2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 173–180. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

A GENERALIZATION OF EKELAND’S VARIATIONAL PRINCIPLE WITH APPLICATIONS

ABDEL R. EL AMROUSS, NAJIB TSOULI

Abstract. In this paper, we establish a variant of Ekeland’s variational principle. This result suggest to introduce a generalization of the famous Palais-Smale condition. An example is provided showing how it is used to give the existence of minimizer for functions for which the Palais-Smale condition and the one introduced by Cerami are not satisfied.

1. Introduction

Let E be a complete metric space with metric d and Φ : E → ℝ ∪ {∞} a lower semicontinuous function which is bounded from below and not identically to +∞. The Ekeland’s variational principle, see [1], allows for each ε > 0, each δ > 0 and each x ∈ E such as

Φ(x) ≤ inf_{E} Φ + ε,

to build an element v ∈ E minimizing the functional Φ_{v} given by

Φ_{v}(x) = Φ(x) + \frac{ε}{δ}d(x, v).

This principle has wide applications in optimization and nonlinear analysis [1, 2, 4].

If E is a Banach space and Φ : E → ℝ is Gâteaux differentiable, lower semicontinuous and bounded from below, then the Ekeland’s variational principle provides the existence of a minimizing sequence (u_{n}) such as Φ′(u_{n}) → 0, when n → ∞. It is well known that if Φ satisfies the Palais-Smale condition then Φ reaches its minimum. But, it is possible to find a minimizing sequence (u_{n}) such as Φ′(u_{n}) → 0, when n → ∞, not having any convergent subsequence. Let us take the example of the function Φ(s) = arctan(s).

Ekeland [2] prove that if Φ is bounded below and satisfies the Cerami condition for every c ∈ ℝ, introduced by [3], then Φ has a minimal point.

In this note, we prove a variant of Ekeland’s variational principle. This result suggest to introduce a generalization of the classical Palais-Smale condition. An example is provided showing how it is used to give the existence of minimizer for functions for which the Palais-Smale condition or the Cerami condition are not satisfied. We also generalize some results cited in [1], [5], which the Palais-Smale condition or Cerami condition has failed.

2000 Mathematics Subject Classification. 58E05, 35J65, 49B27.

Key words and phrases. Ekeland’s principle variational; Palais-Smale condition; optimization.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

2. Variants of Ekeland’s variational principle

In this section we will prove the following variant of Ekeland’s variational principle. We start with a definition.

Definition 2.1. We say that α : [0, ∞[→]0, ∞[ is a comparison function of order k if for every q ≥ k there exist c, d ≥ 0 such that

\frac{α((t + 1)s)}{α(t)} ≤ cs^{q} + d, ∀t, s ∈ ℝ^{+}.

Examples:

(1) α(s) = (1 + s)^{k}

(2) α(s) = (1 + s)^{k}Log(2 + s)

Let (E, d) be a complet space metric and u ∈ E. Denote by B̄(u, r) = {x ∈ E | d(u, x) ≤ r} the closed boule and B(u, r) = {x ∈ E | d(u, x) < r} the open boule.

Theorem 2.2. Let (E, d) be a complete space metric, x_{0} ∈ E fixed, Φ : E → ℝ a lower semi-continuous and bounded below. Let α : [0, ∞[→]0, ∞[ be a comparison function of order k continuous nondecreasing. Thus for each ε > 0, each δ > 0 and each u ∈ E such that

Φ(u) ≤ inf_{E} Φ + ε

there exists a convergent sequence (z_{n})_{n≥1} of E satisfies:

(i) z_{1} = u, z_{n} ∈ B̄(u, γ(u)) with γ(u) be a positive constant such that u ↦ \frac{γ(u)}{1+d(x_{0},u)} is bounded in E

(ii) The sequence (d(x_{0}, z_{n}))_{n≥1} is nondecreasing (iii) ∑_{n=1}^{j} \frac{d(z_{n},z_{n+1})}{α(d(x_{0},z_{n+1}))} < 2δ, for all j ≥ 1 (iv) for v = lim_{n→∞} z_{n}, Φ(v) ≤ Φ(u)

(v) d(u, v) ≤ min{δα(d(x_{0}, v)), γ(u)} (vi) for every w ∈ B̄(u, γ(u)) \ B(u, d(x_{0}, u)),

Φ(w) ≥ Φ(v) − \frac{ε}{δα(d(x_{0}, w))}d(v, w).

Proof. Let us define a partial order in E by letting

Φ(r) ≤ Φ(s) − \frac{ε}{δα(d(x_{0}, r))}d(r, s)  (2.1)

and

d(x_{0}, r) ≥ d(x_{0}, s).  (2.2)

This relation is easily seen to be reflexive, antisymmetry and transitive. Indeed, it is clear that r ≺ r, for every r ∈ E. The partial order ≺ is antisymmetry. Indeed, if r ≺ s and s ≺ r then d(x_{0}, r) = d(x_{0}, s),

Φ(r) ≤ Φ(s) − \frac{ε}{δα(d(x_{0}, r))}d(r, s) ≤ Φ(r) − \frac{2ε}{δα(d(x_{0}, r))}d(r, s).

However d(r, s) = 0 and so r = s. ≺ is transitive, because if r ≺ s and s ≺ t then d(x_{0}, r) ≥ d(x_{0}, s) ≥ d(x_{0}, t) and

Φ(r) ≤ Φ(s) − \frac{ε}{δα(d(x_{0}, r))}d(r, s), Φ(s) ≤ Φ(t) − \frac{ε}{δα(d(x_{0}, s))}d(t, s). (2.3)

EJDE/CONF/14 A GENERALIZATION OF EKELAND’S VARIATIONAL PRINCIPLE 175

From d(t, s) ≤ d(t, r) + d(r, s), (2.3) becomes

Φ(r) ≤ Φ(s) − \frac{ε}{δα(d(x_{0}, r))}[d(t, r) − d(t, s)], Φ(s) ≤ Φ(t) − \frac{ε}{δα(d(x_{0}, s))}d(t, s).

This implies

Φ(r) ≤ Φ(t) + [\frac{ε}{δα(d(x_{0}, r))} − \frac{ε}{δα(d(x_{0}, s))}]d(t, s) − \frac{ε}{δα(d(x_{0}, r))}d(r, t).

Since α(.) is nondecreasing and d(x_{0}, r) ≥ d(x_{0}, s), we obtain

r ≺ s and s ≺ t ⇒ \{ \begin{matrix}Φ(r) ≤ Φ(t) − \frac{ε}{δα(d(x_{0},r))}d(r, t) \\ d(x_{0}, r) ≥ d(x_{0}, t) \end{matrix}
⇒ r ≺ t.

Moreover, if we denote S = {r ∈ E | r ≺ s}, by lower semi-continuity of E, S is closed.

Let ε, δ, u and γ(u) given by theorem. Now we define a sequence S_{n} of subsets as follows. Start with z_{1} = u and define

S_{1} = {w ∈ E | w ≺ z_{1}} ∩ B̄(u, γ(u)),

and inductively

S_{n} = {w ∈ E | w ≺ z_{n}} ∩ B̄(u, γ(u)), z_{n+1} ∈ S_{n}

such that

Φ(z_{n+1}) ≤ inf_{S_{n}} Φ + \frac{1}{(n + 1)α(d(x_{0}, z_{n}))}.  (2.4)

Clearly by transitivity of ≺ the sequence (S_{n})_{n} is a decreasing sequence of non empty closed sets. Hence also (d(x_{0}, z_{n}))_{n} is a bounded nondecreasing sequence and converges in [d(x_{0}, u), d(x_{0}, u) + γ(u)].

Now we prove that the diameters of these sets go to zero: diamS_{n} → 0. Indeed, on one hand w ∈ S_{n+1} implies

Φ(w) ≤ Φ(z_{n+1}) − \frac{ε}{δα(d(x_{0}, w))}d(w, z_{n+1}) and d(x_{0}, w) ≥ d(x_{0}, z_{n+1}).

From (2.4), it results

Φ(w) ≤ inf_{S_{n}} Φ + \frac{1}{(n + 1)α(d(x_{0}, z_{n}))} − \frac{ε}{δα(d(x_{0}, w))}d(w, z_{n+1}).

This implies

d(w, z_{n+1}) ≤ \frac{δ}{ε(n + 1)} \frac{α(d(x_{0}, w))}{α(d(x_{0}, z_{n}))}.

On the other hand, we have that w belongs to B̄(u, γ(u)), we obtain

d(w, z_{n+1}) ≤ \frac{δ}{ε(n + 1)} \frac{α(γ(u) + d(x_{0}, u))}{α(d(x_{0}, z_{n}))}.  (2.5)

Since the function u ↦ \frac{γ(u)}{1+d(x_{0},u)} is bounded, then there exists M > 0 such that

γ(u) ≤ M(1 + d(x_{0}, u)).  (2.6)

From (2.5), (2.6) and α(.) is a nondecreasing function, it results

d(w, z_{n+1}) ≤ \frac{δ}{ε(n + 1)} \frac{α((M + 1)(1 + d(x_{0}, z_{n})))}{α(d(x_{0}, z_{n}))}.  (2.7)

By (2.7) and α(.) is a comparison function of order k, there exist c, d > 0 such that

d(w, z_{n+1}) ≤ \frac{δ}{ε(n + 1)}(c(M + 1)^{k} + d), n ∈ ℕ

which gives diam S_{n+1} go to 0, when n → ∞.

Now we claim that the unique point v ∈ E in the intersection of the S_{n}’s satisfies conditions (iii)–(vi) of Theorem 2.2. Let then ∩_{n}S_{n} = {v} and z_{n} converges to v. Since z_{j} ≺ z_{j−1} ≺ ⋯ ≺ z_{1}; and by (2.1), we have

Φ(z_{j+1}) ≤ Φ(z_{j}) − \frac{ε}{δα(d(x_{0}, z_{j+1}))}d(z_{j}, z_{j+1})

≤ Φ(z_{1}) − ∑_{n=1}^{j} \frac{εd(z_{j}, z_{j+1})}{δα(d(x_{0}, z_{j+1}))}

or

∑_{n=1}^{j} \frac{εd(z_{j}, z_{j+1})}{δα(d(x_{0}, z_{j+1}))} ≤ Φ(u) − Φ(z_{j+1})

≤ inf_{E} Φ + ε − Φ(z_{j+1}) ≤ ε.

Thus assertion (iii) is shown. Since v ∈ S_{1}, (iv) is clear. It also results from it that

\frac{ε}{δα(d(x_{0}, v))}d(v, u) ≤ Φ(u) − Φ(v) ≤ inf_{E} Φ + ε − Φ(v) ≤ ε.

The assertion (v) is shown.

Finally, we prove (vi), let w ∈ E such that w ≺ v and w ∈ B̄(u, γ(u)), then we have w ≺ z_{n} for every n. This gives w ∈ ∩_{n}S_{n} and w = v, which means that v be an minimal element in B̄(u, γ(u)), i.e.

w ∈ B̄(u, γ(u)) and w ≺ v ⇒ w = v.

Consequently,

Φ(w) > Φ(v) − \frac{ε}{δα(d(x_{0}, w))}d(v, w)

for every w ∈ B̄(u, γ(u)) \ B(x_{0}, d(x_{0}, v)). The proof is complete. □

3. Applications

In this section H denotes a Hilbert space, recall that a function Φ : H → ℝ is called Gâteaux differerentiable if at every point x_{0}, there exists a continuous linear functional f′(x_{0}) such that, for every e ∈ X,

lim_{t→0} \frac{f(x_{0} + te) − f(x_{0})}{t} = ⟨f′(x_{0}), e⟩.

We always assume that α : [0, ∞[→]0, ∞[ is a continuous nondecreasing comparison function of order k. For the rest of the text we will write

Φ^{c} = {u ∈ H : Φ(u) ≤ c},

for the sublevel sets as usual.

Definition 3.1. We say that Φ satisfies (C_{c}^{α}) if: Every sequence (u_{n})_{n} ⊂ H such that Φ(u_{n}) → c and Φ′(u_{n})α(∥u_{n}∥) → 0 possesses a convergent subsequence.

Remark 3.2. Note that if α(s) = cte, the (C_{c}^{α}) condition is just the famous Palais-Smale condition and if α(s) = s + 1, (C_{c}^{α}) is (C) condition introduced by Cerami in [3].

We can now state the following result.

Theorem 3.3. Let H be a Hilbert space, Φ : H → ℝ lower semi-continuous, bounded below and Gâteaux differentiable. Let α : [0, ∞[→]0, ∞[ a continuous nondecreasing comparison function of order k. Assume that for every ε > 0,

Φ^{a+ε} ∩ K ≠ ∅,

with K is bounded in H and Φ satisfies (C_{a}^{α}), with a = inf_{H} Φ, then Φ has a minimal point.

For the proof of this theorem we will use the following lemmas.

Lemma 3.4. Under the conditions of Theorem 3.3, for every ε > 0, every u ∈ H such that Φ(u) ≤ inf_{H} Φ + ε and every δ > 0 such that

δ ≤ \frac{∥u∥ + 1}{2α(3(1 + ∥u∥))}

there exists v ∈ H that satisfies (1) Φ(v) ≤ Φ(u)

(2) \frac{∥v−u∥}{α(∥v∥)} ≤ δ

(3) for every h ∈ H, t ∈ ℝ such that∥h∥ = 1, |t| ≤ 1 and t < v, h >≥ 0 we have

Φ(v + th) ≥ Φ(v) − \frac{ε}{δα(∥v + th∥)}|t|.

Proof. Let in Theorem 2.2, x_{0} = 0, γ(u) = 2(∥u∥ + 1) and d(x, y) = ∥x − y∥ for every x, y ∈ H. Then, by iv) and v) of Theorem 2.2, there exists v ∈ H ( v = lim_{n→∞} z_{n}, (z_{n}) the sequence built in theorem 2.2) such that

Φ(v) ≤ Φ(u) and ∥v − u∥ ≤ δα(∥v∥).  (3.1)

Thus assertions 1. and 2. follow.

Now we prove the assertion 3. Let h ∈ H such that ∥h∥ = 1 and |t| ≤ 1 we have v ∈ B̄(u, ∥u∥ + 1). Indeed, if not ∥u∥ + 1 < ∥v − u∥. Since α(.) is nondecreasing, δ ≤ \frac{∥u∥+1}{2α(3(1+∥u∥))} and by (3.1), it results

∥u∥ + 1 < ∥v − u∥ ≤ δα(∥v∥) ≤ δα(3(1 + ∥u∥))) ≤ \frac{∥u∥ + 1}{2}.

This is a contradiction. Furthermore, we have

∥v + th∥ ≤ ∥v∥ + |t|∥h∥ = ∥v∥ + |t|
≤ 2∥u∥ + 1 + 1 = γ(u).

On the other hand, it is clear , since t⟨v, h⟩ ≥ 0, that

∥v + th∥ = [∥v∥^{2} + ∥th∥^{2} + 2t < v, h >]^{1/2} ≥ ∥v∥.

Thus, by (iv), (v), (vi) of Theorem 2.2, assertions 1, 2, 3 of the lemma follow. □

Lemma 3.5. Under the conditions of Theorem 3.3, we have

|⟨Φ′(v), h⟩| ≤ \frac{ε}{δα(∥v∥)}, ∀h ∈ H, ∥h∥ = 1.  (3.2)

Proof. Let h ∈ H such that ∥h∥ = 1 and consider two cases: Case 1. If ⟨v, h⟩ ≥ 0 and t > 0, from 3. of Lemma 3.4 and Φ being Gâteaux differentiable, letting t approach 0, we obtain

⟨Φ′(v), h⟩ ≥ −\frac{ε}{δα(∥v∥)}.

Case 2. In the similar way, if ⟨v, h⟩ ≤ 0 and t < 0 goes to 0, we have

⟨Φ′(v), h⟩ ≤ \frac{ε}{δα(∥v∥)}, ∀h, ∥h∥ = 1.

Thus the Lemma 3.5 follows. □

Proof of Theorem 3.3. For ε = \frac{1}{n}, with n ≥ 1, there exists a sequence (u_{n}) ⊂ K such that

Φ(u_{n}) ≤ a + \frac{1}{n}

and, since (u_{n}) is bounded, there exists δ > 0 such that

δ ≤ \frac{∥u_{n}∥ + 1}{α(3(1 + ∥u_{n}∥))}, ∀n ≥ 1.

Consequently, by Lemma 3.4 and Lemma 3.5, there exists a sequence (v_{n}) satisfying (i) Φ(v_{n}) ≤ Φ(u_{n})

(ii) ∥Φ′(v_{n})∥α(∥v_{n}∥) → 0, as n → ∞.

The (C_{a}^{α}) condition implies that (v_{n}) has a subsequence (v_{n_{k}}) convergent to some point u. Since Φ is lower semi-continuous, we get

inf_{H} Φ ≤ Φ(u) ≤ lim inf_{k→∞} Φ(v_{n_{k}}) ≤ inf_{H} Φ.

Therefore, Φ(u) = inf_{H} Φ. □

Now, we illustrate Theorem 3.3 by an example where the function Φ checks the conditions of Theorem 3.3, but the Palais-Smale condition and Cerami condition do not hold. Example. Consider

f(s) = \begin{cases}arctan(s) & if s ≤ 0 \\ sin(s) & if 0 ≤ s ≤ 2π \\ arctan(s − 2π) & if s ≥ 2π. \end{cases}

and Φ(u) = f(2π + Log(∥u∥^{2} + 1) − (∥u∥^{2} + 1)^{\frac{1}{2}} ) for u ∈ H. It is clear that Φ is C^{1} functional and a = inf_{H} Φ = −1. Take

K = \{u ∈ H : log(∥u∥^{2} + 1) − (∥u∥^{2} + 1)^{\frac{1}{2}} ∈ [−2π, 0]\},

it is easy to see that Φ^{−1+ε} ∩ K ≠ ∅ for every ε > 0. On the other hand, Φ satisfies (C_{c}^{α}), with α(s) = s^{2} + 1, and by Theorem 3.3, Φ has a minimal point u_{0} which Φ′(u_{0}) = 0.

Theorem 3.6. Let Φ : H → ℝ be Gâteaux differentiable and bounded below, says a inf_{H} Φ. Assume that α : [0, ∞[→]0, ∞[ be a continuous nondecreasing function such that ∫_{1}^{∞} \frac{1}{α(s)} ds = +∞. If Φ satisfies (C_{a}^{α}) then the set Φ^{a+β} is bounded, for some β > 0.

The main point to prove Theorem 3.6 is the following.

Lemma 3.7. Under the conditions of Theorem 3.6, for every ε > 0, every u ∈ H such that Φ(u) ≤ inf_{H} Φ + ε and every δ > 0 there exists v ∈ H satisfies

(1) Φ(v) ≤ Φ(u)

(2) \frac{∥v−u∥}{α(∥v∥)} ≤ δ

(3) for every h ∈ H such that ∥h∥ = 1, we have

|⟨Φ′(v), h⟩| ≤ \frac{ε}{δα(∥v∥)}.

Proof. Let in Theorem 2.2, x_{0} = 0 and d(x, y) = ∥x − y∥ for every x, y ∈ H. From theorem 2.2 there exists a sequence (z_{n})_{n≥1} satisfying (∥z_{n}∥) is nondecreasing and

∑_{n=1}^{j} \frac{∥z_{n} − z_{n+1}∥}{α(∥z_{n+1}∥)} < 2δ, ∀j ≥ 1.  (3.3)

However, since ∫_{1}^{∞} \frac{1}{α(s)} ds = +∞ there exists γ > 0 such that

δ ≤ \frac{1}{2} ∫_{∥u∥}^{∥u∥+γ} \frac{1}{α(s)} ds.  (3.4)

Put v = lim_{n→∞} z_{n} and γ(u) = 2∥u∥ + γ + 1 in Theorem 2.2. Thus, by (iv)-(v) of Theorem 2.2, we obtain

Φ(v) ≤ Φ(u) and ∥v − u∥ ≤ δα(∥v∥).

For the proof of assertion 3, it is enough to verify that h ∈ H such that ∥h∥ = 1 we have v + th ∈ B̄(u, γ(u)) for every t sufficiently small. Now we prove that

∥z_{n}∥ ≤ ∥u∥ + γ, ∀n ≥ 1.  (3.5)

If not, there exists j ≥ 1 such that ∥z_{j+1}∥ > ∥u∥ + γ. However, by (3.4) and α is nondecreasing, we obtain

2δ ≤ ∫_{∥z_{1}∥}^{∥z_{j+1}∥} \frac{1}{α(s)} ds

≤ ∑_{n=1}^{j} ∫_{∥z_{n}∥}^{∥z_{n+1}∥} \frac{1}{α(s)} ds

≤ ∑_{n=1}^{j} \frac{∥z_{n+1}∥ − ∥z_{n}∥}{α(∥z_{n+1}∥)}

≤ ∑_{n=1}^{j} \frac{∥z_{n} − z_{n+1}∥}{α(∥z_{n+1}∥)}.

This contradicts (3.3). Using (3.5), we have

∥v − u∥ ≤ 2∥u∥ + γ.  (3.6)

Thus, for |t| ≤ 1 and h ∈ H such that ∥h∥ = 1 and by (3.6), it results

∥v + th − u∥ ≤ 2∥u∥ + γ + 1 = γ(u).

Finally, the Lemma 3.5 allows to conclude. The proof is complete. □

Proof of theorem 3.6. Suppose, by contradiction, that Φ^{a+β} is unbounded for all β > 0. Then, there exists (u_{n}) such that ∥u_{n}∥ ≥ n and

Φ(u_{n}) ≤ a + \frac{1}{n}.

and Lemma 3.7 with ε = (\frac{1}{n})^{2}, δ = \frac{1}{n} implies the existence of (v_{n}) satisfying (i) Φ(v_{n}) ≤ Φ(u_{n}) (ii) ∥v_{n} − u_{n}∥ ≤ \frac{1}{n}α(∥v_{n}∥) (iii) ∥Φ′(v_{n})∥α(∥v_{n}∥) → 0, as n → ∞.

We reach a contradiction with (C_{a}^{α}), since (i)-(iii) give respectively (1) Φ(v_{n}) → a, as n → ∞,

(2) ∥v_{n}∥ → ∞, as n → ∞,

(3) ∥Φ′(v_{n})∥α(∥v_{n}∥) → 0, as n → ∞.

□

As an immediate consequence of the above results we have the following result.

Corollary 3.8. Let H be a Hilbert space, Φ : H → ℝ lower semi-continuous, bounded below and Gâteaux differentiable. Assume that α : [0, ∞[→]0, ∞[ be a continuous nondecreasing function such that ∫_{1}^{∞} \frac{1}{α(s)} ds = +∞. If Φ satisfies (C_{a}^{α}), with a = inf_{H} Φ, then Φ has a minimal point.

References

[1] Ekeland I., On the variational principle, J. Math. Anal. Applic., 47 (1974) 324-357.

[2] Ekeland I., Convexity methods in Hamiltonian mechanics, Springer, Berlin, (1990).

[3] Cerami G., Un criterio de esistenza per i punti critici su varietá ilimitate, Rc. Ist. Lomb. Sci. Lett. 121,(1978) 332-336.

[4] Clark, F. H. Optimisation and Nonsmooth Analysis, Wiley, NewYork1983.

[5] Costa, D. G., Elves De B. eSilva; The Palais-Smale condition versus coercivity, Nonlinear Analysis, T. M. A. 16 (1991) 371-381.

Abdel R. El Amrouss University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

  E-mail address: amrouss@sciences.univ-oujda.ac.ma

Najib Tsouli University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

  E-mail address: tsouli@sciences.univ-oujda.ac.ma