})}{M_{i}^{∗∗}}\} , x ∈ (r_{i}^{−}, M_{i}].

Thus

f(x) ≤ x max \{\frac{f(M_{i}^{∗})}{M_{i}^{∗}}, \frac{f(M_{i}^{∗∗})}{M_{i}^{∗∗}}\}

for x ∈ (r_{i}^{−}, M_{i}]. In view of Lemma 1, there are two possible cases: (1) f is strictly concave in a neighborhood of M_{i}^{∗}, and in a neighborhood of M_{i}^{∗∗}, and (2) M_{i}^{∗} is the inflection point p_{i}^{−} of f, and f is strictly concave in a neighborhood of M_{i}^{∗∗}. Both cases imply p_{i}^{−} ≤ M_{i}^{∗} < M_{i}^{∗∗} < p_{i}^{+}. On the other hand, since g is continuous on [M_{i}^{∗}, M_{i}^{∗∗}], it has a local minimum g(β_{i}) for some β_{i} ∈ (M_{i}^{∗}, M_{i}^{∗∗}). Thus

\frac{f(x)}{x} ≥ \frac{f(β_{i})}{β_{i}}, x ∈ (M_{i}^{∗}, M_{i}^{∗∗}).

In view of (the remark following) Lemma 1, either β_{i} = p_{i}^{−} is an inflection point of f, or else f is strictly convex on a neighborhood of β_{i}. The former case cannot hold since p_{i}^{−} ≤ M_{i}^{∗} < β_{i} < M_{i}^{∗∗} < p_{i}^{+}. The latter case cannot happen neither since p_{i}^{−} ≤ M_{i}^{∗} < M_{i}^{∗∗} < p_{i}^{+}, and f is strictly concave in (p_{i}^{−}, p_{i}^{+}). This shows that g can have only one positive maximum in (r_{i}^{−}, M_{i}], and thus no relative minimum in (r_{i}^{−}, M_{i}).

The proof of the fact that g has only one (negative) minimum in (r_{i−1}^{−}, m_{i}] and no local maximum in (r_{i−1}^{+}, m_{i}) is similar to that above and is thus omitted. The proof is complete.

THEOREM 1. Suppose f satisfies (H2), (H3) and (H4). The the function g defined by (2) has the following properties: (1) lim_{x→0^{+}} g(x) = ∞,

(2) |g(x)| < C for all large x, (3) the set of roots of g is {r_{0}^{+}, r_{1}^{−}, r_{1}^{+}, r_{2}^{−}, r_{2}^{+}, ...}, and (4) the set of all extremal points of g in (0, ∞) is of the form {m_{1}^{∗}, M_{1}^{∗}, m_{2}^{∗}, M_{2}^{∗}, ...} such that

g(m_{i}^{∗}) < 0, g(M_{i}^{∗}) > 0, i = 1, 2, ...,

and

0 < r_{0}^{+} < m_{1}^{∗} < r_{1}^{−} < M_{1}^{∗} < r_{1}^{+} < m_{2}^{∗} < r_{2}^{−} < M_{2}^{∗} < r_{2}^{+} < ... .

Indeed, the properties (1), (2) and (3) are easily seen. To see that (4) holds. First note that g is decreasing on (0, r_{0}^{+}) by Lemma 2. By Lemma 3, g has only one local (negative) minimum g(m_{1}^{∗}) in [r_{0}^{+}, m_{1}] and no local maximum in (r_{0}^{+}, m_{1}). By Lemma 2, g is increasing on [m_{1}, r_{1}^{−}], and hence it has no local extremum in (m_{1}, r_{1}^{−}). Similarly, g has only one local (positive) maximum g(M_{1}^{∗}) in [r_{1}^{−}, M_{1}] and no other local extremum in (r_{1}^{−}, r_{1}^{+}). Thus we see that r_{0}^{+} < m_{1}^{∗} < r_{1}^{−} < M_{1}^{∗} < r_{1}^{+} is true. The other cases are similarly proved.

Note that the property (4) of Theorem 1 also implies the strict monotonicity of g on the intervals (0, m_{1}^{∗}], [m_{i}^{∗}, M_{i}^{∗}] and [M_{i}^{∗}, m_{i+1}^{∗}] for i ≥ 1.

Once we have shown that g has the properties described in Theorem 1, let us consider the equation

g(x) = λ, x ∈ (0, ∞),  (3)

where λ ∈ R. Assume that the positive relative extrema of g can be arranged in the form

g(M_{1}^{∗}) > g(M_{2}^{∗}) > g(M_{3}^{∗}) > ... .

Then for each λ ∈ (g(M_{1}^{∗}), ∞), by means of the monotonicity of g, there is a unique positive solution in (0, r_{0}^{+}) of (3). When λ = g(M_{1}^{∗}), there will be exactly two solutions, one in (0, r_{0}^{+}) and the other is equal to M_{1}. For λ ∈ (g(M_{2}^{∗}), g(M_{1}^{∗})), there will be exactly three solutions, one in (0, r_{0}^{+}), and two in (r_{1}^{−}, r_{1}^{+}). When λ = g(M_{2}^{∗}), there will be exactly four solutions, and so on. As an example, let f(x) = cos x for x ∈ (−∞, ∞). Then the function g defined by

g(x) = \frac{cos x}{x}, x ∈ (0, ∞),

has the properties listed in Theorem 1. And since the function | cos x| is periodic,

|\frac{cos(x + nπ)}{x + nπ}| < |\frac{cos(x + (n + 1)π)}{x + (n + 1)π}|

for x ∈ [nπ, (n + 1)π] and n = 0, 1, 2, ... . Accordingly,

−g(m_{1}^{∗}) > g(M_{1}^{∗}) > −g(m_{2}^{∗}) > g(M_{2}^{∗}) > g(m_{3}^{∗}) > g(M_{3}^{∗}) > ... .

Since lim_{x→∞}(cos x)/x = 0, we also see that the sequence of absolute values of the relative extrema tends to zero. We can actually calculate the relative minima and maxima. Indeed,

g(m_{1}^{∗}) ≃ −0.336508, g(M_{1}^{∗}) ≃ 0.161228,
g(m_{2}^{∗}) ≃ −0.106708, g(M_{2}^{∗}) ≃ 0.0798312,
g(m_{3}^{∗}) ≃ −0.0637916, g(M_{3}^{∗}) ≃ 0.0531265,

etc. as can be verified by means of standard commercial software such as the Mathematica. Let λ_{2i} = g(M_{i}^{∗}) and λ_{2i−1} = −g(m_{i}^{∗}) for i = 1, 2, 3, ... . Then in view of our previous discussions, we see that if λ > λ_{1}, then the equation (3) has a unique solution. If λ = λ_{i} where i ≥ 1, then (3) has exactly 2i solutions. If λ ∈ (λ_{i}, λ_{i+1}) where i ≥ 1, then (3) has exactly 2i+1 solutions. Similar conclusions can be drawn for λ = −λ_{i} and λ ∈ (−λ_{i+1}, λ_{i}).

In terms of the symmetric function f, we may now make the following conclusion.

THEOREM 2. There exists a strictly decreasing, positive, and zero convergent sequence {λ_{1}, λ_{2}, ...} , such that (i) each λ ∈ (−∞, ∞) is an eigenvalue of the equation cos x = λx, (ii) if λ > λ_{1} or −λ_{1} > λ, then there is exactly one corresponding eigensolution, (iii) if λ = ±λ_{i}, then the number of corresponding eigensolutions is 2i, and (iv) if λ ∈ (−λ_{i+1}, −λ_{i}) ⋃(λ_{i}, λ_{i+1}) where i ≥ 1, then the number of eigensolutions is 2i + 1.

References

[1] J. B. Keller and S. Antman, Bifurcation Theory and Nonlinear Eigenvalue Problems, W. A. Benjamin, Inc., new ork, 1969.