����������� ������ 49 (1997), 207–214

UDK 517.983.5

���������� ������ ��� research paper

ON OPERATORS IN BOCHNER SPACES

Nina A. Yerzakova

Abstract. Estimates for the measure of noncompactness of bounded subsets of spaces of (Bochner-) integrable functions are obtained, a new class of condensing operators is discussed, and the solvability of a certain operator equation in a Hilbert space is proved.

In this paper we discuss a new class of condensing operators, and we prove the solvability of a certain operator equation. An extension of some results from [8] is obtained.

Let us recall some definitions. The measure of noncompactness β(U) = β_{E}(U)

[1] of a bounded set U in a normed space E is defined as the supremum of all numbers r > 0 such that there exists a sequence {u_{n}} in U with ∥u_{n} − u_{m}∥ ≥ r for every n ≠ m. Given two Banach spaces G and E, a continuous operator S: G → E is called β-condensing if

β_{E}(SU) < β_{G}(U)

for every bounded U ⊂ G with noncompact closure.

There exists a large amount of literature devoted to measure of noncompactness and condensing operators (see, for example, [1,2,4, 6–8]).

Let Ω be a domain in R^{n}. Let E be a regular space of μ-measurable functions on a domain Ω; here regularity means that every element in E has an absolutely continuous norm. Let P_{D} denote the operator of multiplication by the characteristic function χ_{D} of a measurable subset D ⊂ Ω, i.e. P_{D}u = χ_{D}u. For bounded U ⊂ E put

ν(U) = ν_{E}(U) = \overline{lim}_{μ(D)→0} sup_{u∈U} ∥P_{D}u∥_{E},

for U ⊂ E.

AMS Subject Classification: 47 H 09, 46 E 35, 47 B 07, 47 B 38

Keywords and phrases: Hilbert space, regular space, Lebesgue space, Bochner space, Sobolev space, embedding operator, measure of noncompactness

Communicated at the 4th Symposium on Mathematical Analysis and Its Applications, Aran - delovac 1997.

Grant RFBR 97-01-10530

The measure ν has been studied in [2,6]. In particular, it was shown in [6] (see also [1]) that

β(U) = 2^{1/2}ν(U)  (1)

for every μ-compact (i.e., compact in measure) subset U of a separable Hilbert space E.

Let Δ be a bounded interval on the real axis and E some Banach space. For 1 ≤ p < ∞, we denote by L_{p}(0, T; E) the set of all Bochner-measurable functions with the property that the function t ↦ ∥u(t)∥_{E} belongs to L_{p}(0, T).

For any partition Δ = D_{1} ∪ ⋯ ∪ D_{l} of Δ into Lebesgue-measurable disjoint subsets D_{i}, we denote by Ṽ the set of all functions

ũ(t) = ∑_{i=1}^{l} b_{i}χ_{D_{i}}(t),

where χ_{D_{i}} is the characteristic function of D_{i} as above, and b_{i} are elements from E (1 ≤ i ≤ l).

Lemma 1. Let Ũ ⊆ Ṽ be bounded in L_{p}(Δ; E). For arbitrary t_{0} ∈ Δ, let

Ũ(t_{0}) =^{def} {ũ(t_{0}) : ũ ∈ Ũ}.

Then the function β_{E}(Ũ(t)) is simple, i.e.

β_{E}(Ũ(t)) = ∑_{i=1}^{l} a_{i}χ_{D_{i}}(t),

and

β_{L_{p}(Δ;E)}(Ũ) ≤ (∫_{Δ} β_{E}^{p}(Ũ(t)) dt)^{1/p}.

Proof. The proof of this assertion is analogous to the proof of Lemma 2.1 from [8].

We denote by U some subset of L_{p}(Δ; E) which allows an ϵ-approximation, for every ϵ > 0, through a set

Ũ_{ϵ} =^{def} {ũ : ũ(t) = ∑_{i=1}^{l_{ϵ}} b_{i}χ_{D_{i}}(t) (b_{i} ∈ E)}.

More precisely, we require that

ρ_{E}(U(t), Ũ_{ϵ}(t)) ≤ k_{1}ϵ  (2)

for almost all t ∈ Δ, where ρ_{E} denotes the Hausdorff distance in E, the constant k_{1} > 0 is independent of ϵ, but the integer l_{ϵ} < ∞ may depend on ϵ.

Theorem 1. Let U be a bounded set in L_{p}(Δ; E) which allows an ϵapproximation (2) for every ϵ. Then

β_{L_{p}(Δ;E)}(U) ≤ ∥β_{E}(U)∥_{L_{p}(Δ)}.

Proof. The proof of this assertion is analogous to the proof of Theorem 2.1 from [8].

Let H be some Hilbert space. As usual, we identify H with its conjugate space H^{∗}. Let W^{1,2}(Δ; H) = W^{1,2}(b, d; H) (Δ = (b, d)) for some −∞ < b < d < ∞ denote the space of all functions u: Δ → H such that both u and u′_{t} belong to L_{2}(Δ; H), equipped with the norm

∥u∥_{W^{1,2}(Δ;H)} = ∥u∥_{L_{2}(Δ;H)} + ∥u′_{t}∥_{L_{2}(Δ;H)}.

By Lemma 1.11 from [3], W^{1,2}(Δ; H) is embedded in C(\overline{Δ}; H), i.e.,

∥u∥_{C(\overline{Δ};H)} ≤ c∥u∥_{W^{1,2}(Δ;H)}.  (3)

Let W_{0}^{1,2}(b, d; H) be the subspace of all functions u ∈ W^{1,2}(b, d; H) such that u(b) = u(d) = 0 (0 is zero of H). In W_{0}^{1,2}(b, d; H) we have

∥u∥_{L_{2}(b,d;H)} ≤ k(∫_{b}^{d} ∥u′(t)∥_{H}^{2} dt)^{1/2} = k∥u′_{t}∥_{L_{2}(b,d;H)}.  (4)

Finally, for u, v ∈ W_{0}^{1,2}(b, d; H) we put

∥u∥_{1,2,0} = (∫_{b}^{d} ∥u′_{t}∥_{H}^{2} dt)^{1/2}.

Some notations are in order.

Let Δ ⊆ (b, d) be an interval, \overline{Δ} its closure, |Δ| its length, Δ_{δ} the δneighbourhood of Δ, u_{Δ} an approximation of a function u on Δ, u_{δ} ∈ W_{0}^{1,2}(Δ_{δ}; H) an extension of u ∈ W^{1,2}(Δ; H) preserving the norm in C(\overline{Δ}; H) for δ = |Δ|/2, and U_{δ} the set of all extensions u_{δ} of functions u ∈ U ⊂ W^{1,2}(Δ; H).

Let L_{2}(Ω) denote the Lebesgue space and W^{1,2}(Ω) the Sobolev space. We shall now consider two particular cases of H, namely H_{1} = L_{2} (Ω) and H_{2} = W^{1,2} (Ω), here Ω is a domain in R^{n} of finite measure but, in general, with irregular boundary. In both cases the space W^{1,2}(0, T; H_{i}) (i = 1, 2) consists of all functions (t, x) ↦ u(t, x) such that u(t, .), u′_{t}(t, .) ∈ H_{i} for each t ∈ Δ.

Lemma 2. Let f ∈ L_{1}(Δ; H_{2}), and let Ψ: H_{i} ↦ H_{i} (i = 1, 2) be an operator satisfying the inequality

∥Ψ (ϕ)∥_{H_{i}} ≤ c_{1} + ∑_{j=1}^{J} c̆_{j} ∥ϕ∥_{H_{i}}^{α_{i,j}}  (5)

for all ϕ ∈ H_{i}, where c_{1} ≥ 0, c̆_{j} ≥ 0, and α_{i,j} > 1 are real constants which may depended on H_{i}. Then there exist operators F_{Δ,i}: W^{1,2}(Δ; H_{i}) → W^{1,2}(Δ; H_{i}), such that the equality

∫_{Δ}⟨(F_{Δ,i}u)′_{t}(t), v′_{t}(t)⟩_{H_{i}} dt = ∫_{Δ}⟨f(t) − Ψ(u(t)), v(t)⟩_{H_{i}} dt

is true for arbitrary functions v ∈ W_{0}^{1,2}(Δ; H_{i}). Moreover,

∥F_{Δ,i}u∥_{C(\overline{Δ};H_{i})} ≤ c_{0}∥f∥_{L_{1}(Δ;H_{2})} + č_{1}|Δ|^{1/2} + |Δ|^{1/2} ∑_{j=1}^{J} c̃_{j}∥u∥_{C(\overline{Δ};H_{i})}^{α_{i,j}}, (6)

∥F_{Δ,i}u∥_{W^{1,2}(Δ;H_{i})} ≤ c_{0}∥f∥_{L_{1}(Δ;H_{2})} + č_{1}|Δ|^{1/2} + |Δ|^{1/2} ∑_{j=1}^{J} ˜ c̃_{j}∥u∥_{W^{1,2}(Δ;H_{i})}^{α_{i,j}}, (7)

and

β_{W^{1,2}(Δ;H_{i})}(F_{Δ,i}U) ≤ c_{3}β_{L_{2}(Δ_{δ};H_{i})}(Ψ(U_{δ}))(U ⊂ W^{1,2}(Δ; H_{i})),  (8)

where the constants c_{0}, č_{1}, c̃_{j}, ˜ c̃_{j}, and c_{3} are independent of u, U, Δ and δ.

Proof. Let u_{δ}, v_{δ} ∈ W_{0}^{1,2}(Δ_{δ}; H_{i}). The estimates

∥Ψ(u_{δ})∥_{L_{2}(Δ_{δ};H_{i})} ≤ ∥c_{1} + ∑_{j=1}^{J} c̆_{j} ∥u_{δ}∥_{H_{i}}^{α_{i,j}} ∥_{L_{2}(Δ_{δ};H_{i})} ≤

≤ (c_{1} + ∑_{j=1}^{J} c̆_{j} ∥u∥_{C(\overline{Δ};H_{i})}^{α_{i,j}})(2|Δ|)^{1/2} ≤ (c_{1} + c^{α_{i,j}} ∑_{j=1}^{J} c̆_{j} ∥u∥_{W^{1,2}(Δ;H_{i})}^{α_{i,j}}) (2|Δ|)^{1/2} (9)

can easily be deduced from assumptions (5) on Ψ and the embedding (3). Putting f_{δ}(t) = P_{Δ}f(t) we have

∫_{Δ_{δ}} |⟨f_{δ}(t), v_{δ}(t)⟩_{H_{i}}| dt = ∫_{Δ} |⟨f(t), v_{δ}(t)⟩_{H_{i}}| dt ≤ ∥v_{δ}∥_{C(\overline{Δ};H_{i})}∥f∥_{L_{1}(Δ;H_{2})}

≤ c∥v_{δ}∥_{W^{1,2}(Δ;H_{i})}∥f∥_{L_{1}(Δ;H_{2})} ≤ c(k + 1)∥v_{δ}∥_{W_{0}^{1,2}(Δ_{δ};H_{i})}∥f∥_{L_{1}(Δ;H_{2})}.

This shows that the linear functional

R (v_{δ}) = ∫_{Δ_{δ}}⟨f_{δ}(t) − Ψ(u(t)), v_{δ}(t)⟩_{H_{i}} dt

is bounded in module for all v_{δ} ∈ W_{0}^{1,2}(Δ_{δ}; H_{i}). By the Riesz representation theorem, there exists a bounded (generally speaking, nonlinear) operator F_{δ,i}: W_{0}^{1,2}(Δ_{δ}; H_{i}) → W_{0}^{1,2}(Δ_{δ}; H_{i}) such that

⟨F_{δ,i}u_{δ}, v_{δ}⟩_{1,2,0} = ∫_{Δ_{δ}}⟨(F_{δ,i}u_{δ})′_{t}(t), (v_{δ})′_{t}(t)⟩_{H_{i}} dt
= ∫_{Δ_{δ}}⟨f_{δ}(t) − Ψ(u_{δ}(t)), v_{δ}(t)⟩_{H_{i}} dt

≤ (c(k + 1)∥f∥_{L_{1}(Δ;H_{2})} + k∥Ψ(u_{δ})∥_{L_{2}(Δ_{δ};H_{i})})∥v_{δ}∥_{W_{0}^{1,2}(Δ_{δ};H_{i}).} Putting in the last equality v_{δ} = F_{δ,i}u_{δ}, we conclude that

∥F_{δ,i}u_{δ}∥_{W_{0}^{1,2}(Δ_{δ};H_{i})} ≤ c(k + 1)∥f(t)∥_{L_{1}(Δ;H_{2})} + k∥Ψ(u_{δ})∥_{L_{2}(Δ_{δ};H_{i})}.  (10)

We define an operator (F_{Δ,i}u) as approximation of F_{δ,i}u_{δ} on Δ. Taking into consideration (3), (4), (9), and (10), we obtain then (6) and (7), since

∥F_{Δ,i}u∥_{C(\overline{Δ};H_{i})} ≤ c∥F_{Δ,i}u∥_{W^{1,2}(Δ;H_{i})} ≤ c(k + 1)∥F_{δ,i}u_{δ}∥_{W_{0}^{1,2}(Δ_{δ};H_{i})}

≤ c^{2}(k + 1)^{2}∥f∥_{L_{1}(Δ;H_{2})} + ck(k + 1)∥Ψ(u_{δ})∥_{L_{2}(Δ_{δ};H_{i})}

≤ c^{2}(k + 1)^{2}∥f∥_{L_{1}(Δ;H_{2})} + ck(k + 1)(c_{1} + ∑_{j=1}^{J} c̆_{j} ∥u∥_{C(\overline{Δ};H_{i})}^{α_{i,j}})(2|Δ|)^{1/2}

≤ c^{2}(k + 1)^{2}∥f∥_{L_{1}(Δ;H_{2})} + ck(k + 1)(c_{1} + c^{α_{i,j}} ∑_{j=1}^{J} c̆_{j} ∥u∥_{W^{1,2}(Δ;H_{i})}^{α_{i,j}}) (2|Δ|)^{1/2} .

The inequality

∥F_{δ,i}u_{δ} − F_{δ,i}v_{δ}∥_{W_{0}^{1,2}(Δ_{δ};H_{i})} ≤ k∥Ψ(u_{δ}) − Ψ(v_{δ})∥_{L_{2}(Δ_{δ};H_{i})}

for arbitrary u_{δ}, v_{δ} ∈ W_{0}^{1,2}(Δ_{δ}; H_{i}) is proved analogously to (10). Therefore, by the definition of β and (4) we have for any subset U of W^{1,2}(Δ; H_{i})

β_{W^{1,2}(Δ;H_{i})}(F_{Δ,i}U) ≤ (k + 1)β_{W_{0}^{1,2}(Δ_{δ};H_{i})}(F_{δ,i}U_{δ}) ≤ k(k + 1)β_{L_{2}(Δ_{δ};H_{i})}(ΨU_{δ}),

as claimed.

Corollary 1. Let Δ ⊆ Δ_{1} ⊆ (b, d), ũ be some fixed function from W^{1,2}(Δ_{1}; H_{i}), ũ_{Δ} its approximation on Δ, and Ũ the set of all functions u from W^{1,2}(Δ_{1}; H_{i}) which coincide on Δ with ũ. Then for arbitrary u ∈ Ũ the approximation F_{Δ_{1},i}u differs on interval Δ from F_{Δ,i}ũ_{Δ} only by a constant depending on u.

Corollary 2. Let the assumptions of Corollary 1 be satisfied. Suppose that, for every ϕ, ϕ_{1} ∈ H_{1}, we have

Ψ(ϕ + ϕ_{1}) = Ψ(ϕ)  (11)

if and only if ϕ_{1} = 0. Let

∫_{Δ}⟨(ũ)′_{t}(t), v′(t)⟩_{H_{1}} dt = ∫_{Δ}⟨f(t) − Ψ(ũ(t)), v(t)⟩_{H_{1}} dt

and for some u ∈ Ũ and ϕ ∈ H_{1}

∫_{Δ}⟨(F_{Δ_{1},2}(u + ϕ))′_{t}(t), v′(t)⟩_{H_{1}} dt = ∫_{Δ}⟨f(t) − Ψ(u(t) + ϕ), v(t)⟩_{H_{1}} dt

for all v ∈ W_{0}^{1,2}(Δ; H_{1}). Then ϕ = 0.

Lemma 3. Let f(t, .) ∈ H_{2} for all t ∈ Δ, and let Ψ : H_{i} → H_{i} be an operator which satisfies (7) (i = 1, 2). Let F_{Δ,i}: W^{1,2}(Δ; H_{i}) → W^{1,2}(Δ; H_{i}) be the corresponding operators, defined in Lemma 2. Then for all u ∈ W^{1,2}(Δ; H_{2}) we have F_{Δ,1}(u(t, x)) ≡ F_{Δ,2}(u(t, x)) on Δ.

Proof. The proof of this assertion is analogous to the proof of Lemma 3.2 from [8].

We shall show now that in a particular case of Lemma 3 we are led to condensing operators. Let us denote by \overline{B(0, r)} the closure of the set ϕ ∈ H_{2}, with ∥ϕ∥_{H_{2}} ≤ r in the norm H_{1}.

Theorem 2. Let the assumption (5) be satisfied. Given r_{0} assume that, for each r ≤ r_{0} and all functions ϕ, ϕ_{1} from \overline{B(0, r)} for the operator Ψ: H_{1} → H_{1} the next inequalities are true:

|Ψ(ϕ)| ≤ |ϕ_{0} + c̃∥ϕ∥_{H_{1}}^{α−1}ϕ|,  (12)
∥Ψ(ϕ) − Ψ(ϕ_{1})∥_{H_{1}} ≤ k̃(r)∥ϕ − ϕ_{1}∥_{H_{1}}.  (13)

are true. Then there exists r > 0 sufficiently small such that, for every bounded set U ⊂ W^{1,2}(Δ; H_{1}) with values in \overline{B(0, r)}, the inequality

β_{W^{1,2}(Δ;H_{1})}(F_{Δ,1}U) ≤ aβ_{W^{1,2}(Δ;H_{1})}(U),  (14)

holds with some 0 < a < 1, i.e. F_{Δ,1} is a condensing map.

Proof. Let U be any bounded subset W^{1,2}(Δ; H_{1}); in particular, U satisfies the inequality (2). From (13) it follows that the set Ψ(U) satisfies the inequality (2), too. By [5, Theorem 4.8.4], every subset of W^{1,2}(Ω) is μ-compact (μ being the Lebesgue measure). Consequently, by our assumptions on Ψ and our choice of the set U, the set Ψ(U(t_{0})) = {Ψ(u(t_{0}, .)) : u ∈ U} is also μ-compact for fixed t_{0} ∈ Δ. Thus from (1), (2), (8), (12) and Theorem 1 we obtain

β_{W^{1,2}(Δ;H_{1})}(F_{Δ,1}U) ≤ c_{3}β_{L_{2}(Δ_{δ};H_{1})}(Ψ(U_{δ})) ≤ c_{3}(∫_{Δ_{δ}} β_{H_{1}}^{2}(Ψ(U_{δ}(t))) dt)^{1/2}

≤ \sqrt{2}c_{3}(∫_{Δ_{δ}} ν_{H_{1}}^{2} \{ϕ_{0} (x) + c̃∥u_{δ} (t, x) ∥_{H_{1}}^{α_{1}−1}u_{δ} (t, x) : u_{δ} ∈ U_{δ}\} dt)^{1/2}

≤ c_{3}r^{α_{1}−1}c̃(∫_{Δ_{δ}} β_{H_{1}}^{2}(U_{δ}(t)) dt)^{1/2} ≤ cc_{3}r^{α_{1}−1}c̃(2|Δ|)^{1/2}β_{W^{1,2}(Δ;H_{1})}(U).

Taking r sufficiently small we arrive at the inequality (14).

As example of an application of our results we study now the existence of solutions u ∈ W_{0}^{1,2}(0, T; H_{1}) of the ordinary operator differential equation

−u′′_{tt}(t) + Ψ(u(t)) = f(t), t ∈ (0, T), u(0) = u(T) = 0,  (15)

where 0 < T < ∞ and f ∈ L_{1}(0, T; H_{2}) are given. We say that ũ ∈ W_{0}^{1,2}(0, T; H_{1}) is a generalized solution of (15) if

∫_{0}^{T} ⟨u′(t), v′(t)⟩_{H_{1}} dt + ∫_{0}^{T} ⟨Ψ(ũ(t)), v(t)⟩_{H_{1}} dt = ∫_{0}^{T} ⟨f(t), v(t)⟩_{H_{1}} dt  (16)

for any v ∈ W_{0}^{1,2}(0, T; H_{1}).

Theorem 3. Let the assumptions (5), (11), (12) and (13) be satisfied. Then the equation (15) has a generalized solution in the space W_{0}^{1,2}(0, T; H_{1}) for each f ∈ L_{1}(0, T; H_{2}).

Proof. Let F_{Δ,i}u be the operators defined in Lemma 2 for H_{i} (i = 1, 2). By (7) we have

∥F_{Δ,i}u∥_{W^{1,2}(Δ;H_{i})} ≤ c_{0}∥f∥_{L_{1}(Δ;H_{2})} + č_{1}|Δ|^{1/2} + |Δ|^{1/2} ∑_{j=1}^{J} ˜ c̃_{j}∥u∥_{W^{1,2}(Δ;H_{i})}^{α_{i,j}} ≤ r_{0}

if ∥u∥_{W^{1,2}(Δ;H_{i})} ≤ r_{0} for some 0 < r_{0} < 1, and |Δ| ≤ τ for τ sufficiently small. We take as Δ the interval (0, τ) and consider the Hilbert space W^{1,2}(Δ; H_{1}) of functions satisfying u(0, x) = 0 for all x ∈ Ω. From Lemma 3 it follows that F_{Δ,1}u(t, x) ≡ F_{Δ,2}u(t, x) if u ∈ W^{1,2}(Δ; H_{2}). By (6) there exist τ_{1} ≤ τ and r > 0 sufficiently small such that for Δ = (0, τ_{1}) we have

∥F_{Δ,i}u∥_{C(\overline{Δ};H_{i})} ≤ c_{0}∥f∥_{L_{1}(Δ;H_{2})} + č_{1}|Δ|^{1/2} + |Δ|^{1/2} ∑_{j=1}^{J} c̃_{j}∥u∥_{C(\overline{Δ};H_{i})}^{α_{i,j}} ≤ r

if ∥u∥_{C(\overline{Δ};H_{2})} ≤ r. By Theorem 2 we may choose r > 0 such that the inequality

β_{W^{1,2}(Δ;H_{1})}(F_{Δ,1}U) < β_{W^{1,2}(Δ;H_{1})}(U)

is true for the operator F_{Δ,1} and for every bounded not precompact subset U ⊂ W^{1,2}(Δ; H_{1}) with values in \overline{B(0, r)}. This shows that F_{Δ,1} is a condensing map with respect to the measure of noncompactness β. Moreover, the set of all functions u ∈ W^{1,2}(Δ; \overline{B(0, r)}), with ∥u∥_{W^{1,2}(Δ;H_{1})} ≤ r_{0} is closed, convex, nonempty and invariant with respect to F_{Δ,1}. Thus, by an analogue to Schauder’s fixed point principle for β-condensing maps [1], the operator F_{Δ_{1},1} has a fixed point u_{1} ∈ W^{1,2}(Δ; H_{1}). By Corollary 2, applied to Δ_{1} = (τ_{1}/2, 3/2τ_{1}) the set Ũ of all functions u ∈ W^{1,2}(Δ_{1}; \overline{B(0, r)}), ∥u∥_{W^{1,2}(Δ_{1};H_{1})} ≤ r_{0}, which coincide on Δ ∩ Δ_{1} with u_{1} + ϕ for some ϕ ∈ H_{2} depending on u, is invariant with respect to the operator F_{Δ_{1},1}. Consequently, the operator F_{Δ_{1},1} has also a fixed point u_{2} ∈ Ũ which, by Corollary 2, coincides with u_{1} on Δ ∩ Δ_{1}. Now let

ũ(t, x) = \{ \begin{matrix}u_{1}(t, x), & by t ∈ (0, τ), \\ u_{2}(t, x), & by t ∈ (τ, 3τ/2). \end{matrix}

Then the equality (16) is true for all v ∈ W_{0}^{1,2}(Δ; H_{1}) with supp v ⊆ (0, 3τ/2), since every function v ∈ W_{0}^{1,2}(Δ; H_{1}) with supp v ⊆ (0, 3τ/2) can be decomposed into a sum v_{1} + v_{2}, where v_{1} ∈ W_{0}^{1,2}(Δ; H_{1}) with supp v_{1} ⊆ (0, τ), and v_{2} ∈ W_{0}^{1,2}(Δ; H_{1}) with supp v_{2} ⊆ (τ/2, 3τ/2). Applying this procedure a finite number of times, we obtain the solution on the whole (0, T).

Theorem 3 is illustrated in the next example.

Example. Let H_{1} = L_{2}(Ω). Let Ψ: H_{1} → H_{1} be given by

Ψ(ϕ) = ϕ ∑_{j=1}^{J} c̆_{j} ∥ϕ∥_{H_{1}}^{α_{j}−1} (ϕ ∈ H_{1}),

where c̆_{j} ≥ 0, and α_{j} > 1 are real constants. Let α_{0} = min{α_{1},…, α_{J}}, and c̆_{0} = max{c̆_{1},…, c̆_{J}}. Then the condition (12) is true. Since there exists 0 < r_{0} < 1 such that

|Ψ(ϕ)| ≤ Jc̆_{0}∥ϕ∥_{H_{1}}^{α_{0}−1}|ϕ|

if ϕ from \overline{B(0, r_{0})}. It can easily be verified that (5), (11), (13) are satisfied too.

Theorem 3 ensures the the existence of a generalized solution of the boundary value problem (15) in the Bochner space W_{0}^{1,2}(0, T; H_{1}) for each f ∈ L_{1}(0, T; H_{2}).

Remark. The operator F_{Δ,1} with the function Ψ, considered in Example is, in general, not compact [8].

REFERENCES

[1] Akhmerov, R. R., Kamenskij, M. I., Potapov, A. S., Rodkina, A. E., Sadovskij, B. N., Measures of Noncompactness and Condensing Operators (in Russian), Novosibirsk, Nauka 1986; Engl. transl.: Basel, Birkhäuser 1992.

[2] Appell, J., Zabrejko, P. P., Nonlinear Superposition Operators, London, Cambridge University Press, 1990.

[3] Gajewski, H., Gröger, K., Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin, Akademie-Verlag 1974.

[4] Jovanović, I., Rakočević, V., Multipliers of mixed-norm sequence spaces and measures of noncompactness, Publ. de l’Inst. Math. 56(70) (1994), 61–68.

[5] Maz’ja, V. G., Sobolev Spaces, Berlin, Springer 1985.

[6] Yerzakova, N. A., On properties of the β-measure of noncompactness (in Russian), Deposited in VINITI No. 6132-82, Voronezh 1982.

[7] Yerzakova, N. A., On measures of non-compactness in regular spaces, Zeitschr. Anal. Anw. 15, 2 (1996), 299–307.

[8] Yerzakova, N. A., Investigation of some operators and operater equations in a space of integrable functions (in Russian), Khabarovsk, Inst. Appl. Math. FEB RAS. Preprint No. 3 (1997), 1–23.

(received 15.07.1997.)

Department of Mathematics, Khabarovsk Technical University, R-680035 Khabarovsk, Russian Federation e-mail: erz@centr.khstu.khabarovsk.su