On the Palais-Smale condition for nonsmooth functions over closed and convex sets of restrictions

by Helmut Dietrich Martin-Luther-University Halle-Wittenberg Institute of Optimization and Stochastics 06099 Halle Germany

Abstract. The aim of the paper is to generalize the Palais-Smale condition for nonsmooth functions over closed and convex sets in terms of the lower Dini directional derivative. For an important family of functions we prove the Palais-Smale condition on every closed and convex set. As an example we consider a boundary value problem with lower and upper solutions. Key Words : Nondifferentiable optimization, Dini subdifferential, Palais-Smale condition, Ekeland’s variational principle. Mathematics Subject Classification 1991: Primary : 49J52, 49K20 Secondary : 90C26, 49K24

1 Introduction

Many of the laws of mathematics and physics can be formulated in terms of extremum principles. In applications we frequently encounter functionals on closed and convex subsets of Banach spaces. We will derive Palais-Smale conditions for such nonsmooth problems in the sense of the constrained optimization and by means of the methods of optimization. In the case of mainly continuously differentiable functionals, Struwe [25, 26] has developed at first systematically variational methods for functionals on closed and convex sets, for example the Ljusternik-Schnirelman theory on convex sets. In section 2 we give an arrangement and a completion of some results about Dini’s lower subdifferentials. For lower semicontinuous functions on a nonempty, closed and convex subset of a Banach space we derive in section 3 optimality conditions by means of Dini’s lower directional derivatives and define from it the Palais-Smale condition (P S)_{M}. In section 4 we prove a corresponding minimization theorem and the condition (P S)_{M} for an important family of constrained optimization problems. As an application of the results we consider a nonlinear boundary value problem with a weakly lower and upper solution and prove the existence of a weak solution under most general assumptions.

The known relationship between the Palais-Smale condition and the coercivity of the function is retained also for the condition (P S)_{M}. Finally we generalize another method of the consideration of a convex and closed set of restrictions for locally Lipschitz functions.

2 About Dini’s subdifferential

Let X be a Banach space and let f : X ⟶ R ∪ {+∞} be a lower semicontinuous (l.s.c.) function. Then for x ∈ domf

\underline{f}′(x; y) = lim inf_{{t→+0 \atop h→y}} \frac{f(x + th) − f(x)}{t}  (1)

is called the lower Dini derivative of f at x ( or upper contingent derivative, cf. Aubin, Cellina [2] 6.1, or semiderivatives, cf. Penot [23] ). The set

∂^{−}f(x) = \{x^{∗} ∈ X^{∗} :< x^{∗}, y >≤ \underline{f}′(x; y) ∀y ∈ X\}  (2)

will be called the Dini subdifferential of f at x. ( Here and below X^{∗} is the dual space of X and < x^{∗}, x > with x ∈ X and x^{∗} ∈ X^{∗} is the canonical pairing between the both spaces. ) By construction \underline{f}′(x; ·) is a l.s.c. positively homogeneous function from X into R̄ := R ∪ {±∞} and the corresponding lower Dini subdifferential ∂^{−}f(x) represents a sharp positively homogeneous approximation of {(x + ty, f(x + ty)|t ≥ 0}. Dini’s subdifferential ∂^{−}f is also called Fréchet subdifferential, compare Mordukhovich [21]. We note that ∂^{−}f has the alternative description

x^{∗} ∈ ∂^{−}f(x) ⟺ lim inf_{y→x} \frac{f(y) − f(x)− < x^{∗}, y − x >}{∥y − x∥} ≥ 0,  (3)

as Burke, Qi [4] inform, at least for locally Lipschitz functions. Indeed, with the substitution y = x + th′ condition (3) implies always

\underline{f}′(x; h) = lim inf_{{t→+0 \atop h′→h}} \frac{f(x + th′) − f(x)}{t} ≥< x^{∗}, h > ∀h ∈ X,

and therefore x^{∗} ∈ ∂^{−}f(x). On the other hand the equality

\underline{f}′(x; h) = lim inf_{t→+0} \frac{f(x + th) − f(x)}{t}

is valid if the function f is locally Lipschitz near x ∈ X, compare Ioffe [18]. Then \underline{f}′(x; h) ≥< x^{∗}, h > for all h ∈ X implies that lim inf_{y→x} \frac{f(y)−f(x)−<x^{∗},y−x>}{∥y−x∥} ≥ 0. For the equivalence (3), compare also Treiman [27]. In this manner Dini’s subdifferential is used by Marino and Tosques [20] and Degiovanni, Marino and Tosques [11], compare also Weckesser [29].

Dini’s subdifferential is of major importance in optimization theory, cf. Correa and Hiriart-Urruty [8] and Penot [22] for example. It is also a fundamental concept in the theory of viscosity solutions for partial differential equations, see among other things Crandall and Lions [10], and in the investigation of nonsmooth and nonconvex variational problems, cf.Marino and Tosques [20]. The existing calculus of Dini subdifferentials is very poor compared with the calculus of convex or Clarke’s subdifferentials, cf. Ioffe [18], Marino and Tosques [20], Degiovanni, Marino and Tosques [11]. Rules for estimating ∂^{−}f(x) are important for analyzing when 0 ∈ ∂^{−}f(x) since 0 ∈ ∂^{−}f(x) is the subgradient optimality condition. We prove a simple chain rule for a linear and continuous operator.

Theorem 1 Let X and Y be Banach spaces, let A : X → Y be a linear and bounded operator on X and let f : Y → R ∪ {+∞} a l.s.c.function. Then

A^{∗}∂^{−}f(Ax) ⊂ ∂^{−}(f ∘ A)(x).  (4)

If additionally the range of A, R(A) = Y, and A is an invertible operator, then

A^{∗}∂^{−}f(Ax) = ∂^{−}(f ∘ A)(x).  (5)

Proof : The inclusion y^{∗} ∈ ∂^{−}f(Ax) implies always < y^{∗}, Aξ >≤ \underline{f}′(Ax; Aξ) ∀ξ ∈ X and therefore A^{∗}∂^{−}f(Ax) ⊂ ∂^{−}(f ∘ A)(x). On the other hand from x^{∗} ∈ ∂^{−}(f ∘ A)(x) we have < x^{∗}, ξ >≤ \underline{f}′(Ax; Aξ) ∀ξ ∈ X. Since A is continuously invertible, R(A^{∗}) = Y^{∗} holds, where A^{∗} : Y^{∗} → X^{∗} is the adjoint operator of A and R(A^{∗}) its range. Therefore there exists an element y^{∗} ∈ Y^{∗} with x^{∗} = A^{∗}y^{∗} and we have < y^{∗}, Aξ >≤ \underline{f}′(Ax; Aξ) ∀ξ ∈ X. Since R(A) = Y, we conclude y^{∗} ∈ ∂^{−}f(Ax) and x^{∗} = A^{∗}y^{∗} ∈ A^{∗}∂^{−}f(Ax), also ∂^{−}(f ∘ A)(x) ⊂ A^{∗}∂^{−}f(Ax). It is useful to see that the subdifferential of the sum of two functions f, g : X ⟶ R ∪ {+∞} is the sum of its subdifferentials. Unfortunately in general the rule of sums is invalid for Dini’s lower subdifferentials, see for example Marino and Tosques [20]. But if x ∈ domf ∩ domg, we have

\underline{f}′(x; h) + \underline{g}′(x; h) ≤ \underline{(f + g)}′(x; h) ∀h ∈ X

and therefore also by definition

∂^{−}f(x) + ∂^{−}g(x) ⊂ ∂^{−}(f + g)(x),  (6)

if both subdifferentials ∂^{−}f and ∂^{−}g are not empty. Furthermore if g is local Lipschitz and Gâteaux-differentiable at x, the equality holds. In fact, if x^{∗} ∈ ∂^{−}(f + g)(x) , i.e.

lim inf_{{t→+0 \atop h′→h}}[\frac{f(x + th′) − f(x)}{t} + \frac{g(x + th′) − g(x)}{t}] ≥< x^{∗}, h > ∀h ∈ X,

we get

\underline{f}′(x; h) ≥< x^{∗} − g′(x), h > ∀h ∈ X,

by this means ∂^{−}(f + g)(x) ⊂ ∂^{−}f(x) + g′(x), and the equality is proved:

∂^{−}f(x) + ∂^{−}g(x) = ∂^{−}(f + g)(x).  (7)

We give an example that a convex and l.s.c.function, which is discontinuous, possesses a nonempty subdifferential in the sense of Dini and of convex analysis.

Lemma 1 Suppose A : D(A) ⊂ X ⟶ X^{∗} is a closed, selfadjoint linear operator whose domain D(A) is dense in the reflexive Banach space X. Assume that A is strictly monotone

< Ax, x > > 0 ∀x ∈ D(A) \ {0}

and coercive in the following sense: lim_{{∥x∥→+∞ \atop x∈D(A)}} < Ax, x >= +∞. Define the function

f : X ⟶ R ∪ {+∞} by

f(x) := \{ \begin{matrix}\frac{1}{2} < Ax, x > : & if x ∈ D(A) \\ +∞ : & if x ∈ X \ D(A) \end{matrix} .

Then ∂^{−}f(x) = ∂f(x) = {Ax} for x ∈ D(A), else the empty set.

Proof : We prove immediately that Ax ∈ ∂f(x) for x ∈ D(A) is valid. From the following inequality for x ∈ D(A)

< Ay − Ax, y − x >≥ 0 ∀y ∈ D(A)

at once we get by consideration of A = A^{∗}

\frac{1}{2} < Ay, y > −\frac{1}{2} < Ax, x >≥< Ax, y − x > ∀y ∈ D(A),

i.e. Ax ∈ ∂f(x). On the other hand, the function ϕ : X ⟶ R ∪ {+∞}

ϕ(y) := \{ \begin{matrix}\frac{1}{2} < Ay, y > − < x^{∗}, y > : & if y ∈ D(A) \\ +∞ : & otherwise \end{matrix}

with x^{∗} ∈ X^{∗} - an arbitrary and fixed element - is a strictly convex and coercive l.s.c. function, which on the reflexive Banach space X always possesses an unique determined minimizer x ∈ D(A). Therefore the inequality

\frac{1}{2} < Ay, y > −\frac{1}{2} < Ax, x >≥< x^{∗}, y − x > ∀y ∈ D(A)

and with it x^{∗} ∈ ∂f(x) hold. Assume that x^{∗} ≠ Ax, then there exists a y ∈ D(A), y ≠ x, so that x^{∗} = Ay because of R(A) = X^{∗} is valid. The least inequality implies with A = A^{∗} the contradiction 0 ≥< Ay − Ax, y − x >> 0 for y ≠ x.

Remark 1 If for the function f by lemma 1 is merely supposed that the operator A is linear, closed, selfadjoint and densely defined in the reflexive Banach space X, then it always possesses the directional derivative f′(x; h) =< Ax, h > for all x, h ∈ D(A).

3 Optimality condition and the Palais-Smale condition

Given a l.s.c. function f : X ⟶ R ∪ {+∞} on the Banach space X . Let M ⊂ X be a nonempty, closed and convex subset of X and we consider the minimization problem

α := inf_{x∈M} f(x) > −∞.  (8)

We prove at first the existence of approximate solutions for problem (8) by means of Ekeland’s variational principle and obtain a necessary optimality condition in terms of the lower Dini ( directional ) derivative, respectively of the Dini subdifferential, cf. Ioffe [18].

Lemma 2 Suppose that the assumptions above are satisfied. For given positive numbers ϵ, λ, there exist elements x_{ϵ}, x̄ ∈ M with the following three properties:

(i) α = inf_{x∈M} f(x) ≤ f(x_{ϵ}) ≤ α + ϵ (ii) ∥x_{ϵ} − x̄∥ ≤ λ (iii) 0 ≤ \underline{f}′(x̄; x − x̄) + \frac{ϵ}{λ}∥x − x̄∥ ∀ x ∈ M.

Proof : Through

f̃(x) := \{ \begin{matrix}f(x) : & x ∈ M \\ +∞ : & else \end{matrix}

we define a lower semicontinuous function f̃: X ⟶ R∪{+∞} on the Banach space X, which is bounded below because of the assumption α > −∞. By Ekeland’s variational principle it follows that there exist elements x_{ϵ}, x̄ ∈ M with the properties (i) and (ii) as well as

f̃(x̄) ≤ f̃(x̃) + \frac{ϵ}{λ} ∥ x̃ − x̄ ∥ ∀x̃ ∈ X.

We obtain the inequality

f(x̄) ≤ f(x̃) + \frac{ϵ}{λ}∥x̃ − x̄∥ ∀x̃ ∈ M,

and due to the convexity of M with x̃ = x̄ + t(h − x̄), t ∈ (0, 1] and h ∈ M,

−\frac{ϵ}{λ}∥h − x̄∥ ≤ \frac{f(x̄ + t(h − x̄)) − f(x̄)}{t}.

Because

\underline{f}′(x̄; x − x̄) = lim inf_{{t→+0 \atop h→x}} \frac{f(x̄ + t(h − x̄)) − f(x̄)}{t},

(iii) is valid. Here, \underline{f}′ : X × X ⟶ R̄ means Dini’s lower derivative of f.

Lemma 2 is a generalization of a result of Aubin [1] for continuously differentiable functions to l.s.c. functions.

Remark 2 One can obtain variants of lemma 2 by using of other directional derivatives and subdifferentials.If the function f : X ⟶ R is locally Lipschitzian for example, then we get instead of lemma 2 (iii)

0 ≤ f^{0}(x̄; x − x̄) + \frac{ϵ}{λ}∥x − x̄∥ ∀ x ∈ M.

Here, f^{0} : X × X ⟶ R means Clarke’s generalized directional derivative of f.

In the same way we prove the following necessary optimality condition for a solution of problem (8).

Lemma 3 Under the assumptions above, the implication

min_{x∈M} f(x) = f(x_{0}) ⟹ \underline{f}′(x_{0}; x − x_{0}) ≥ 0 (f^{0}(x_{0}; x − x_{0}) ≥ 0) ∀x ∈ M

(9)

holds.

Proof : Suppose that f(x̃) ≥ f(x_{0}) ∀x̃ ∈ M is valid and set x̃ = x_{0} + t(h − x_{0}) for t ∈ (0, 1] and h ∈ M. Then the condition

0 ≤ lim inf_{{t→0 \atop h→x}} \frac{f(x_{0} + t(h − x_{0})) − f(x_{0})}{t} = \underline{f}′(x_{0}; x − x_{0}) ∀x ∈ M

immediately holds.

Struwe [26] utilizes diverse Palais-Smale conditions for the investigation of critical points for functionals, which are chiefly continuously differentiable. We now generalize the Palais-Smale condition of Struwe [26] for continuously differentiable functionals over convex and closed sets to arbitrary l.s.c. functionals. This generalized Palais-Smale condition is different from the appertaining conditions of Ghoussoub [13], Costa,Goncalves [9] and Chang [7] and answer the practice of the optimization. Suppose M is a closed and convex subset of a Banach space X, and f : X ⟶ R∪{+∞} is a proper l.s.c. function. For x ∈ M define the function F : M ⊂ X ⟶ R̄

F (x) := inf\{\underline{f}′(x; y − x) : ∥y − x∥ < 1, y ∈ M\}  (10)

as a measure of the slope of f in x ∈ M. This definition of F is unessential different from the attached definition of Struwe for continuously differentiable functionals and of Shi [24]. In the case of a locally Lipschitz function f : X ⟶ R for (10) we can also define

F̃(x) = inf{f^{0}(x; y − x) : ∥y − x∥ < 1, y ∈ M},

where f^{0} denote Clarke’s directional derivative of f at x. Clearly, if the function f is locally Lipschitzian, M = X and

< x^{∗}, h >= max_{ξ^{∗}∈∂f(x)} < ξ^{∗}, h >= f^{0}(x; h),

then F̃(x) ≥ −∥x^{∗}∥_{∗} . In general F ≤ F̃ holds, see Treiman [28],theorem 3. Immediately from the definition it follows that

F (x) ≤ 0 ∀x ∈ M  (11)

is valid, we notice for it that \underline{f}′(x; 0) is either 0 or −∞, cf. Ioffe [18]. If f is a locally Lipschitzian, then

lim sup_{y→x} F̃(y) ≤ F̃(x), x, y ∈ M.  (12)

In the case that f is a continuously differentiable functional on the convex and closed set M, the function F is continuous on M, see Struwe [26]. For locally Lipschitz functions f, the function F̃ is in general merely upper semicontinuous because of the continuity properties of Clarke’s subdifferential as set-valued mapping. A result in this direction is the following observation.

Lemma 4 If f :⟶ R is locally Lipschitz, M ⊂ X is a nonempty, convex and closed subset of a finite dimensional Banach space X and the function f is Gâteauxdifferentiable at x ∈ M,then there exists to any sequence {x_{n}} ⊂ M with x_{n} → x a subsequence {x_{n_{i}}} ⊂ {x_{n}} with x_{n_{i}} → x ( as i → ∞ ) so that

F̃(x) = lim_{i→∞} F̃(x_{n_{i}})

holds.

Proof : By (12) the function F̃ is upper semicontinuous on M. Suppose x_{n} → x (n → +∞) where x_{n}, x ∈ M. Then for any n ∈ N we choose y_{n} ∈ M such that ∥y_{n} −x_{n}∥ < 1 and

F̃(x_{n}) ≤ f^{0}(x_{n}; y_{n} − x_{n}) ≤ F̃(x_{n}) + \frac{1}{n}

hold. As Struwe [26] we set

z_{n} := ∥x_{n} − x∥x_{n} + (1 − ∥x_{n} − x∥)y_{n}

and note that z_{n} ∈ M holds because of the convexity of M. Further

∥z_{n} − x∥ = ∥x − x_{n}∥ + (1 − ∥x_{n} − x∥)∥y_{n} − x_{n}∥ < 1

and ∥z_{n}−y_{n}∥ ≤ ∥x_{n}−x∥ → 0 as n → +∞ are valid. Now F̃(x) ≤ lim inf_{n→+∞} f^{0}(x; z_{n}− x) holds because of ∥z_{n} − x∥ < 1 and we obtain the inequality

F̃(x) ≤ lim inf_{n→+∞}[F̃(x_{n}) + \frac{1}{n} + f^{0}(x; z_{n} − x_{n}) + f^{0}(x; x_{n} − x)+ < f′(x) − x_{n}^{∗}, y_{n} − x_{n} >]

with x_{n}^{∗} ∈ ∂f(x_{n}) and the Gâteaux-derivative f′(x). Because of ∥x_{n}^{∗}∥_{∗} ≤ K for n sufficiently large and with the local Lipschitz constant K > 0 of f at x ∈ M, there exists a subsequence x_{n}^{∗}_{i} → f′(x) as i → +∞ and it follows the assertion.

Struwe [26] uses the function F to characterize the critical points of problem (8) for continuously differentiable functions with convex and closed restrictions. We prove that one can also define critical points of problem (8) for l.s.c. functions through F. For this purpose we use Ekeland’s variational principle for the derivation of a local compactness condition of the Palais-Smale type so similar as Costa,Goncalves [9] do.

Lemma 5 For a minimizing sequence {x_{n}} ⊂ M of problem (8) which is constructed by means of Lemma 1 with ϵ = \frac{1}{n} and λ = 1 we obtain

lim_{n→+∞} F (x_{n}) = 0.  (13)

Proof : For the minimizing sequence {x_{n}} of problem (8) by lemma 2 we get the inequality

0 ≤ \underline{f}′(x_{n}; x − x_{n}) + \frac{1}{n}∥x − x_{n}∥ ∀x ∈ M

and further by (11)

−\frac{1}{n} sup\{∥x − x_{n}∥ : ∥x − x_{n}∥ < 1, x ∈ M\} ≤ F (x_{n}) ≤ 0.

The last inequality implies (13).

Lemma 3 and 5 justify the following definitions.

Definition 1 For a given l.s.c. function f : X ⟶ R ∪ {+∞} and the nonempty, closed and convex subset M ⊂ X of the Banach space X we denote a point x ∈ M with F (x) = 0 a critical point, otherwise a regular point. If f(x) = c ∈ R holds for some critical point x ∈ M, the value c is called critical, otherwise regular. The set

K_{c} := {x ∈ M : f(x) = c and F (x) = 0}  (14)

is called the set of critical points of f on the level c ∈ R with respect to M.

This definition coincides with the definition of regular or critical points, respectively values, of a functional given on M = X and in the case of differentiability of f, cf. Struwe [26].

Definition 2 Let M ⊂ X be a nonempty, closed and convex subset of a Banach space X and let f : X ⟶ R ∪ {+∞} be an arbitrary l.s.c. function. We say that f satisfies the Palais-Smale condition (P S)_{M} on the set M if and only if for arbitrary given sequences {x_{n}} ⊂ M and ϵ_{n} → +0, ϵ_{n} > 0, with

(i) |f(x_{n})| ≤ K and

(ii) 0 ≤ \underline{f}′(x_{n}; y − x_{n}) + ϵ_{n}∥y − x_{n}∥ ∀y ∈ M

there exists a subsequence {x_{n}_{i}}, such that {x_{n}_{i}} is strongly convergent x_{n}_{i} → x as i → +∞ to an element x ∈ M.

The condition item (ii) implies F (x_{n}) → 0, cf.lemma 5. From technical causes we use (ii) and not F (x_{n}) → 0 analog to Struwe [26] for continuous differentiable functions, compare also Costa and Goncalves [9]. The condition (P S)_{M} is therefore a generalization of Struwe [26]. On the other hand the condition (P S)_{M} follows from the condition (P S)_{c}^{∗} of Costa and Goncalves [9] through a modification and the consideration of the closed and convex set M. For a locally Lipschitz function f : X ⟶ R one can modify definition 2(ii)

0 ≤ f^{0}(x_{n}; y − x_{n}) + ϵ_{n}∥y − x_{n}∥ ∀y ∈ M.

4 The minimization theorem and properties of the Palais-Smale condition

Theorem 2 Let X be a Banach space and let M ⊂ X be a nonempty, closed and convex subset. Suppose that the closed function f : X ⟶ R ∪ {+∞} is bounded below on M with α = inf_{x∈M} f(x) > −∞ and satisfies (P S)_{M}. Then the minimum problem (8) has a solution x_{0} ∈ M which also is a critical point of f, i.e. x_{0} ∈ K_{α}.

Proof : According to lemma 1 and 4 for ϵ = 1/n and λ = 1 exists a sequence {x_{n}} with f(x_{n}) → α and F (x_{n}) → 0. Due to the Palais-Smale condition (P S)_{M} there exists a convergent subsequence {x_{n}_{i}} with x_{n}_{i} → x_{0} as i → +∞ and x_{0} ∈ M. Consequently, f(x_{0}) ≤ α is valid because of the lower semicontinuity of f. Thus, x_{0} ∈ M is a solution of (8). On the other hand by lemma 2 the condition \underline{f}′(x_{0}; x − x_{0}) ≥ 0 ∀ x ∈ M and therefore F (x_{0}) ≥ 0 and x_{0} ∈ K_{α} hold.

For a special and important class of functions we prove that the Palais-Smale condition (P S)_{M} is valid. Let f : X ⟶ R be a local Lipschitz and Gâteaux-differentiable function, whose Gâteaux-derivative is uniformly monotone in the sense of Kluge [19], i.e.

< f′(x) − f′(y), x − y > ≥ a(∥x − y∥) ∀x, y ∈ X  (15)

with a continuous and increasing function a : R^{+} → R^{+} and a(0) = 0. Let g : X ⟶ R ∪ {+∞} be a closed function and denote G := f + g. Consider the optimization problem

inf_{x∈M⊂X} G(x) > −∞  (16)

with the closed and convex set M of restrictions in the reflexive Banach space X. Suppose additionally that M is either a bounded set or G is coercive on M, i.e. lim_{{∥x∥→+∞ \atop x∈M}} G(x) = +∞.

Theorem 3 Under the given assumptions the function G fulfils the Palais-Smale condition (P S)_{M} for every closed and convex set M, if merely for Dini’s lower directional derivative of g the condition

for every sequence \{ξ_{n}\} with ξ_{n} ⇀ ξ ⟹ lim inf_{n→+∞} \underline{g}′(ξ_{n}; ξ − ξ_{n}) ≤ 0  (17)

is valid.

Proof : Let {x_{n}} ⊂ M and {ϵ_{n}} ⊂ (0, +∞) be such that ϵ_{n} → +0, |G(x_{n})| ≤ K and

0 ≤ \underline{G}′(x_{n}; y − x_{n}) + ϵ_{n}∥y − x_{n}∥ ∀y ∈ M

hold. By (7) it follows that

0 ≤< f′(x_{n}), y − x_{n} > +\underline{g}′(x_{n}; y − x_{n}) + ϵ_{n}∥y − x_{n}∥ ∀y ∈ M.

Because of the boundedness of M, respectively the coercivity of G, there exists a subsequence {x_{n}_{i}} which is weakly convergent x_{n}_{i} ⇀ x in the reflexive Banach space X to an element x ∈ M. We also obtain with y = x ∈ M and (15) the inequality

a(∥x − x_{n}_{i}∥) ≤ < f′(x_{n}_{i}) − f′(x), x_{n}_{i} − x > ≤
≤ \underline{g}′(x_{n}_{i}; x − x_{n}_{i})+ < f′(x), x − x_{n}_{i} > +ϵ_{n}_{i}∥x − x_{n}_{i}∥,

and as i → +∞ in virtue of (17), that x_{n}_{i} → x is strongly convergent to x.

Remark 3 a) If the function g is Gâteaux-differentiable, then (17) is the known condition (P) of Hess [16],

x_{n} ⇀ x ⟹ lim sup_{n→+∞} < g′(x_{n}), x_{n} − x > ≥ 0,

compare also Zeidler [30, 31].

b) It is known that g is a weakly lower semicontinuous (w.l.s.c.) function if for example

g′ satisfy (P) and is locally bounded, or
g′ satisfy (P) and is continuous, or
g′ satisfy (P) and g is continuous,

cf.Zeidler [30]. One can expect that (17) is valid not merely for w.l.s.c. functions because of the discontinuity in general of the Dini’s lower subdifferential mapping and the necessary assumptions for a mean valued theorem of lower Dini-subdifferentiable functions, cf. Weckesser [29].

c) If the function g is locally Lipschitz, then theorem 3 is also valid in the case that one used (P S)_{M} in the form with Clarke’s generalized directional derivative.

d) Theorem 3 remains valid if X is a locally uniformly convex Banach space and the function f instead of (15) is merely d-monotone, i.e.

< f′(x) − f′(y), x − y > ≥ (d(∥x∥) − d(∥y∥))(∥x∥ − ∥y∥) ∀x, y ∈ X

holds, where the function d : R → R is supposed to be strictly increasing, c.f. Kluge [19].

Let us consider for application of the theorem above a problem with lower and upper solutions. Let Ω ⊂ R^{n} , n ≥ 1 , be a bounded domain with smooth boundary ∂Ω and we would like to investigate the boundary value problem

− Δu = f(·, u) in Ω, u = 0 on ∂Ω  (18)

with a discontinuous nonlinearity f. We use the variational method and connect problem (18) with an optimization problem with a closed and convex set of restrictions given by an interval between a lower and upper solution of (18). It is sensible to consider for discontinuous nonlinearities f instead of (18), cf. Carl,Dietrich [6], the following inclusion problem

− Δu(x) ∈ [\underline{f}(x, u(x)), f̄(x, u(x))] in Ω, u(x) = 0 on ∂Ω,  (19)

where

\underline{f}(x, t) = lim inf_{s→t} f(x, s) and f̄(x, t) = lim sup_{s→t} f(x, s) ,  (20)

as we will see. A function u ∈ W_{0}^{1,2}(Ω) is called a weak solution of the nonlinear Dirichlet problem (19) if there is a function v^{∗} ∈ L^{2}(Ω) so that

v^{∗}(x) ∈ [\underline{f}(x, u(x)), f̄(x, u(x))] a.e. x ∈ Ω and
∫_{Ω} ∇u∇ϕdx = ∫_{Ω} v^{∗}ϕdx for all ϕ ∈ W_{0}^{1,2}(Ω).  (21)

In natural fashion, we call ū ∈ W^{1,2}(Ω) an upper solution of (19) if

f̄(·, ū) ∈ L^{2}(Ω) and ū ≥ 0 on ∂Ω as well as
∫_{Ω} ∇ū∇ϕdx ≥ ∫_{Ω} f̄(·, ū)ϕdx for all ϕ ∈ W_{0}^{1,2}(Ω) ∩ L_{+}^{2}(Ω).  (22)

Hence let L_{+}^{2}(Ω) be the set of all nonnegative elements of L^{2}(Ω). A function \underline{u} ∈ W^{1,2}(Ω) is called a lower solution of (19) if

\underline{f}(·, \underline{u}) ∈ L^{2}(Ω) and \underline{u} ≤ 0 on ∂Ω as well as
∫_{Ω} ∇\underline{u}∇ϕdx ≤ ∫_{Ω} \underline{f}(·, \underline{u})ϕdx for all ϕ ∈ W_{0}^{1,2}(Ω) ∩ L_{+}^{2}(Ω).  (23)

We want to suppose that there exist an upper solution ū and a lower solution \underline{u} to (19) such that \underline{u} ≤ ū. As restriction set we then consider the interval

M = \{u ∈ W_{0}^{1,2}(Ω) | \underline{u} ≤ u ≤ ū a.e. in Ω\}.

Furthermore suppose upon the nonlinearity f, that

f : Ω × R ⟶ R is Baire measurable  (24)

and there exists a function k ∈ L^{2}(Ω) so that

|f(x, s)| ≤ k(x) a.e. x ∈ Ω and for s ∈ [\underline{u}(x), ū(x)].  (25)

Condition (24) implies that the composed function x ⟼ f(x, u(x)) is measurable whenever the function x ⟼ u(x) is measurable, cf. Chang [7]. Let us introducing according to the boundary value problem (19) the function Φ : W_{0}^{1,2}(Ω) ⟶ R given by

Φ(u) = \frac{1}{2} ∫_{Ω} |∇u|^{2}dx − Ψ(u),  (26)

where Ψ : W_{0}^{1,2}(Ω) ⟶ R is the restriction of the function Ψ̃ : L^{2}(Ω) ⟶ R defined by

Ψ̃(v) = ∫_{Ω}∫_{\underline{u}(x)}^{v(x)} f(x, t)dtdx.  (27)

Then we consider the optimization problem

β := inf_{u∈M} Φ(u).  (28)

Let I : W_{0}^{1,2}(Ω) ⟶ L^{2}(Ω) be the embedding operator between the both spaces, v = Iu and Ψ = Ψ̃∘I are valid. The function Ψ̃ above is globally Lipschitzian on [\underline{u}, ū] because of (25)

|Ψ̃(u) − Ψ̃(v)| = | ∫_{Ω} ∫_{v(x)}^{u(x)} f(x, t)dtdx|
≤ ∫_{Ω} k(x)|u(x) − v(x)|dx ≤ ∥k∥∥u − v∥

and we get for Clarke’s subdifferential

∂Φ(u) = −Δu − ∂(Ψ̃ ∘ I)(u) ⊂ −Δu − I^{∗}∂Ψ̃(Iu)  (29)

with the adjoint embedding operator I^{∗} : L^{2}(Ω) ⟶ W^{−1,2}(Ω), if we apply the rule of sums and the chain rule. We obtain further by definition

∂Ψ̃(v) = {w ∈ L^{2}(Ω) | Ψ̃^{0}(v; z) ≥ ∫_{Ω} wzdx ∀z ∈ L^{2}(Ω)},

where

Ψ̃^{0}(v; z) = lim sup_{{h→v \atop t→+0}} \frac{Ψ̃(h + tz) − Ψ̃(h)}{t} h, z ∈ L^{2}(Ω),

is Clarke’s generalized directional derivative. By Chang [7], theorem 2.1, we have with v = Iu

v^{∗}(x) ∈ ∂Ψ̃(v(·)) ⊂ [\underline{f}(x, v(x)), f̄(x, v(x))] a.e.  (30)

Corollary 1 Under the assumptions above the minimization problem (28) possesses a solution.

Proof : At first we prove, that Φ fulfils (P S)_{M}. We have with suitable c > 0 and d > 0 the following inequality because of (25) : Φ(u) ≥ \frac{1}{2}∥u∥_{1}^{2} − c∥u∥_{1} − d ∀u ∈ M, where ∥ · ∥_{1} is the norm in W_{0}^{1,2}(Ω). Therefore Φ is coercive on M and we can apply theorem 3 in the form with Clarke’s subdifferential (29). The operator −Δ : W_{0}^{1,2}(Ω) ⟶ W^{−1,2}(Ω) is even strongly monotone and it remains to prove that the set-valued mapping I^{∗}∂Ψ̃(I(·)) : W_{0}^{1,2}(Ω) ⟶ W^{−1,2}(Ω) fulfils (17), the generalized condition (P), i.e.

if u_{n} ⇀ u then lim inf_{n→+∞} < u_{n}^{∗}, u − u_{n} >_{1} ≤ 0,

hence u_{n}^{∗} ∈ I^{∗}∂Ψ̃(Iu_{n}) and < ·, · >_{1} is the duality pairing between W^{−1,2}(Ω) and W_{0}^{1,2}(Ω). The set ∪_{n=1}^{+∞}∂Ψ̃(Iu_{n}) ⊂ L^{2}(Ω) is bounded as a known property of Clarke’s subdifferential and therefore there is a strongly convergent subsequence u_{n}^{∗}_{i} → u^{∗} as i → +∞ because of the compact embedding I^{∗} : L^{2}(Ω) ⟶ W^{−1,2}(Ω). For this subsequence {u_{n}_{i}}, lim inf_{i→+∞} < u_{n}^{∗}_{i}, u − u_{n}_{i} >= 0 holds. By the minimization theorem 2, problem (28) possesses a solution u ∈ W_{0}^{1,2}(Ω) with \underline{u} ≤ u ≤ ū. The next theorem include a result of Graham-Eagle [15], our proof is similar but different.

Theorem 4 A solution of the minimization problem (28) is also a solution of the boundary value problem (19).

Proof : If we suppose that (28) possesses a solution u ∈ M, then by lemma 3

0 ≤ Φ^{0}(u; y − u) ∀y ∈ M

holds and further with

Φ^{0}(u; y − u) = max_{ũ^{∗}∈∂Φ(u)} < ũ^{∗}, y − u >_{1}=< u^{∗}, y − u >_{1}

0 ≤< −Δu − I^{∗}v^{∗}, y − u >_{1} ∀y ∈ M  (31)

is valid with v^{∗} ∈ ∂Ψ̃(v) , v = Iu. Now remains to show that (31) implies u is a weak solution of (19) in the sense of (21). For that purpose we following in effect Graham-Eagle [15] and define with ϕ ∈ W_{0}^{1,2}(Ω) and ϵ > 0 the function y ∈ M by

y = \{ \begin{matrix}ū : & u + ϵϕ ≥ ū, \\ u + ϵϕ : & \underline{u} < u + ϵϕ < ū, \\ \underline{u} : & u + ϵϕ ≤ \underline{u}. \end{matrix}

Substituting this element into (31), we get

0 ≤ ϵ ∫_{\underline{u}<u+ϵϕ<ū}[∇u∇ϕ − v^{∗}Iϕ]dx
+ ∫_{u+ϵϕ≥ū}[∇u∇(ū − u) − v^{∗}I(ū − u)]dx
+ ∫_{u+ϵϕ≤\underline{u}}[∇u∇(\underline{u} − u) − v^{∗}I(\underline{u} − u)]dx,  (32)

and further after some transformations

0 ≤ ϵ ∫_{Ω}[∇u∇ϕ − v^{∗}Iϕ]dx
− ∫_{u+ϵϕ≥ū}[∇ū∇(u + ϵϕ − ū) − f̄(x, ū)(u + ϵϕ − ū)]dx
+ ∫_{u+ϵϕ≤\underline{u}}[∇\underline{u}∇(\underline{u} − u − ϵϕ) − \underline{f}(x, \underline{u})(\underline{u} − u − ϵϕ)]dx
− ∫_{u+ϵϕ>ū} |∇u − ∇ū|^{2}dx − ∫_{u+ϵϕ<\underline{u}} |∇u − ∇\underline{u}|^{2}dx

− ∫_{u+ϵϕ>ū}(v^{∗} − f̄(x, ū))(ū − u − ϵϕ)dx − ∫_{u+ϵϕ<\underline{u}}(v^{∗} − \underline{f}(x, \underline{u}))(\underline{u} − u − ϵϕ)dx

−ϵ ∫_{u+ϵϕ>ū}(∇u − ∇ū)∇ϕdx − ϵ ∫_{u+ϵϕ<\underline{u}}(∇u − ∇\underline{u})∇ϕdx.  (33)

The second and third lines of this inequality are nonpositive because \underline{u} is a lower solution and ū is an upper solution, the fourth and fifth lines are trivially nonpositive. The sixth line is o(ϵ) since the integrands are integrable and the measure of the domains of integration tends to zero as ϵ → +0. Dividing by ϵ and letting ϵ → +0 we obtain

0 ≤ ∫_{Ω}[∇u∇ϕ − v^{∗}Iϕ]dx ∀ϕ ∈ W_{0}^{1,2}(Ω).  (34)

for arbitrary ϕ ∈ W_{0}^{1,2}(Ω). By (21) u ∈ W_{0}^{1,2}(Ω) is in fact a weak solution of (19).

A relationship between the Palais-Smale condition and the coercivity of the associate function is proved by Caklovic, Li and Willem [5] and generalized by Goeleven [14] for certain nonsmooth functions. We prove now that this connection is retained also for l.s.c. functions over closed and convex sets.

Theorem 5 Let X be a Banach space, let f : X ⟶ R ∪ {+∞} be a l.s.c. function and let M ⊂ X be a nonempty, closed, unbounded and convex subset and consider (8). Define c := lim inf_{{∥x∥→+∞ \atop x∈M}} f(x) and suppose that c ∈ R. Then there exist sequences {x_{n}} ⊂ M and {ϵ_{n}} ⊂ (0, +∞) with ϵ_{n} → +0 such that the assertions

(i) ∥x_{n}∥ → +∞

(ii) f(x_{n}) → c

(iii) 0 ≤ \underline{f}′(x_{n}; x − x_{n}) + ϵ_{n}∥x − x_{n}∥ ∀x ∈ M

are valid.

Proof : For every n ∈ N, there exists an element y_{n} ∈ M so that f(y_{n}) ≤ c + 1/n and ∥y_{n}∥ ≥ 2n. By Ekeland’s variational principle there exists x_{n} ∈ M such that with δ_{n} := c + 1/n − α, f(x_{n}) ≤ f(y_{n}) ≤ α + δ_{n}, ∥y_{n} − x_{n}∥ ≤ n and

f(x_{n}) − (1/n)δ_{n}∥y − x_{n}∥ ≤ f(y) ∀y ∈ M

hold. Let y = th + (1 − t)x_{n}, where h ∈ M and t ∈ (0, 1]. We obtain further

0 ≤ \underline{f}′(x_{n}; x − x_{n}) + ϵ_{n}∥x − x_{n}∥ ∀x ∈ M

with ϵ_{n} := (1/n)δ_{n} → +0 as n → +∞. Moreover, by ∥y_{n}∥ ≥ 2n, we get

∥x_{n}∥ ≥ ∥y_{n}∥ − ∥x_{n} − y_{n}∥ ≥ 2n − n = n.

Because now f(x_{n}) ≤ c+1/n for ∥x_{n}∥ → +∞ and by the assumption c ≤ lim inf_{n→+∞} f(x_{n}) hold (i), (ii) and (iii) are satisfied.

Corollary 2 Let X be a Banach space, let M ⊂ X be a nonempty, closed, unbounded and convex set and let f : X ⟶ R ∪ {+∞} be a function satisfying the Palais-Smale condition (P S)_{M}. If f is bounded below on M, then f is coercive on M.

Proof : Assume on the contrary that c = lim inf_{{∥x∥→+∞ \atop x∈M}} f(x) ∈ R and apply the previous theorem. Then there exists a sequence {x_{n}} ⊂ M such that ∥x_{n}∥ → +∞, f(x_{n}) → c and

\underline{f}′(x_{n}; x − x_{n}) + ϵ_{n}∥x − x_{n}∥ ≥ 0 ∀x ∈ M

with ϵ_{n} → +0 as n → +∞ hold. Further F (x_{n}) → 0 as n → +∞ is valid. Thus we obtain a contradiction to the Palais-Smale condition (P S)_{M}.

Another possibility to consider a convex and closed set of restrictions is the following method, c.f. Hofer [17], De Figueiredo and Solimini [12] and compare also Ghoussoub [13]. Now we generalize this method for locally Lipschitz functions.

Theorem 6 Let X be a locally uniformly convex Banach space, let X^{∗} be its dual space and let F : X ⟶ X^{∗} be the corresponding duality mapping. If

(i) M ⊂ X is a nonempty, convex and closed set, (ii) f : X ⟶ R is a locally Lipschitzian which is bounded below on M and

(I − F^{−1} ∘ ∂f)(M) ⊂ M

holds with Clarke’s subdifferential ∂f : X ⟶ 2^{X^{∗}} and (iii) f fulfil the Palais-Smale condition in the sense of Chang [7], then the minimum problem (8) above has a solution x̄ ∈ M.

Proof : Let {ξ_{n}} ⊂ M be a minimizing sequence of α = inf_{x∈M} f(x) and ϵ_{n} > 0 for all n ∈ N with ϵ_{n} → +0 so that

−∞ < α ≤ f(ξ_{n}) ≤ α + ϵ_{n}^{2}.

Then there exists by Ekeland’s variational principle a second minimizing sequence {x_{n}} ⊂ M with the properties

(j) f(x_{n}) ≤ f(ξ_{n}) (jj) ∥x_{n} − ξ_{n}∥ ≤ ϵ_{n} and (jjj) f(x_{n}) ≤ f(y) + ϵ_{n}∥y − x_{n}∥ ∀y ∈ M. Because of the convexity of M and (ii)

y_{n} := (1 − t)x_{n} + t(x_{n} − F^{−1}x_{n}^{∗}) = x_{n} − tF^{−1}x_{n}^{∗} ∈ M

for t ∈ (0, 1] implies

0 ≤ f^{0}(x_{n}; −F^{−1}x_{n}^{∗}) + ϵ_{n}∥x_{n}^{∗}∥_{∗}.

With min_{ξ_{n}^{∗}∈∂f(x_{n})} ∥ξ_{n}^{∗}∥_{∗} = ∥x_{n}^{∗}∥_{∗} further we get ∥x_{n}^{∗}∥_{∗} ≤ ϵ_{n}. In virtue of the boundedness below of the sequence {f(x_{n})} it follows by Chang’s Palais-Smale condition that there exists a convergent subsequence {x_{n}_{i}}, x_{n}_{i} → x̄ ∈ M and x̄ ∈ M is an solution of the minimum problem (8).

Acknowledgement

The author expresses his thanks to Prof.S.Carl for helpful discussions on weak lower and upper solutions of elliptic boundary value problems with discontinuous nonlinearities.

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