4.1 Preliminaries
Let us recall some basic definitions on multi-valued maps [28, 29].
For a normed space (X, ‖.‖), let , , , and . A multi-valued map is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. The map G is bounded on bounded sets if is bounded in X for all (i.e., . G is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set G(x0) is a nonempty closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood of x0 such that . G is said to be completely continuous if is relatively compact for every . If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., x
n
→ x*, y
n
→ y*, y
n
∈ G(x
n
) imply y* ∈ G(x*). G has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map is said to be measurable if for every , the function
is measurable.
Let C([0, 1]) denotes a Banach space of continuous functions from [0, 1] into ℝ with the norm . Let L1([0, 1], ℝ) be the Banach space of measurable functions x : [0, 1] → ℝ which are Lebesgue integrable and normed by .
Definition 4.1
A multivalued map
is said to be Carathéodory if
(i) t ↦ F (t, x) is measurable for each x ∈ ℝ;
(ii) x ↦ F (t, x) is upper semicontinuous for almost all t ∈ [0, 1];
Further a Carathéodory function F is called L
1
-Carathéodory if
(iii) for each α > 0, there exists such that
for all ‖x‖∞ ≤ α and for a. e. t ∈ [0, 1].
For each , define the set of selections of F by
Let X be a nonempty closed subset of a Banach space E and be a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set {y ∈ X : G(y) ∩ B ≠ ∅} is open for any open set B in E. Let A be a subset of [0, 1] × ℝ. A is measurable if A belongs to the σ-algebra generated by all sets of the form , where
is Lebesgue measurable in [0, 1] and
is Borel measurable in ℝ. A subset
of L1([0, 1], ℝ) is decomposable if for all and measurable , the function , where stands for the characteristic function of
.
Definition 4.2 Let Y be a separable metric space and let be a multivalued operator. We say N has a property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with F as
which is called the Nemytskii operator associated with F.
Definition 4.3 Let be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator
is lower semi-continuous and has nonempty closed and decomposable values.
Let (X, d) be a metric space induced from the normed space (X; ‖.‖). Consider given by
where d(A, b) = infa∈Ad(a; b) and d(a, B) = infb∈Bd(a; b). Then (P
b,cl
(X), H
d
) is a metric space and (P
cl
(X), H
d
) is a generalized metric space (see [30]).
Definition 4.4 A multivalued operator N : X → P
cl
(X) is called:
(a) γ-Lipschitz if and only if there exists γ > 0 such that
(b) a contraction if and only if it is γ-Lipschitz with γ < 1.
The following lemmas will be used in the sequel.
Lemma 4.5 (Nonlinear alternative for Kakutani maps) [31]. Let E be a Banach space, C is a closed convex subset of E, U is an open subset of C and 0 ∈ U. Suppose that is a upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets of C. Then either
(i) F has a fixed point in , or
(ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF(u).
Lemma 4.6 [32] Let X be a Banach space. Let be an L1-Carathéodory multivalued map and let θ be a linear continuous mapping from L1([0, 1], X) to C([0, 1], X). Then the operator
is a closed graph operator in C([0, 1], X) × C([0, 1], X).
Lemma 4.7 [33] Let Y be a separable metric space and let be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) such that g(x) ∈ N(x) for every x ∈ Y .
Lemma 4.8 [34] Let (X, d) be a complete metric space. If N : X → P
cl
(X) is a contraction, then FixN ≠ ∅.
Definition 4.9 A function x ∈ C2([0, 1], ℝ) is a solution of the problem (1.2) if , and there exists a function f ∈ L1([0, 1], ℝ) such that
f(t) ∈ F (t, x(t)) a.e. on [0, 1] and
(4.1)
4.2 The Carathéodory case
Theorem 4.10
Assume that:
(H1) is Carathéodory and has nonempty compact and convex values;
(H2) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function such that
(H3) there exists a constant M > 0 such that
Then the boundary value problem (1.2) has at least one solution on [0, 1].
Proof. Define the operator by
for f ∈ S
F,x
. We will show that Ω
F
satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that Ω
F
is convex for each x ∈ C([0, 1], ℝ). This step is obvious since S
F,x
is convex (F has convex values), and therefore we omit the proof.
In the second step, we show that Ω
F
maps bounded sets (balls) into bounded sets in C([0, 1], ℝ). For a positive number ρ, let B
ρ
= {x ∈ C([0, 1], ℝ): ‖x‖ ≤ ρ} be a bounded ball in C([0, 1], ℝ). Then, for each h ∈ Ω
F
(x), x ∈ B
ρ
, there exists f ∈ S
F,x
such that
Then for t∈[0, 1] we have
Thus,
Now we show that Ω
F
maps bounded sets into ;equicontinuous sets of C([0, 1], ℝ).
Let t', t'' ∈ [0, 1] with t' < t'' and x ∈ B
ρ
. For each h ∈ Ω
F
(x), we obtain
Obviously the right-hand side of the above inequality tends to zero independently of x ∈ B
ρ
as t'' - t' → 0. As Ω
F
satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that Ω
F
has a closed graph. Let , and . Then we need to show that . Associated with h
n
∈ Ω
F
(x
n
), there exists such that for each t ∈ [0, 1],
Thus it suffices to show that there exists such that for each t ∈ [0, 1],
Let us consider the linear operator θ: L1([0, 1], ℝ) → C([0, 1], ℝ) given by
Observe that
as n → ∞.
Thus, it follows by Lemma 4.6 that θ ο S
F
is a closed graph operator. Further, we have . Since , therefore, we have