Journal of Convex Analysis Volume 4 (1997), No. 1, 69–89

Γ-convergence for a Class of Functionals with Deviating Argument

Mikhail E. Drakhlin

Research Institute, College of Judea and Samaria, Kedumim-Ariel, D.N. Efraim 44820, Israel.

Eugene Stepanov

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. e-mail: stepanov@cibs.sns.it

Received November 20, 1995 Revised manuscript received April 4, 1996

The paper deals with the problem of Γ-convergence of functionals involving nonlocal transformation of argument, in particular, the argument deviation. The property of Γ-convergence of the functionals is related with various types of convergence of inner superposition (composition) operators. The study is conducted in Orlicz spaces. Γ-convergence of functionals with impulsive constraints is also studied as one of the applications.

Keywords: Calculus of Variations, Γ-convergence, Functional Differential Equation, Orlicz Space.

1991 Mathematics Subject Classification: 49J25, 49J45, 49N25, 47A55, 47B38 .

1. Introduction

When dealing with minimization problem for the given functional I: X → ĪR ≡ IR ∪ {+∞}, X being some Banach space of admissible functions, it is important to know the behavior of its solutions (i.e. minimum values and points of minima) with respect to small perturbation of parameters of the problem. In other words, we are to study the problem of Γ-convergence of the functionals I_{ν} to I. This paper is concerned with minimization problem for functional with deviating argument defined by the relationship

I(x) = ∫_{0}^{1} f(t, x(h_{1}(t)), …, x(h_{k}(t)), ẋ(g_{1}(t)), …, ẋ(g_{l}(t))) dt,

x(t) = 0, ẋ(t) = 0, whenever t ∉ [0, 1].

(V_{∞})

where t ∈ [0, 1], x(t) ∈ IR^{n} is a vector function, {k, l} ⊂ IN, h_{i}: [0, 1] → IR, g_{j}: [0, 1] → IR and f: [0, 1]×IR^{nk} ×IR^{nl} → ĪR. The problem of continuous dependence of its solutions on

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70 M. Drakhlin, E. Stepanov / Γ-convergence for a class of functionals ...

parameters can be stated as follows: suppose we have a sequence of perturbed functionals of the same type

I_{ν}(x) = ∫_{0}^{1} f_{ν}(t, x(h_{1}^{ν}(t)), …, x(h_{k}^{ν}(t)), ẋ(g_{1}^{ν}(t)), …, ẋ(g_{l}^{ν}(t))) dt,

x(t) = 0, ẋ(t) = 0, whenever t ∉ [0, 1],

(V_{ν})

h_{i}^{ν}: [0, 1] → IR, g_{j}^{ν}: [0, 1] → IR, f_{ν}: [0, 1] × IR^{nk} × IR^{nl} → ĪR tending in certain sense to h_{i}, g_{j} and f respectively as ν → ∞. One is interested in finding out whether in this case the solutions to the minimization problems (V_{ν}) (minimum values of the functionals I_{ν} and the points of their minima x_{ν}) approach the solution to the minimization problem (V_{∞}) (respectively, minimum value of I and the points x of minimum of the latter). The concept of Γ-convergence was introduced in the 1970s by E. De Giorgi (see, e.g., [1]) and afterwards received significant contributions by many authors. In the sequel we mainly follow its recent developments in control theory by G. Dal Maso and G. Buttazzo [2,3]. It happens that the standard Sobolev space W^{1,p} ((0, 1); IR^{n}), being defined as the space of equivalent classes of vector functions, is not well adapted to the study of nonlocal problems of the above type, and it is much more convenient to conduct the study in the spaces of absolutely continuous vector functions. In this paper we use the spaces of absolutely continuous functions with the derivatives in some Orlicz spaces, which allow also to consider the functionals with rapidly (e.g. exponentially) growing integrands. After reiterating with minor changes and generalizations for this case the standard auxiliary results for classical functionals (without transformation of the argument) in section 4, we start with introducing the inner superposition (composition) operators which play the key role in the further discussion and study various types of their convergence in section 5. In the last section we relate the properties of convergence of inner superposition operators with Γ-convergence of the functionals with deviating argument thus solving the above-posed problem of stability of solutions. Developing the technique introduced in the paper we consider also some generalizations, e.g. for the functionals with impulsive constraints, which most often appear in the optimal control problems for impulsive functional differential equations.

2. Notation and preliminaries

Recall that the Fenchel conjugate I′: X′ → IR ∪ {±∞} for the functional I: X′ → IR ∪ {±∞} is defined by the formula

I′(x′) := sup_{x∈X}(⟨x, x′⟩ − I(x)).

For the general properties of the conjugate the reader is referred to [4, p. 35]. In what follows we also use the Fenchel conjugate in finite-dimensional space referring to it as the Legendre transformation. Let L^{p}(0, 1) stand for the standard Lebesgue space of functions integrable on the interval (0, 1) with the power 1 ≤ p < ∞ (or essentially bounded when p = ∞). In the sequel we make an extensive use of the space L^{p} ((0, 1); IR^{n}) (further on denoted by L_{n}^{p} for the sake of brevity) of IR^{n}-valued vector functions with the components in L^{p}(0, 1), equipped by its usual norm. By AC_{n}^{p} denote the space of vector functions with absolutely continuous components and with the derivatives in L_{n}^{p}, equipped with the norm ||x||_{AC_{n}^{p}} := |x(0)| + ||ẋ||_{p}, where |·| stands for the norm in IR^{n} (upon the necessity to emphasize the dimension of the space, the latter will be also denoted by | · |_{n}). Also by C_{n}[0, 1] (C_{n}, for short) we denote the space of all IR^{n}-valued vector functions with continuous components. When dealing with rapidly increasing (e.g. exponential) nonlinearities, it is often convenient to use Orlicz spaces. While introducing them here we follow the lines of [5, chapter 4]. Let P: IR → [0, +∞) be the Young function (i.e. even, convex, lower semicontinuous with P (0) = 0). Of course, the trivial situations P (x) ≡ 0 and P (x) ≡ +∞ are excluded. In the sequel we denote by L_{n}^{P} the Orlicz space of vector functions x: [0, 1] → IR^{n}, the components of which are from the Orlicz space L^{P} (0, 1) generated by P (x). Equip L_{n}^{P} with the usual Luxembourg norm

||x||_{P} := inf {λ > 0 : ∫_{0}^{1} P (|x(τ)|/λ) dτ ≤ 1} .

Recall that the Lebesgue spaces L_{n}^{p} are particular examples of the Orlicz spaces introduced. The corresponding space of absolutely continuous vector functions AC_{n}^{P} is introduced in the same way as AC_{n}^{p}. In what follows we will use as well the notion of the

subspace AC_{n}^{P}^{∘} (resp. AC_{n}^{p}^{∘}) of AC_{n}^{P} (resp. AC_{n}^{p}) consisting of the functions satisfying the restriction x(0) = x(1) = 0.

Consider the associated Young function P′ obtained by the Legendre transform of P [5, p. 122]. It generates the associate Orlicz space L_{n}^{P′} which is closed in the topological dual (L_{n}^{P})′ and coincides with the latter if and only if P (x) is a Δ_{2}-function, that is, for each λ > 0 there is α > 0 such that P (λx) ≤ αP (x). For the particular case L_{n}^{P} = L_{n}^{p} one

gets L_{n}^{P′} = L_{n}^{p′}, where 1/p + 1/p′ = 1. Furthermore, L_{n}^{P} is reflexive if both P and P′ are Δ_{2}-functions [5, pp. 124-125]. Another important property of Orlicz spaces generated by Δ_{2}-functions is their separability.

Note that AC_{n}^{P} ⊂ C_{n} (as sets). Using the standard arguments involving Young’s inequality and the estimate for the Orlicz norm of a characteristic function of the set (formula (4.40) in [5]) one can easily show that the latter immersion is compact provided that P^{−1}(x)/x → 0 when x → +∞. This surely holds when P′ is a Δ_{2}-function. It is also worth noting that the inclusions L_{n}^{∞} ⊂ L_{n}^{P} ⊂ L_{n}^{1} hold for any Young function P (x).

3. Basic facts about Γ-convergence

Throughout this section let (X, τ) be a topological space with a topology τ, I_{ν}: X → ĪR, ν ∈ IN be a sequence of functionals, I: X → ĪR.

Definition 3.1. I is called the (sequential) Γ^{−}(τ)-limit of I_{ν} over X (written I = Γ^{−}(τ) lim_{ν} I_{ν}), if

I = Γ^{−}(τ) lim inf_{ν} I_{ν} = Γ^{−}(τ) lim sup_{ν} I_{ν}, where
Γ^{−}(τ) lim inf_{ν} I_{ν}(x) := inf { lim inf_{ν} I_{ν}(x_{ν}) :  x_{ν}→^{τ}x},
Γ^{−}(τ) lim sup_{ν} I_{ν}(x) := inf { lim sup_{ν} I_{ν}(x_{ν}) :  x_{ν}→^{τ}x}.

From now on we restrict ourselves to the following 3 situations: (C1) X is a Banach space with τ the topology of its norm. (C2) X is a reflexive Banach space with separable topological dual X′, τ is the weak

topology. (C3) X = V′ is the topological dual of a separable Banach space V, τ is the ∗-weak

topology. In the case (C1) we speak about strong Γ^{−}-limits, in the cases (C2) and (C3) about weak and ∗-weak Γ^{−}-limits, respectively. The reference to the topology τ will be omitted in the sequel whenever it cannot cause misunderstanding. Recall that the functional I_{0}: X → ĪR is said to be coercive, when I_{0}(x) → +∞ as ||x||_{X} → ∞. The sequence {I_{ν}} is then called equicoercive, if I_{ν}(x) ≥ I_{0}(x) for each ν ∈ IN. We will make an extensive use of following criterion for the Γ^{−}-convergence which is the reformulation of the propositions 1.2 and 1.4 from [6].

Proposition 3.2. In the case (C1) I = Γ^{−} lim_{ν} I_{ν}, if and only if two conditions are satisfied simultaneously:

∀{x_{ν}} ⊂ X, x ∈ X, x_{ν} → x ⇒ I(x) ≤ lim inf_{ν} I_{ν}(x_{ν});  (1)
∀x ∈ X, ∃{x_{ν}} ⊂ X, x_{ν} → x such that I(x) = lim_{ν} I_{ν}(x_{ν}).  (2)

The same is true in the cases (C2) and (C3) under the additional requirement that the sequence {I_{ν}} be equicoercive.

The proposition below gives the well-known main property of Γ^{−}-limits (see [2], corollary 7.20 and theorem 7.8, corollary 1.25).

Proposition 3.3. Let I = Γ^{−} lim_{ν} I_{ν} and {x_{ν}} ⊂ X be such that

lim_{ν} I_{ν}(x_{ν}) = lim_{ν}(inf_{X} I_{ν}).

I. If x_{ν} → x, then x provides global minimum for I on X, while

lim_{ν}(inf_{X} I_{ν}) = min_{X} I.

II. If {I_{ν}} is equicoercive, I ≢ +∞, then I is coercive, while there exists a subsequence x_{ν_{μ}} → x, the above being valid. Moreover, if I has unique point of minimum x, then x_{ν} → x.

Remark 3.4. In particular, for x_{ν} one can take the points of global minima of the respective I_{ν} over X. It is well-known that the weak Γ^{−}-convergence of a sequence of functionals in a reflexive Banach space with separable dual is closely related to the strong Γ^{−}-convergence of their Fenchel conjugates. To state the exact assertion to be used in the sequel, recall first that the functional I: X → IR ∪ {±∞} is said to be proper, if I(x) ≢ ±∞, and lower semicontinuous, if x_{ν} → x in X implies I(x) ≤ lim inf_{ν} I(x_{ν}).

Proposition 3.5. Let X be a reflexive Banach space with the separable dual X′, while the functionals I, I_{ν}, I_{0}: X → ĪR and their Fenchel conjugates I′, I′_{ν}, I′_{0}: X′ → IR∪{±∞} satisfy the following assumptions:

(i) I and all I_{ν} are convex and lower semicontinuous; (ii) I_{ν}(x) ≥ I_{0}(x) for all x ∈ X, where I_{0} is coercive, I′_{0} being proper; (iii) lim_{ν} I′_{ν}(x′) = I′(x′) for all x′ ∈ X′, the sequence {I′_{ν}} being equicontinuous.

Then I_{ν} → I Γ^{−}-weakly in X.

Remark 3.6. In the condition (iii) it is enough to require that lim_{ν} I′_{ν}(x′) = I′(x′) only for each x′ ∈ D′, where cl D′ = X′, whenever {I′_{ν}} is equilipschitzian.

Proof. The condition (ii) implies, by virtue of the proposition 5.9 of [2], that the pointwise convergence of I′_{ν} to I′ over X′, provided by (iii), is equivalent to the Γ^{−}-convergence of I′_{ν} to I′ in the strong (normed) topology of X′. From condition (i) follows that the second conjugates I′′ = I and I′′_{ν} = I_{ν}. Thus, using the fact that all I′_{ν} are proper and the theorem 3.2.4 of [7], one yields the desired result.

4. Auxiliary lemmata

Recall that g: [0, 1]×IR^{n} → ĪR is said to be a Carathéodory function, if g(t, y) is continuous in y ∈ IR^{n} for a.e. t ∈ [0, 1] and Lebesgue measurable in t ∈ [0, 1] for all y ∈ IR^{n}. Consider its Legendre transform

g′(t, y′) := sup_{y∈IR^{n}}(yy′ − g(t, y)).

Introduce the functional J: L_{n}^{P} → IR ∪ {±∞} by the relationship

J(y) = ∫_{0}^{1} g(t, y(t)) dt.  (3)

The statement below which gives the representation of its Fenchel conjugate, is the slight variation of the proposition 2.1 in [8, chapter IX].

Lemma 4.1. Let g(t, y) be a Carathéodory function, while the functional J given by (3) be proper in L_{n}^{P}. Then the conjugate J′: (L_{n}^{P})′ → IR ∪ {±∞} of the latter admits representation J′(y′) = ∫_{0}^{1} g′(t, y′(t)) dt.

Proof. According to the conditions, there exists y_{0} ∈ L_{n}^{P} such that J(y_{0}) ≠ ±∞. Given y′ ∈ (L_{n}^{P})′, consider the functions

g′_{j}(t) := sup_{|y|≤j|y_{0}(t)|}(yy′(t) − g(t, y)).

One observes that the sequence {g′_{j}} is nondecreasing and convergent for a.e. t ∈ [0, 1] to g′(t, y′(t)). Moreover, g′_{j}(t) ≥ y_{0}(t)y′(t) − g(t, y_{0}(t)) for all j ∈ IN, the right hand side being integrable due to the original assumption. According to the Krasnosel’skiǐ-La-dyzhenskiǐ lemma on measurable selections [5, theorem 6.2], for each j ∈ IN there is a measurable vector function y_{j}: [0, 1] → IR^{n}, such that |y_{j}(t)| ≤ j|y_{0}(t)| and

g′_{j}(t) = y_{j}(t)y′(t) − g(t, y_{j}(t)).

Noting that g′(t, y′(t)) is measurable as a limit of measurable functions and bounded from below by an integrable function, it is easy to conclude that

\begin{aligned}∫_{0}^{1} g′(t, y′(t)) dt & = sup_{j} (∫_{0}^{1}(y_{j}(t)y′(t) − g(t, y_{j}(t))) dt) \\ & ≤ sup_{x∈L_{n}^{P}} (∫_{0}^{1} (y(t)y′(t) − g(t, y(t))) dt) = J′(y′). \end{aligned}

The analogous upper estimate J′(y′) ≤ ∫_{0}^{1} g′(t, y′(t)) dt follows from Young’s inequality.

Consider the sequence of functionals J_{ν}: L_{n}^{P} → IR ∪ {±∞} defined by the formula

J_{ν}(y) = ∫_{0}^{1} g_{ν}(t, y(t)) dt.  (4)

Another auxiliary assertion to be used in the sequel concerns its pointwise convergence to a functional J: L_{n}^{P} → IR ∪ {±∞} given by the relationship (3), and follows closely lemma 3.1 of [6].

Lemma 4.2. Let g_{ν}, g, g_{0}, g^{0}: [0, 1] × IR^{n} → ĪR be Carathéodory functions satisfying the assumptions

(i) For a.e. t ∈ [0, 1], for all y ∈ IR^{n} holds one has g^{0}(t, y) ≤ g_{ν}(t, y) ≤ g_{0}(t, y), g_{ν}(t, ·) and g(t, ·) being convex, while g^{0}(·, y), g_{0}(·, y) ∈ L^{1}(0, 1);

(ii) g_{ν}(·, y) ⇀ g(·, y) weakly in L^{1}(0, 1). Then J_{ν}(y) → J(y) for each y ∈ C_{n}.

Proof. First observe that as g^{0}(t, ·) and g_{0}(t, ·) are finite, then g_{ν}(t, ·) and g(t, ·) are equilipschitzian according to corollary 2.4 of [8, chapter I], that is, from |y_{1}| < R, |y_{2}| < R follows that for a.e. t ∈ [0, 1]

|g_{ν}(t, y_{1}) − g_{ν}(t, y_{2})| ≤ l_{R}(t)|y_{1} − y_{2}|, where

l_{R}(t) = \frac{max_{|y|≤R_{0}} g_{0}(t, y) − min_{|y|≤R_{0}} g^{0}(t, y)}{R_{0} − R}, R_{0} > R.

Obviously l_{R} ∈ L^{1}(0, 1). It is also easy to show that g(t, y) has the same upper and lower bounds, and therefore, it has the same estimate for a Lipschitz constant. For y ∈ C_{n} consider the finite ε-net {y_{i}}_{i=1}^{m} ⊂ IR^{n} of y([0, 1]) and the respective open balls B_{ε}(y_{i}): |y − y_{i}| < ε covering the latter. Let {h_{i}(t)} stand for the partition of unity on y^{−1}(B_{ε}(y_{i})) and denote Ω_{i} := [0, 1] ∩ y^{−1}(B_{ε}(y_{i})). Also let R stand for the radius of the ball covering all y([0, 1]). The estimate

|J_{ν}(y) − J(y)| ≤ ∑_{i=1}^{m} |∫_{Ω_{i}}(g_{ν}(t, y(t)) − g(t, y(t)))h_{i}(t) dt| ≤

∑_{i=1}^{m}( |∫_{Ω_{i}}(g_{ν}(t, y(t)) − g_{ν}(t, y_{i}))h_{i}(t) dt| + |∫_{Ω_{i}}(g_{ν}(t, y_{i}) − g(t, y_{i}))h_{i}(t) dt| +

|∫_{Ω_{i}}(g(t, y(t)) − g(t, y_{i}))h_{i}(t) dt|) ≤ (2l_{R} + 1)ε

for sufficiently large ν shows the statement.

5. Convergence of sequences of inner superposition operators

In this section we discuss various notions of continuous convergence of a sequence of mappings A_{ν}: X → Y between two Banach spaces. In the sequel we will be particularly interested in studying the convergence of sequence of linear inner superposition operators {S_{g^{ν}}} acting on some space of vector functions x: [0, 1] → IR^{n} and defined formally by the relationships

(S_{g^{ν}}x)(t) = \{ \begin{matrix}x(g^{ν}(t)), & g^{ν}(t) ∈ [0, 1], \\ 0, & g^{ν}(t) ∉ [0, 1], \end{matrix}  (5)

where g_{ν}: [0, 1] → IR ∪ {±∞}. Denote by S_{g} the operator

(S_{g}x)(t) = \{ \begin{matrix}x(g(t)), & g(t) ∈ [0, 1], \\ 0, & g(t) ∉ [0, 1], \end{matrix}  (6)

with g: [0, 1] → IR ∪ {±∞}. From now on we assume that both g(t) and all g^{ν}(t) are almost everywhere finite and measurable. For the sake of brevity we denote g^{0} := g. On the σ-algebra of measurable subsets e ⊂ [0, 1] define for all ν ∈ IN ∪ {0} the functions μ_{g^{ν}}(e) := meas (g^{ν})^{−1} (e) and, when the latter are absolutely continuous with respect to

the Lebesgue measure, the respective Radon-Nikodym derivatives \frac{dμ_{g^{ν}}}{dm}. The methods of the explicit calculation of the latter in various particular cases are given in [9, pp. 21-23]. In an analogous way, for each τ ∈ [0, 1], ν ∈ IN ∪ {0} consider the set functions μ_{g^{ν}}(τ, e) := meas (g^{ν})^{−1} (e) ∩ [0, τ] and their Radon-Nikodym derivatives

\frac{dμ_{g^{ν}}(τ)}{dm}(t) := lim_{ε→+0} \frac{μ_{g^{ν}}(τ, [t − ε, t + ε])}{2ε}, for a.e. t ∈ (0, 1).

Further on we as usual omit the superscript if ν = 0. To calculate the above functions one can use the relationship

\frac{dμ_{g^{ν}}(τ)}{dm}(t) = \frac{d}{dt} meas (g^{ν})^{−1} ([0, t]) ∩ [0, τ].

It is also worth noting that \frac{dμ_{g^{ν}} (1)}{dm} ≡ \frac{dμ_{g^{ν}}}{dm}. Also for every τ ∈ [0, 1] consider the sets

E_{ν}(τ) := (g^{ν})^{−1} ([0, τ]).

Further on we will write E(τ) instead of E_{0}(τ) and omit the reference to τ when τ = 1. Let from now on F(X, Y) denote the set of all mappings between two Banach spaces X and Y, L(X, Y) stand for the space of linear bounded operators acting in the same spaces. If X = Y, we write as usual F(X), L(X) respectively.

5.1. Pointwise and continuous convergence

Definition 5.1. The sequence of maps A_{ν} ∈ F(X, Y) is said to converge continuously

to A ∈ F(X, Y), A_{ν}→^{C}A, if {x_{ν}} ⊂ X, x_{ν} → x ∈ X ⇒ A_{ν}x_{ν} → Ax ∈ Y.

One notes that whether A_{ν} are continuous or not, A_{ν}→^{C}A implies that A is continuous (see Kuratowski [10], Chapter II, § 20, Section VII). Moreover, if all A_{ν} are linear, then it also implies their boundedness as well as the linearity and boundedness of the limit operator A. Furthermore, it is well-known that for linear operators between two Banach spaces continuous convergence is equivalent to the strong (pointwise) convergence. Turn now to the particular case of a sequence of inner superposition operators. At first, we study the strong (pointwise) and, therefore, continuous convergence of the sequence S_{g^{ν}}: C_{n} → L_{n}^{P} to the S_{g}: C_{n} → L_{n}^{P}. Observe that for these operators to be well-defined by the formulae (5) and (6) respectively it suffices to require the measurability of all g_{ν} and g. The assertion below extends the analogous one for Lebesgue space (see theorem 2.1 in § 4.2 of [9]).

Proposition 5.2. Let the Young function P satisfy the Δ_{2}-condition. The sequence of operators S_{g^{ν}}: C_{n} → L_{n}^{P} defined by (5) converges pointwise to S_{g}: C_{n} → L_{n}^{P} defined by (6) if and only if the following two assumptions hold as ν → ∞:

(i) g^{ν} → g in measure on E; (ii) meas E_{ν}ΔE → 0, where Δ stands for symmetric difference between the sets.

Remark 5.3. The same result is valid for L_{n}^{P} = L_{n}^{∞} if the assumptions are replaced by (i) g^{ν} → g in L^{∞} norm on E; (ii) meas E_{ν}ΔE = 0 starting from some ν ∈ IN.

Proof. The necessity has been proved in the theorem 2.1 in § 4.2 of [9]. To prove sufficiency recall that according to the same theorem S_{g^{ν}}x → S_{g}x in measure for any x ∈ C_{n} under the conditions (i) and (ii). To show the norm convergence note that

||S_{g^{ν}}x − S_{g}x||_{P} = inf λ,

where infimum is taken over all λ > 0 such that

\begin{matrix}∫_{E_{ν}\E} P (|(S_{g^{ν}}x) (t)| /λ) dt + ∫_{E\E_{ν}} P (|(S_{g}x) (t)| /λ) dt+ \\ ∫_{E∩E_{ν}} P (|(S_{g^{ν}}x) (t) − (S_{g}x) (t)| /λ) dt ≤ 1. \end{matrix}

For each fixed λ > 0 the first two integrals in this sum tend to zero when ν → ∞ due to the the boundedness by the value P (max_{[0,1]} |x(t)|/λ) meas EΔE_{ν}, the last one tending to zero by virtue of the Lebesgue monotone convergence theorem, which shows the statement.

For the sake of completeness we mention here that the strong convergence of the sequence of inner superposition operators acting between two Lebesgue spaces has been studied in [11].

5.2. Weak pointwise convergence

Definition 5.4. The sequence of maps A_{ν} ∈ F(X, Y) is said to converge weakly pointwise to A ∈ F(X, Y), if for all x ∈ X A_{ν}x ⇀ Ax ∈ Y.

This type of convergence is clearly strictly weaker than the pointwise convergence unless Y is a finite-dimensional space. However, for a sequence of linear operators it still implies their boundedness as well the linearity and the boundedness of the limit operator A. One notes also that the weak pointwise convergence of the linear operators A_{ν} to A is equivalent to the weak pointwise convergence of their adjoints A′_{ν}: Y′ → X′ to A′: Y′ → X′. We discuss now the case of inner superposition operators.

Proposition 5.5. Let the Young functions P and Q satisfy the Δ_{2}-condition, while Φ(x) = P (Q^{−1}(x)) be the Young function with the Legendre transform Φ′. The sequence of operators S_{g^{ν}}: L_{n}^{P} → L_{n}^{Q} defined by (5) converges weakly pointwise to S_{g}: L_{n}^{P} → L_{n}^{Q} defined by (6), if

(i) ∃A : ∥\frac{dμ_{g^{ν}}}{dm} ∥_{Φ′} ≤ A for all ν ∈ IN ∪ \{0\};

(ii) meas E_{ν}(τ)ΔE(τ) → 0 when ν → ∞ for any τ ∈ [0, 1].

Remarks 5.6.

1. Under the conditions of this proposition necessarily L_{n}^{P} ⊂ L_{n}^{Q}.

2. The same result holds when instead of Q its Legendre transform Q′ satisfies the Δ_{2}-condition. In this case weak convergence should be replaced by the ∗-weak one.

Proof. It is enough to show that (I) The sequence S_{g^{ν}} has uniformly bounded norms; (II) S_{g^{ν}}x ⇀ S_{g}x for all x ∈ X_{0}, where span X_{0} is dense in L_{n}^{P}. Clearly, (I) follows from (i). One should demonstrate also that (ii) implies (II). For this purpose take

X_{0} = {χ_{[0,τ]}e_{j} | τ ∈ [0, 1], j = 1, …, n} ⊂ L_{n}^{P} ,

where χ_{[0,τ]} stands for the characteristic function of the interval [0, τ], e_{j} is the j-th coordinate unit vector in IR^{n}. Observe that cl span X_{0} = L_{n}^{P}, because P is a Δ_{2}-function. Taking into account that

S_{g^{ν}}χ_{[0,τ]}e_{j} = χ_{E_{ν}(τ)}e_{j} and S_{g}χ_{[0,τ]}e_{j} = χ_{E(τ)}e_{j},

and that for any f ∈ L^{Q′}(0, 1)

|∫_{0}^{1} χ_{E_{ν}(τ)}(t)f(t) dt − ∫_{0}^{1} χ_{E(τ)}(t)f(t) dt| ≤ |∫_{E_{ν}(τ)\E(τ)} f(t) dt| + |∫_{E(τ)\E_{ν}(τ)} f(t) dt| → 0,

according to (ii), one shows (II).

It is clear from the proof that condition (ii) of the above proposition is also necessary for the weak pointwise convergence of the sequence of inner superposition operators between two Orlicz spaces. In fact, the choice x = χ_{[0,τ]}e_{j} implies then χ_{E_{ν}(τ)} ⇀ χ_{E(τ)} in L^{Q}(0, 1), wherefrom (ii) follows. On the contrary, condition (i) is only sufficient. For instance, if

P ≡ Q, then L^{Φ′}(0, 1) = L^{∞}(0, 1), while the uniform essential boundedness of \frac{dμ_{g^{ν}}}{dm} is generally speaking not necessary for the uniform boundedness of the operators S_{g^{ν}}. However for the particular case of Lebesgue spaces we can improve at this point proposition 5.5.

Corollary 5.7. The sequence of operators S_{g^{ν}}: L_{n}^{p} → L_{n}^{q}, 1 < q ≤ p < ∞, defined by (5), converges weakly pointwise to S_{g}: L_{n}^{p} → L_{n}^{q} defined by (6) if and only if

(i) ∃A : ∥\frac{dμ_{g^{ν}}}{dm} ∥_{p/(p−q)} ≤ A;

(ii) meas E_{ν}(τ)ΔE(τ) → 0 when ν → ∞ for any τ ∈ [0, 1].

Proof. One observes that if L_{n}^{P} = L_{n}^{p}, L_{n}^{Q} = L_{n}^{q}, 1 < q ≤ p < ∞, then L_{n}^{Φ′} = L_{n}^{p/(p−q)}. It remains to note that the condition (i) is also necessary for the uniform boundedness of the operators S_{g^{ν}} (which is an immediate consequence of theorem 1 of [12]).

5.3. Weak continuous convergence

Definition 5.8. The sequence of maps A_{ν} ∈ F(X, Y) is said to converge weakly

continuously to A ∈ F(X, Y), A_{ν}⇀^{C_{σ}}A, if {x_{ν}} ⊂ X, x_{ν} ⇀ x ∈ X ⇒ A_{ν}x_{ν} ⇀ Ax ∈ Y.

Obviously this type of convergence implies weak pointwise convergence. The reverse however is not true even for linear operators. Consider for example, the sequence of linear operators A_{ν}: L^{2}(0, 1) → L^{2}(0, 1), (A_{ν}x)(t) = x(t) sin νπt. It converges weakly pointwise to the zero operator, while does not converge in the weak continuous sense. To show the latter, take the sequence x_{ν}(t) = sin νπt (to be more precise, the restrictions of the above functions to the interval [0, 1]), and observe that A_{ν}x_{ν} does not converge weakly to zero in L^{2}(0, 1). One also notes that due to the statement of Kuratowski ([10], Chapter II,

§ 20, Section VII) from A_{ν}⇀^{C_{σ}}A follows the weak continuity of A (i. e. A maps any weakly convergent sequence into a weakly convergent one).

For linear operators it is obvious that for A_{ν}⇀^{C_{σ}}A in L(X, Y) it is sufficient that A′_{ν} → A′ strongly (pointwise) in L(Y′, X′). If, in addition, X is reflexive and Y is uniformly convex (in particular, Hilbert) space, then this condition is also necessary. To show this, assume

A_{ν}⇀^{C_{σ}}A in L(X, Y) and consider an arbitrary subsequence of a sequence of operators {A_{ν}} (for brevity we will use the same index ν). As A′_{ν}y′ ⇀ A′y′ for each y′ ∈ Y′, then ||A′y′||_{X′} ≤ lim inf_{ν} ||A′_{ν}y′||_{X′}. On the other hand, according to the Hahn-Banach theorem there is such a sequence {x_{ν}} ∈ X, that ||x_{ν}||_{X} = 1 and < x_{ν}, A′_{ν}y′ >= ||A′_{ν}y′||_{X′}. Extracting a weakly converging subsequence x_{ν_{μ}} ⇀ x, ||x||_{X} ≤ 1, and recalling that the continuous convergence of a sequence of operators implies the continuous convergence of any its subsequence to the same limit ([10], Chapter II, § 20, Section VII), one sees that ||A′_{ν_{μ}}y′||_{X′} → ⟨x, A′y′⟩ ≤ ||A′y′||_{X′} and hence A′_{ν_{μ}}y′ → A′y′. Therefore, the whole weakly convergent sequence {A′_{ν}y′} ⊂ X′ is compact and thus A′_{ν}y′ → A′y′. Now return to the particular case of inner superposition operators acting between two Orlicz spaces.

Proposition 5.9. Let both the Young functions P, Q and their Legendre transforms P′, Q′ satisfy the Δ_{2}-condition, while Φ(x) = P (Q^{−1}(x)) be the Young function with the Legendre transform Φ′. If the following requirements are satisfied

(i) ∃A > 0: ||\frac{dμ_{g^{ν}}}{dm} ||_{Φ′} ≤ A for all ν ∈ IN ∪ \{0\};

(ii) ∥\frac{dμ_{g^{ν}} (τ)}{dm} ∥_{P′} → ∥\frac{dμ_{g}(τ)}{dm} ∥_{P′} for each τ ∈ [0, 1];

(iii) meas E_{ν}(τ)ΔE(τ) → 0 when ν → ∞ for each τ ∈ [0, 1],

then S_{g^{ν}} ⇀^{C_{σ}}S_{g} in L(L_{n}^{P} , L_{n}^{Q}). Moreover, if L_{n}^{P} = L_{n}^{p}, L_{n}^{Q} = L_{n}^{q}, 1 < q ≤ p < +∞, then

also the reverse is true.

Proof. We will show that the adjoint operators S′_{g^{ν}}: L_{n}^{Q′} → L_{n}^{P′} converge strongly

(pointwise) to S′_{g}: L_{n}^{Q′} → L_{n}^{P′}. For this purpose it is enough to prove that

(I) The sequence S_{g^{ν}} has uniformly bounded norms;

(II) S′_{g^{ν}}y′ → S′_{g}y′ for all y ∈ X_{0} ⊂ L_{n}^{Q′}, where X_{0} is defined in the proof of the

proposition 5.5. Obviously (I) holds due to the fulfillment of the condition (i). Now note that the operators S_{g^{ν}} converge to S_{g} weakly pointwise due to the proposition 5.5, and thus

S′_{g^{ν}}χ_{[0,τ]}e_{j} = \frac{dμ_{g^{ν}}(τ)}{dm}e_{j} ⇀ \frac{dμ_{g}(τ)}{dm}e_{j} = S′_{g}χ_{[0,τ]}e_{j},

which together with (ii) implies (II). It is clear that the conditions (ii) and (iii) are also necessary for the weak continuous convergence of the operators S_{g^{ν}} to S_{g}. However, condition (i) is necessary only in the case of Lebesgue spaces, which can be proved with reference to theorem 1 of [12].

5.4. Strong continuous and uniform convergence

Definition 5.10. The sequence of maps A_{ν} ∈ F(X, Y) is said to converge strongly

continuously to A ∈ F(X, Y), A_{ν}⇀^{C}A, if {x_{ν}} ⊂ X, x_{ν} ⇀ x ∈ X ⇒ A_{ν}x_{ν} → Ax ∈ Y.

Of course, the strong continuous convergence implies all the other types of convergences discussed before. The reference to Kuratowski ([10], Chapter II, § 20, Section VII) also

shows that from A_{ν}⇀^{C}A follows the strong continuity of A (i. e. A maps any weakly convergent sequence into a norm convergent one). In particular, if X is reflexive, then A is necessarily compact and continuous. Furthermore, one can easily show (see again [10, p. 109]) that in the case of reflexive X the strong continuous convergence implies the uniform convergence. It has been shown (see Theorem 3.3 in § 1.3 of [9]) that an inner superposition operator acting between two Orlicz spaces cannot be compact and continuous unless it is identically zero. Hence one concludes that the sequence of inner superposition operators defined on a reflexive Orlicz space cannot converge strongly continuously to an inner superposition operator different from zero. Finally we point out that if the sequence {A_{ν}} is compact in totality (i. e. ∪_{ν}A_{ν}(B) is

compact in Y for any bounded B ∈ X), then A_{ν}⇀^{C_{σ}}A implies A_{ν}⇀^{C}A.

5.5. Condition T

In the sequel we need also the following property of the sequence of mappings {A_{ν}} ⊂ F(X, Y), closely related to the property of weak continuous convergence:

A_{ν}x_{ν} ⇀ y ∈ Y ⇒ ∃x ∈ X : x_{ν} ⇀ x.  (T)

The statement below concerns the verification of this condition for the sequence of operators {S_{g^{ν}}} defined by the expression (5) and acting in some Orlicz space.

Proposition 5.11. Assume that the Young function P and its Legendre transform P′ satisfy the Δ_{2}-condition. Let for the operator S_{g}: L_{n}^{P} → L_{n}^{P} and the sequence of operators S_{g^{ν}}: L_{n}^{P} → L_{n}^{P} the following conditions hold:

(i) There is a measurable function γ: g(E) → E satisfying γ(g(t)) = t for a.e. t ∈ E,

while

e ⊂ g(E), meas e = 0 ⇒ meas γ^{−1}(e) = 0;

(ii) There are numbers a, A such that for all ν ∈ IN a.e. on [0, 1] holds

0 < a ≤ \frac{dμ_{g^{ν}}}{dm} ≤ A, 0 < a ≤ S_{g} \frac{dμ_{g}}{dm} ≤ A;

(iii) Conditions (ii) and (iii) of proposition 5.9 hold. Then the sequence {S_{g^{ν}}} satisfies property (T).

Remark 5.12. The condition (i) is usually referred to as the ω-condition (see [9, pp. 42–44]).

Proof. Let S_{g^{ν}}x_{ν} ⇀ y ∈ L_{n}^{P} for some {x_{ν}} ⊂ L_{n}^{P}. Thus there is a constant C > 0 satisfying ||S_{g^{ν}}x_{ν}||_{P} ≤ C. It is easy to show that condition (ii) implies the uniform boundedness from below of the operator norms of S_{g^{ν}} and thus the uniform boundedness of the sequence {x_{ν}}. On the other hand, it is easy to show by the method used in [13], that due to the conditions (i) and (ii) the operator S_{g} is continuously invertible in L_{n}^{P}. Set then x = S_{g}^{−1}y.

According to the conditions S_{g^{ν}} ⇀^{C_{σ}}S_{g} in L_{n}^{P}. Furthermore, since {x_{ν}} is bounded and thus weakly precompact, (S_{g^{ν}} − S_{g})x_{ν} ⇀ 0. Minding that S_{g^{ν}}x_{ν} ⇀ S_{g}x, we obtain S_{g}x_{ν} ⇀ S_{g}x, which implies x_{ν} ⇀ x due to the continuous invertibility of S_{g}.

6. Γ-convergence for functionals with transformed argument

We start this section with a simple abstract statement which relates various types of convergence of mappings introduced in section 5 with the weak Γ^{−}-convergence of functionals with transformed argument. Let X, Y be reflexive Banach spaces with separable duals and consider the functionals I, I_{ν}: Y → ĪR. Recall that the linear operator A: X → Y is called correctly solvable, if ||Ax||_{Y} ≥ C||x||_{X} for all x ∈ X and for some constant C > 0.

Proposition 6.1. Assume that the sequence I_{ν} be equicoercive and equicontinuous (in norm), while I_{ν} → I Γ^{−}-weakly over Y. If the sequence of linear uniformly in ν correctly

solvable operators A_{ν}: X → Y with dense images in Y satisfies A_{ν}⇀^{C_{σ}}A in L(X, Y) and condition (T), then I_{ν}(A_{ν}(·)) → I(A(·)) Γ^{−}-weakly over X.

Remark 6.2. It will be clear from the proof, that if in addition every A_{ν} is surjective, then the statement is true without the requirement of equicontinuity of I_{ν}.

Proof. From proposition 3.2 follows that

y_{ν} ⇀ y in Y ⇒ I(y) ≤ lim inf_{ν} I_{ν}(y_{ν}),  (α)
∃{ȳ_{ν}} ⊂ Y, ȳ_{ν} ⇀ y in Y ⇒ I(y) = lim_{ν} I_{ν}(ȳ_{ν})  (β)

for every y ∈ Y. The weak continuous convergence of the operators A_{ν} to A and the property (α) imply

x_{ν} ⇀ x in X ⇒ I(Ax) ≤ lim inf_{ν} I_{ν}(Ax_{ν}).  (α_{1})

On the other hand, given x ∈ X, according to (β) there is a sequence y_{ν} ⇀ Ax satisfying I(Ax) = lim_{ν} I_{ν}(y_{ν}). Using the density of the images of each A_{ν} in Y, one finds a sequence {x̄_{ν}} ⊂ X, such that ||A_{ν}x̄_{ν} −y_{ν}||_{Y} → 0 when ν → ∞. Obviously A_{ν}x̄_{ν} ⇀ Ax. Moreover, the estimate

|I(Ax) − I_{ν}(A_{ν}x̄_{ν})| ≤ |I(Ax) − I_{ν}(y_{ν})| + |I_{ν}(A_{ν}x̄_{ν}) − I_{ν}(y_{ν})|

together with condition (T) provide that x̄_{ν} ⇀ x and

I(Ax) = lim_{ν} I_{ν}(A_{ν}x̄_{ν}).  (β_{1})

To verify the equicoercivity of I_{ν}(A_{ν}(·)), it is enough to check according to proposition 7.7 of [2] that the sublevels of the latter are uniformly bounded in X. This can be observed as an immediate consequence of the uniform correct solvability of A_{ν}. To conclude the proof, it remains to refer to proposition 3.2.

One should point out that the conditions on the sequence {A_{ν}} used in the above result are in general not necessary for the Γ^{−}-convergence of the respective functionals with transformed argument. To illustrate this consider a particular example of a functional J and the sequence of its perturbations J_{ν}

J(x) = ∫_{0}^{1} (S_{g}x)^{2} (t) dt, J_{ν}(x) = ∫_{0}^{1} (S_{g^{ν}}x)^{2} (t) dt,

where x ∈ L^{2}(0, 1), S_{g^{ν}} and S_{g} are the inner superposition operators as introduced by the relationships (5) and (6) respectively. Changing the variable under the integration sign and applying proposition 3.5 with lemmata 4.1 (representation of Fenchel conjugate) and 4.2 (pointwise convergence of the conjugates) one obtains that for J_{ν} to converge Γ^{−}-weakly to J on L^{2}(0, 1) it is sufficient that

∃a : \frac{dμ_{g^{ν}}}{dm} ≥ a > 0 and (\frac{dμ_{g^{ν}}}{dm} )^{−1} ⇀ (\frac{dμ_{g}}{dm} )^{−1} weakly in L^{1}(0, 1)  (7)

which is certainly much less than required by proposition 6.1.

6.1. Functionals with deviating argument

Now return to the original problem (V_{∞}) and its perturbations (V_{ν}). The Legendre transforms of f_{ν}, f in the last variables are given by the expressions

f′(t, u, v′) := sup_{v∈IR^{nl}}(vv′ − f(t, u, v)),
f′_{ν}(t, u, v′) := sup_{v∈IR^{nl}}(vv′ − f_{ν}(t, u, v)),

Introduce the following set of conditions. (P0) The Young functions P, Q and their Legendre transforms P′, Q′ satisfy the Δ_{2}-

condition. (Q1) f, f_{ν}: [0, 1] × IR^{nk} × IR^{nl} → ĪR are Carathéodory functions, such that f_{ν}(t, u, ·),

f(t, u, ·) are convex, while for a.e. t ∈ [0, 1], for all (u, v) ∈ IR^{nk} × IR^{nl}, v′ ∈ IR^{nl} and for some v̄ ∈ IR^{nl}, a_{j} ≥ 0, j = 1, …, l, ∑_{j=1}^{l} a_{j}^{2} > 0, b ∈ L^{1}(0, 1) holds

\begin{aligned}f_{ν}(t, u, v_{1}, …, v_{l}) & ≥ ∑_{j=1}^{l} a_{j}P (|v_{j}|) + b(t), \\ f_{ν}(t, u, v̄) & ≤ c(t, u), \end{aligned}

where c: [0, 1] × IR^{nk} → ĪR is some Carathéodory function, c(·, u) ∈ L^{1}(0, 1). (Q2) For a.e. t ∈ [0, 1], for each {u_{1}, u_{2}} ⊂ IR^{nk}, v ∈ IR^{nl} holds

|f_{ν}(t, u_{1}, v) − f_{ν}(t, u_{2}, v)| ≤ ω(t, |u_{1} − u_{2}|_{nk}),

M. Drakhlin, E. Stepanov / Γ-convergence for a class of functionals ... 83

where ω: [0, 1] × [0, +∞) → [0, +∞) is a Carathéodory function nondecreasing in the second variable, with ω(t, 0) ≡ 0, ω(·, u) ∈ L^{1}(0, 1) for each u > 0.

(Q3) For any u ∈ L_{nk}^{Q} the functionals

J_{ν}(v) = ∫_{0}^{1} f_{ν}(t, u(t), v(t)) dt and J(v) = ∫_{0}^{1} f(t, u(t), v(t)) dt

are proper in the space L_{nl}^{P}; (Q4) f′_{ν}(·, u, v′) ⇀ f′(·, u, v′) weakly in L^{1}(0, 1) for all {u, v′} ∈ IR^{nk} × IR^{nl}. In the sequel we denote for the sake of brevity h_{i}^{0} := g_{i}, g_{j}^{0} := g_{j}. The notations below refer to ν ∈ IN∪{0}. Define the inner superposition operators T_{g_{i}^{ν}}: C_{n} → L_{n}^{Q}, i = 1, …, k and S_{g_{j}^{ν}}: L_{n}^{P} → L_{n}^{P}, j = 1, …, l by the relationships

(T_{h_{i}^{ν}}x)(t) = \{ \begin{matrix}x(h_{i}^{ν}(t)), & h_{i}^{ν}(t) ∈ [0, 1], \\ 0, & h_{i}^{ν}(t) ∉ [0, 1], \end{matrix} (S_{g_{j}^{ν}}x)(t) = \{ \begin{matrix}x(g_{j}^{ν}(t)), & g_{j}^{ν}(t) ∈ [0, 1], \\ 0, & g_{j}^{ν}(t) ∉ [0, 1]. \end{matrix}

Also denote

T_{h}^{ν} : x ∈ C_{n} ↦ (T_{h_{1}^{ν}}x, …, T_{h_{k}^{ν}}x) ∈ L_{nk}^{Q}, S_{g}^{ν} : x ∈ L_{n}^{P} ↦ (S_{g_{1}^{ν}}x, …, S_{g_{l}^{ν}}x) ∈ L_{nl}^{P}.

Let for each τ ∈ [0, 1], i = 1, …, k, j = 1, …, l,

E_{i}^{ν} := (h_{i}^{ν})^{−1}([0, 1]), E_{j}^{ν}(τ) := (g_{j}^{ν})^{−1} ([0, τ]).

Again we omit the reference to τ when τ = 1 and to ν when ν = 0. For the above-introduced operators consider the following assumptions. (R1) For any j = 1, …, l the functions g_{j}, g_{j}^{ν}: [0, 1] → IR∪{±∞} are almost everywhere

finite and measurable, while

∃(a, A) : 0 < a ≤ \frac{dμ_{g_{j}^{ν}}}{dm} (t) ≤ A a.e. on [0, 1], ν ∈ IN ∪ \{0\} and

∥\frac{dμ_{g_{j}^{ν}}(τ)}{dm}∥_{P′} → ∥\frac{dμ_{g_{j}}(τ)}{dm}∥_{P′} as ν → ∞.

(R2) meas E_{j}^{ν} = 1, while there are measurable functions γ_{j}^{ν}: IR → [0, 1], j = 1, …, l,

ν ∈ IN ∪ {0} such that e ⊂ [0, 1], meas e = 0 implies meas (γ_{j}^{ν})^{−1} (e) = 0, and

γ_{j}^{ν}(g_{j}^{ν}(t)) = t for a.e. t ∈ E_{j}^{ν}.

(R3) For all i = 1, …, k the functions h_{i}, h_{i}^{ν}: [0, 1] → IR ∪ {±∞} are almost everywhere finite and measurable, while h_{i}^{ν} → h_{i} in measure on E_{i}.

(R4) meas E_{i}^{ν}ΔE_{i} → 0 for each i = 1, …, k and meas E_{j}^{ν}(τ)ΔE_{j}(τ) → 0 for any τ ∈ [0, 1], j = 1, …, l.

Theorem 6.3. Consider the functionals I, I_{ν}: AC_{n}^{P}^{∘} → ĪR defined by the expressions (V_{∞}) and (V_{ν}) respectively. Under the assumptions (P0), (Q1)–(Q4), (R1)–(R4) the fol-

lowing statement holds: if {x_{ν}} ⊂AC_{n}^{P}^{∘} is such that

lim_{ν} I_{ν}(x_{ν}) = lim_{ν}( inf_{AC_{n}^{P}^{∘}} I_{ν}).

(in particular, when each x_{ν} is the point of global minimum of the respective I_{ν} over AC_{n}^{P}^{∘}) and x_{ν} ⇀ x, then

lim_{ν}( inf_{AC_{n}^{P}^{∘}} I_{ν}) = min_{AC_{n}^{P}^{∘}} I = I(x).

Remarks 6.4.

1. If the sequence {x_{ν}} is bounded, then it contains a weakly convergent subsequence {x_{ν_{μ}}} for which the statement of the theorem is valid.

2. The statement is valid in particular in AC_{n}^{p}^{∘}, 1 < p < ∞.

Proof. Step 1. Consider the functionals G_{ν}, G: L_{nk}^{Q} × L_{nk}^{P} → ĪR defined by the formulae

G_{ν}(u, v) = ∫_{0}^{1} f_{ν}(t, u(t), v(t)) dt and G(u, v) = ∫_{0}^{1} f(t, u(t), v(t)) dt.

We will prove (see, for instance, the analogous results of G. Buttazzo and G. Dal Maso [3]) that due to conditions (P0), (Q1)-(Q4) from u_{ν} → u in L_{nk}^{Q} follows

v_{ν} ⇀ v in L_{nl}^{P} ⇒ G(u, v) ≤ lim inf_{ν} G_{ν}(u_{ν}, v_{ν});  (A)
∃{v_{ν}} ⊂ L_{nl}^{P} such that v_{ν} ⇀ v ∈ L_{nl}^{P} and G(u, v) = lim_{ν} G_{ν}(u_{ν}, v_{ν}),  (B)

where in (B) the the sequence {v_{ν}} is independent of the choice of {u_{ν}}. These properties will be proved below in step 4. Step 2. Turn now to the original functionals I_{ν}, I and prove properties (1), (2) of propo-

sition 3.2. Suppose x_{ν} ⇀ x in AC_{n}^{P}^{∘}. Let then

v_{ν} := S_{g}^{ν} ẋ_{ν}, v := S_{g}ẋ, u_{ν} := T_{h}^{ν}x_{ν}, u := T_{h}x.

From conditions (R1)-(R4) and proposition 5.9 follows that v_{ν} ⇀ v in L_{nl}^{P} and u_{ν} → u

in L_{nk}^{Q}, and hence (A) implies that I(x) ≤ lim inf_{ν} I_{ν}(x_{ν}), proving (1).

On the other hand, proposition 5.11 provides the validity of condition (T) for the sequence of operators {S_{g}^{ν}}. Therefore, if a sequence {S_{g}^{ν} ẋ_{ν}} is weakly convergent in L_{nl}^{P}, then

so is {ẋ_{ν}} in L_{n}^{P}, and thus {x_{ν}} is weakly convergent in AC_{n}^{P}^{∘}. Conditions (R1) and (R2) provide the continuous invertibility of S_{g}^{ν}, S_{g}. Hence (B) implies that there is a sequence

S_{g}^{ν} ẋ_{ν} = v_{ν} ⇀ v_{0} in L_{nl}^{P} such that given T_{h}^{ν}x_{ν} = u_{ν} → u_{0} in L_{nk}^{Q}, the relationship holds

G(u_{0}, v_{0}) = lim_{ν} I_{ν}(x_{ν}).

Observing by our previous discussion that x_{ν} ⇀ x in AC_{n}^{P}^{∘}, we obtain v_{0} = S_{g}ẋ, u_{0} = T_{h}x and I(x) = lim_{ν} I_{ν}(x_{ν}), which proves (2). Step 3. At this moment we are going to show that I = Γ^{−}(w) lim_{ν} I_{ν}, where w stands for

the weak topology of AC_{n}^{P}^{∘}. For this purpose it is enough to prove that the sequence {I_{ν}} is equicoercive and refer to proposition 3.2. To prove equicoercivity, use conditions (Q1) and (R1), observing that

I_{ν}(x) ≥ b_{1} + ∑_{j=1}^{l} a_{j} ∫_{0}^{1} P (|(S_{g_{j}^{ν}}x) (t)|) dt ≥ b_{1} + a ∑_{j=1}^{l} a_{j} ∫_{0}^{1} P (|x(t)|) dt ≥ I_{0}(||x||_{P}),

where b_{1} ∈ IR and I_{0}(x) → +∞ for x → +∞. At last, having established that I is the (weak) Γ^{−}-limit of the sequence {I_{ν}}, it remains only to conclude the proof referring to proposition 3.3. Step 4. We prove (A) and (B). Let u_{ν} → u in L_{nk}^{Q} and consider the functionals J_{ν}, J: L_{nl}^{P} → ĪR introduced by the formulae

J_{ν}(v) := G_{ν}(u, v), J(v) := G(u, v).

Proposition 3.5 with the help of lemmata 4.1 and 4.2 implies that J_{ν} → J Γ^{−}-weakly over L_{nl}^{P}. This according to proposition 3.2 provides the following two conditions:

(A′) If v_{ν} ⇀ v in L_{nl}^{P}, then J(u) = G(u, v) ≤ lim inf_{ν} J_{ν}(v_{ν}). (B′) There is a sequence {v_{ν}} ⊂ L_{nl}^{P}, and v ∈ L_{nl}^{P} satisfying v_{ν} ⇀ v, such that J_{ν}(v_{ν}) →

J(v). To prove (A), use (A′) and the lower semicontinuity of each G_{ν}(·, v_{ν}), concluding

G(u, v) ≤ lim inf_{ν} lim inf_{μ} G_{ν}(u_{μ}, v_{ν}) ≤ lim inf_{ν} G_{ν}(u_{ν}, v_{ν}).

Property (B) is proved by (B′) and the estimate

|G(u, v) − G_{ν}(u_{ν}, v_{ν})| ≤ |J(v) − J_{ν}(v_{ν})| + ∫_{0}^{1} ω(t, |u(t) − u_{ν}(t)|_{nl}) dt.

6.2. Nonlocal perturbations of the classical functional

In this section we apply theorem 6.3 to the analysis of the typical situation when the unperturbed functional (V_{∞}) has the classical form, that is, does not involve the argument deviation:

I(x) = ∫_{0}^{1} f̃(t, x(t), ẋ(t)) dt,  (Ṽ_{∞})

where f̃(t, x(t), ẋ(t)) := f(t, x(t), …, x(t), ẋ(t), …, ẋ(t)). In this case the Euler equations written out for the functional (Ṽ_{∞}) are the ordinary differential equations, as it is usual for such well studied variational problems. Thus one may formulate in other words that the purpose of this section is to analyze the stability of solutions to the classical problem of the calculus of variations with respect to the nonlocal perturbations. It is particularly interesting to note that such a setting can be regarded as the “dual” to the well-known situations, when the homogenization of the local problems leads to a nonlocal one (see for instance [14, 15, 16] and references therein). Replace the conditions (R1)–(R4) by the following ones:

(R1′) For any j = 1, …, l the functions g_{j}^{ν}: [0, 1] → IR ∪ {±∞} are almost everywhere finite and measurable, while

∃(a, A) : 0 < a ≤ \frac{dμ_{g_{j}^{ν}}}{dm} (t) ≤ A a.e. on [0, 1], ν ∈ IN ∪ \{0\} and

∥\frac{dμ_{g_{j}^{ν}}(τ)}{dm}∥_{P′} → τP^{−1}(1/τ) as ν → ∞.

(R2′) meas E_{j}^{ν} = 1 and there are measurable functions γ_{j}^{ν}: IR → [0, 1], j = 1, …, l,

ν ∈ IN such that e ⊂ [0, 1], meas e = 0 implies meas (γ_{j}^{ν})^{−1} (e) = 0, and
γ_{j}^{ν}(g_{j}^{ν}(t)) = t for a.e. t ∈ E_{j}^{ν}.

(R3′) For all i = 1, …, k the functions h_{i}^{ν}: [0, 1] → IR ∪ {±∞} are almost everywhere

finite and measurable, while h_{i}^{ν} → h, h(t) ≡ t, in measure. (R4′) meas E_{i}^{ν} → 1 for each i = 1, …, k and meas E_{j}^{ν}(τ)Δ[0, τ] → 0 for any τ ∈ [0, 1],

j = 1, …, l. Now we claim the following direct corollary from theorem 6.3.

Theorem 6.5. Consider the functionals I, I_{ν}: AC_{n}^{P} → ĪR defined by the expressions (Ṽ_{∞}) and (V_{ν}) respectively. Under the assumptions (P0), (Q1)–(Q4), (R1′)–(R4′) the assertion of theorem 6.3 remains valid.

6.3. Functionals with impulsive constraints

In this section we show how the technique developed can be used to tackle the problems of Γ^{−}-convergence for a sequence of functionals defined on some space different from the the standard space of absolutely continuous functions. Such problems most often arise in applications from optimal control theory. In particular, here we deal with an important example of Γ^{−}-convergence of a sequence of functionals with deviating argument subject to impulsive constraints, which very often appears when analyzing continuous dependence on parameters of optimal solutions to controlled impulsive differential equations. To apply the general scheme used in the proof of theorem 6.3 we construct some special space of piecewise absolutely continuous functions with prescribed behavior at discontinuity points. Let the original functional (V_{∞}) be restricted by the system of impulsive constraints:

Δx(t_{i}) ≡ x(t_{i}) − x(t_{i} − 0) = ζ_{i},  i = 1, …, m,  (C_{∞})

where the points of discontinuity {t_{i}}_{i=1}^{m}, 0 = t_{0} < t_{1} < … < t_{m} < t_{m+1} = 1 are fixed, while ζ_{i} ∈ IR^{n} are given vectors. Introduce the space P AC_{n}^{P} [{t_{i}}_{i=1}^{m}] (for short, P AC_{n}^{P}) isomorphic to L_{n}^{P} × IR^{n+mn} by the relationship

J : (z, β) ∈ L_{n}^{P} × IR^{n+mn} ↦ x = Λz + Dβ := ∫_{0}^{t} z(τ) dτ + ω(t)β ∈ P AC_{n}^{P} , (8)

where ω(t) is the n × (n + mn) matrix

ω(t) = (E_{n}, χ_{[t_{1},1]}E_{n}, ⋯, χ_{[t_{m},1]}E_{n}) ,

E_{n} is the unit n × n matrix, χ_{A}(t) is the characteristic function of the set A ⊂ [0, 1]. The inverse of (8) is determined by the equality

J^{−1} : x ∈ P AC_{n}^{P} ↦ (z, β) = (δx, ρx) := (ẋ, Δx) ∈ L_{n}^{P} × IR^{n+mn},  (9)

where Δx := col{x(0), Δx(t_{1}), …, Δx(t_{m})}. Thus any element of P AC_{n}^{P} is uniquely representable in the form

x(t) = ∫_{0}^{t} ẋ(τ) dτ + x(0) + ∑_{i=1}^{m} χ_{[t_{i},1]}Δx(t_{i}).

Introducing the norm ||x||_{P AC} := ||ẋ||_{P} + |Δx|_{n+mn} one turns P AC_{n}^{P} into a Banach space. The isomorphism J given by the expressions (8) and (9) was first suggested by A. Anokhin (see [9, § 6.3]). One easily observes that P AC_{n}^{P} is exactly the desired space of piecewise absolutely continuous vector functions x: [0, 1] → IR^{n} continuous from the right at the given points {t_{i}}_{i=1}^{m} and satisfying the impulsive conditions (C_{∞}) at these points. It is reasonable to expect that the perturbations of the original problem affect also the set of constraints, and therefore, we have to deal with the sequence of perturbed variational problems (V_{ν}) restricted by additional conditions

Δx(t_{i}^{ν}) ≡ x(t_{i}^{ν}) − x(t_{i}^{ν} − 0) = ζ_{i}^{ν},  i = 1, …, m,  (C_{ν})

where again the points of discontinuity are ordered: 0 = t_{0}^{ν} < t_{1}^{ν} < … < t_{m}^{ν} < t_{m+1}^{ν} = 1. Reiterating the above construction we obtain for each functional I_{ν} the respective space of piecewise absolutely continuous vector functions P AC_{n}^{P} [{t_{i}^{ν}}_{i=1}^{m}] (for short, P AC_{n}^{P} (ν)) isometrically isomorphic to L_{n}^{P} × IR^{n+mn}. These spaces are however different and so to study the problem of stability of minimizers of (V_{∞}) subject to (C_{∞}) as the problem of Γ-convergence in P AC_{n}^{P} one needs extra constructions. For this purpose denote

r^{ν}(t) = ∑_{i=0}^{m} (t_{i} + \frac{t_{i+1} − t_{i}}{t_{i+1}^{ν} − t_{i}^{ν}} (t − t_{i}^{ν})) χ_{[t_{i}^{ν},t_{i+1}^{ν}]}(t), t ∈ [0, 1].

It is easy to verify that the inner superposition operator S_{r^{ν}}: L_{n}^{P} → L_{n}^{P} is continuous and continuously invertible, while S_{r^{ν}}x ∈ P AC_{n}^{P} whenever x ∈ P AC_{n}^{P} (ν).

Definition 6.6. The sequence of functions x_{ν} ∈ P AC_{n}^{p}(ν) is said to be weakly P-

convergent to x ∈ P AC_{n}^{P}, x_{ν}⇀^{P}x, if S_{r^{ν}}x_{ν} ⇀ x in P AC_{n}^{P}.

Denote by \overline{P AC}_{n}^{P}, \overline{P AC}_{n}^{P} (ν) the subspaces of P AC_{n}^{P}, P AC_{n}^{P} (ν) respectively, consisting of the functions satisfying the constraints x(0) = x(1) = 0. One can now claim the following extension of theorem 6.3 to the case of functionals with impulsive constraints.

Theorem 6.7. Let t_{i}^{ν} → t_{i} and ζ_{i}^{ν} → ζ_{i} when ν → ∞. Then under the conditions (P0),

(Q1)–(Q4), (R1)–(R4) if \{x_{ν}\} ⊂ \overline{P AC}_{n}^{P} (ν) is such that

lim_{ν} I_{ν}(x_{ν}) = lim_{ν}( inf_{\overline{P AC}_{n}^{P} (ν)} I_{ν}).

(in particular, when each x_{ν} is the point of global minimum of the respective I_{ν} over

\overline{P AC}_{n}^{P} (ν)) and x_{ν}⇀^{P}x, then

lim_{ν}( inf_{\overline{P AC}_{n}^{P} (ν)} I_{ν}) = min_{\overline{P AC}_{n}^{P}} I = I(x).

Acknowledgements. We are grateful to professor G. Buttazzo (University of Pisa, Italy) for fruitful discussions. The first author was financially supported in part by the Israeli Ministry of Science and Arts, the second by the German Mathematical Society.

References

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