International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 1 (2017), 27-33
http://www.etamaths.com

EXISTENCE OF SOLUTIONS FOR A CERTAIN BOUNDARY VALUE PROBLEM

ASSOCIATED TO A FOURTH ORDER DIFFERENTIAL INCLUSION

AURELIAN CERNEA1,2,∗

Abstract. Existence of solutions for a fourth order differential inclusion with cantilever boundary

conditions is investigated. New results are obtained when the right hand side has convex or non

convex values.

1. Introduction

Fourth order differential equations are often used in engineering and physical problems. Boundary
value problems associated to fourth order differential equations appear in elasticity theory describing
stationary states of the deflection of an elastic beam. The same equation can describe the ”effect of the
shear” when investigating transverse vibrations. As a consequence there was an intensive development
of the study of such problems. In the single valued case several results concerning existence, localization
and multiplicity of solutions may be found in [3], [4], [5], [8], [10], [12] etc..

This paper is devoted to the following boundary value problem

x(4) ∈ F(t,x), a.e. ([0, 1]), x(0) = x′(0) = x′′(1) = x′′′(1) = 0, (1.1)

where F(., .) : [0, 1] ×Rn →P(Rn) is a set-valued map.
The aim of our paper is to consider the more general framework of set-valued problems and to

present three existence results for problem 1.1. Our results are obtained under several hypotheses
concerning the regularity of the set-valued map F and are based on a nonlinear alternative of Leray-
Schauder type, on Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with
decomposable values and on Kuratowski and Ryll-Nardzewski selection theorem. We mention that the
methods used are rather known in the theory of differential inclusions, however their exposition in the
framework of problem 1.1 is new.

The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the
sequel, in Section 3 we prove our results using fixed point techniques and in Section 4 we provide a
Filippov type existence result.

2. Preliminaries

In this section we sum up some basic facts that we are going to use later.
Let (X,d) be a metric space with the corresponding norm |.| and let I ⊂ R be a compact interval.

Denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I, by P(X) the family of all
nonempty subsets of X and by B(X) the family of all Borel subsets of X. If A ⊂ I then χA(.) : I →
{0, 1} denotes the characteristic function of A. For any subset A ⊂ X we denote by A the closure of
A.

Recall that the Pompeiu-Hausdorff distance of the closed subsets A,B ⊂ X is defined by dH (A,B) =
max{d∗(A,B),d∗(B,A)}, d∗(A,B) = sup{d(a,B); a ∈ A}, where d(x,B) = infy∈B d(x,y).

As usual, we denote by C(I,X) the Banach space of all continuous functions x(.) : I → X endowed
with the norm |x(.)|C = supt∈I|x(t)| and by L1(I,X) the Banach space of all (Bochner) integrable
functions x(.) : I → X endowed with the norm |x(.)|1 =

∫
I
|x(t)|dt.

Received 4th December, 2016; accepted 6th March, 2017; published 2nd May, 2017.

2010 Mathematics Subject Classification. 34A60, 34B18.

Key words and phrases. differential inclusion; fixed point; decomposable set.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

27



28 CERNEA

A subset D ⊂ L1(I,X) is said to be decomposable if for any u(·),v(·) ∈ D and any subset A ∈L(I)
one has uχA + vχB ∈ D, where B = I\A.

Consider T : X → P(X) a set-valued map. A point x ∈ X is called a fixed point for T(.) if
x ∈ T(x). T(.) is said to be bounded on bounded sets if T(B) := ∪x∈BT(x) is a bounded subset of X
for all bounded sets B in X. T(.) is said to be compact if T(B) is relatively compact for any bounded

sets B in X. T(.) is said to be totally compact if T(X) is a compact subset of X. T(.) is said to
be upper semicontinuous if for any x0 ∈ X, T(x0) is a nonempty closed subset of X and if for each
open set D of X containing T(x0) there exists an open neighborhood V0 of x0 such that T(V0) ⊂ D.
Let E a Banach space, Y ⊂ E a nonempty closed subset and T(.) : Y → P(E) a multifunction with
nonempty closed values. T(.) is said to be lower semicontinuous if for any open subset D ⊂ E, the set
{y ∈ Y ; T(y) ∩D 6= ∅} is open. T(.) is called completely continuous if it is upper semicontinuous and
totally compact on X.

It is well known that a compact set-valued map T(.) with nonempty compact values is upper
semicontinuous if and only if T(.) has a closed graph.

We recall the following nonlinear alternative of Leray-Schauder type proved in [11] and its conse-
quences.

Theorem 2.1. Let D and D be the open and closed subsets in a normed linear space X such that
0 ∈ D and let T : D → P(X) be a completely continuous set-valued map with compact convex values.
Then either

i) the inclusion x ∈ T(x) has a solution, or
ii) there exists x ∈ ∂D (the boundary of D) such that λx ∈ T(x) for some λ > 1.

Corollary 2.1. Let Br(0) and Br(0) be the open and closed balls in a normed linear space X centered

at the origin and of radius r and let T : Br(0) → P(X) be a completely continuous set-valued map
with compact convex values. Then either

i) the inclusion x ∈ T(x) has a solution, or
ii) there exists x ∈ X with |x| = r and λx ∈ T(x) for some λ > 1.

Corollary 2.2. Let Br(0) and Br(0) be the open and closed balls in a normed linear space X centered

at the origin and of radius r and let T : Br(0) → X be a completely continuous single valued map with
compact convex values. Then either

i) the equation x = T(x) has a solution, or
ii) there exists x ∈ X with |x| = r and x = λT(x) for some λ < 1.

If F(., .) : I × X → P(X) is a set-valued map with compact values we define SF : C(I,X) →
P(L1(I,X)) by

SF (x) := {f ∈ L1(I,X); f(t) ∈ F(t,x(t)) a.e. (I)}.
We say that F(., .) is of lower semicontinuous type if SF (.) is lower semicontinuous with nonempty
closed and decomposable values. The next result is proved in [2].

Theorem 2.2. Let S be a separable metric space and G(.) : S →P(L1(I,X)) be a lower semicontin-
uous set-valued map with closed decomposable values.

Then G(.) has a continuous selection (i.e., there exists a continuous mapping g(.) : S → L1(I,X)
such that g(s) ∈ G(s) ∀s ∈ S).

A set-valued map G : I →P(X) with nonempty compact convex values is said to be measurable if
for any x ∈ X the function t → d(x,G(t)) is measurable. A set-valued map F(., .) : I × X → P(X)
is said to be Carathéodory if t → F(t,x) is measurable for any x ∈ X and x → F(t,x) is upper
semicontinuous for almost all t ∈ I. Moreover, F(., .) is said to be L1-Carathéodory if for any l > 0
there exists hl(.) ∈ L1(I,R) such that sup{|v|; v ∈ F(t,x)}≤ hl(t) a.e. (I), ∀x ∈ Bl(0).

Theorem 2.3. Let X be a Banach space, let F(., .) : I ×X →P(X) be a L1-Carathéodory set-valued
map with SF (x) 6= ∅ for all x(.) ∈ C(I,X) and let Γ : L1(I,X) → C(I,X) be a linear continuous
mapping.

Then the set-valued map Γ ◦SF : C(I,X) →P(C(I,X)) defined by
(Γ ◦SF )(x) = Γ(SF (x))



A FOURTH ORDER DIFFERENTIAL INCLUSION 29

has compact convex values and has a closed graph in C(I,X) ×C(I,X).

The proof of theorem above may be found in [9], Note that if dimX < ∞, and F(., .) is as in
Theorem 2.5, then SF (x) 6= ∅ for any x(.) ∈ C(I,X) (e.g., [9]).

We recall also a selection result ( [1]) which is a version of the celebrated Kuratowski and Ryll-
Nardzewski selection theorem.

Lemma 2.1. Consider X a separable Banach space, B is the closed unit ball in X, H : I →P(X) is
a set-valued map with nonempty closed values and g : I → X,L : I → R+ are measurable functions. If

H(t) ∩ (g(t) + L(t)B) 6= ∅ a.e.(I),
then the set-valued map t → H(t) ∩ (g(t) + L(t)B) has a measurable selection.

In what follows I = [0, 1], and V = {x ∈ W4,1(I,R); x(0) = x′(0) = x′′(1) = x′′′(1) = 0} with the
norm ||x||V = ||x(4)||1. By a solution of problem (1.1) we mean a function x(.) ∈ V for which there
exists a function f(.) ∈ L1(I,R) with f(t) ∈ F(t,x(t)), a.e. (I) such that x(4)(t) = f(t) a.e. (I).

The next technical result is proved in [3].

Lemma 2.2. If f(.) : [0, 1] → R is an integrable function, then the solution of the boundary value
problem

x(4) = f(t), a.e. ([0, 1]), x(0) = x′(0) = x′′(1) = x′′′(1) = 0

is given by

x(t) =

∫ 1
0

G(t,s)f(s)ds,

where

G(t,s) :=

{
s2

6
(3t−s), if 0 ≤ s < t ≤ 1,

t2

6
(3s− t), if 0 ≤ t < s ≤ 1

Obviously, |G(t,s)| ≤ 1
2
∀t.s ∈ I.

3. Existence via fixed points

We are able now to present the two existence results for problem (1.1) using fixed point techniques.
We consider first the case when F(., .) is convex valued.

Hypothesis H1. i) F(., .) : I×R →P(R) has nonempty compact convex values and is Carathéodory.
ii) There exist ϕ(.) ∈ L1(I,R) with ϕ(t) > 0 a.e. (I) and there exists a nondecreasing function

ψ : [0,∞) → (0,∞) such that
sup{|v|; v ∈ F(t,x)}≤ ϕ(t)ψ(|x|) a.e. (I), ∀x ∈ R.

Theorem 3.1. Assume that Hypothesis H1 is satisfied and there exists r > 0 such that

r >
1

2
|ϕ|1ψ(r). (3.1)

Then problem 1.1 has at least one solution x(.) such that |x(.)|C < r.

Proof. Let X = W4,1(I,R) and consider r > 0 as in 3.1. It is obvious that the existence of solutions
to problem 1.1 reduces to the existence of the solutions of the integral inclusion

x(t) ∈
∫ 1
0

G(t,s)F(s,x(s))ds, t ∈ I. (3.2)

Consider the set-valued map T : Br(0) →P(W4,1(I,R)) defined by

T(x) := {v(.) ∈ W4,1(I,R); v(t) :=
∫ 1
0

G(t,s)f(s)ds, f ∈ SF (x)}. (3.3)

We show that T(.) satisfies the hypotheses of Corollary 2.1. First, we show that T(x) ⊂ W4,1(I,R)
is convex for any x ∈ W4,1(I,R).

If vi ∈ T(x) then there exist fi ∈ SF (x) such that for any t ∈ I one has vi(t) =
∫ 1
0
G(t,s)fi(s)ds,

i = 1, 2.



30 CERNEA

Let 0 ≤ α ≤ 1. Then for any t ∈ I we have

(αv1 + (1 −α)v2)(t) =
∫ 1
0

G(t,s)[αf1(s) + (1 −α)f2(s)]ds.

The values of F(., .) are convex, thus SF (x) is a convex set and hence αf1 + (1 −α)f2 ∈ T(x).
Secondly, we show that T(.) is bounded on bounded sets of W4,1(I,R).
Let B ⊂ W4,1(I,R) be a bounded set. Then there exist m > 0 such that |x|C ≤ m ∀x ∈ B. If

v ∈ T(x) there exists f ∈ SF (x) such that v(t) =
∫ 1
0
G(t,s)f(s)ds. One may write for any t ∈ I

|v(t)| ≤
∫ 1
0

|G(t,s)|.|f(s)|ds ≤
∫ 1
0

|G(t,s)|ϕ(s)ψ(|x(t)|)ds

and therefore |v|C ≤ 12|ϕ|1ψ(m) ∀v ∈ T(x), i.e., T(B) is bounded.
We show next that T(.) maps bounded sets into equi-continuous sets.
Let B ⊂ W4,1(I,R) be a bounded set as before and v ∈ T(x) for some x ∈ B. There exists

f ∈ SF (x) such that v(t) =
∫ 1
0
G(t,s)f(s)ds. Then for any t,τ ∈ I we have

|v(t) −v(τ)| ≤ |
∫ 1
0

G(t,s)f(s)ds−
∫ 1
0

G(τ,s)f(s)ds| ≤

∫ 1
0

|G(t,s) −G(τ,s)|.|f(s)|ds ≤
∫ 1
0

|G(t,s) −G(τ,s)|ϕ(s)ψ(m)ds.

It follows that |v(t) −v(τ)|→ 0 as t → τ. Therefore, T(B) is an equi-continuous set in W4,1(I,R).
We apply now Arzela-Ascoli’s theorem we deduce that T(.) is completely continuous on W4,1(I,R).
In the next step of the proof we prove that T(.) has a closed graph.
Let xn ∈ W4,1(I,R) be a sequence such that xn → x∗ and vn ∈ T(xn) ∀n ∈ N such that

vn → v∗. We prove that v∗ ∈ T(x∗). Since vn ∈ T(xn), there exists fn ∈ SF (xn) such that
vn(t) =

∫ 1
0
G(t,s)fn(s)ds. Define Γ : L

1(I,R) → W4,1(I,R) by (Γ(f))(t) :=
∫ 1
0
G(t,s)f(s)ds. One has

maxt∈I |vn(t) −v∗(t)| = |vn(.) −v∗(.)|C → 0 as n →∞
We apply Theorem 2.3 to find that Γ ◦ SF has closed graph and from the definition of Γ we

get vn ∈ Γ ◦ SF (xn). Since xn → x∗, vn → v∗ it follows the existence of f∗ ∈ SF (x∗) such that
v∗(t) =

∫ 1
0
G(t,s)f∗(s)ds.

Therefore, T(.) is upper semicontinuous and compact on Br(0). We apply Corollary 2.1 to deduce

that either i) the inclusion x ∈ T(x) has a solution in Br(0), or ii) there exists x ∈ X with |x|C = r
and λx ∈ T(x) for some λ > 1. Assume that ii) is true. With the same arguments as in the second
step of our proof we get r = |x(.)|C ≤ 12|ϕ|1ψ(r) which contradicts 3.1. Hence only i) is valid and
theorem is proved. �

We consider now the case when F(., .) is not necessarily convex valued. Our existence result in
this case is based on the Leray-Schauder alternative for single valued maps and on Bressan Colombo
selection theorem.

Hypothesis H2. i) F(., .) : I × R → P(R) has compact values, F(., .) is L(I) ⊗B(R) measurable
and x → F(t,x) is lower semicontinuous for almost all t ∈ I.

ii) There exist ϕ(.) ∈ L1(I,R) with ϕ(t) > 0 a.e. (I) and there exists a nondecreasing function
ψ : [0,∞) → (0,∞) such that

sup{|v|; v ∈ F(t,x)}≤ ϕ(t)ψ(|x|) a.e. (I), ∀x ∈ R.

Theorem 3.2. Assume that Hypothesis H2 is satisfied and there exists r > 0 such that condition 3.1
is satisfied.

Then problem 1.1 has at least one solution on I.

Proof. We note first that if Hypothesis H2 is satisfied then F(., .) is of lower semicontinuous type
(e.g., [7]). Therefore, we apply Theorem 2.2 with S = W4,1(I,R) and G(.) = SF (.) to deduce that there



A FOURTH ORDER DIFFERENTIAL INCLUSION 31

exists a continuous mapping f(.) : W4,1(I,R) → L1(I,R) such that f(x) ∈ SF (x) ∀x ∈ W4,1(I,R).
We consider the corresponding problem

x(t) =

∫ 1
0

G(t,s)f(x(s))ds, t ∈ I (3.4)

in the space X = W4,1(I,R). It is clear that if x(.) ∈ W4,1(I,R) is a solution of the problem (3.4)
then x(.) is a solution to problem 1.1.

Let r > 0 that satisfies condition 3.1 and define the set-valued map T : Br(0) →P(W4,1(I,R)) by
(T(x))(t) :=

∫ 1
0
G(t,s)f(x(s))ds.

Obviously, the integral equation 3.4 is equivalent with the operator equation

x(t) = (T(x))(t), t ∈ I.

It remains to show that T(.) satisfies the hypotheses of Corollary 2.2.

We show that T(.) is continuous on Br(0). From Hypotheses H2 ii) we have

|f(x(t))| ≤ ϕ(t)ψ(|x(t)|) a.e. (I)

for all x(.) ∈ W4,1(I,R). Let xn,x ∈ Br(0) such that xn → x. Then

|f(xn(t))| ≤ ϕ(t)ψ(r) a.e. (I).

From Lebesgue’s dominated convergence theorem and the continuity of f(.) we obtain, for all t ∈ I

lim
n→∞

(T(xn))(t) =

∫ 1
0

G(t,s)f(xn(s))ds =

∫ 1
0

G(t,s)f(x(s))ds = (T(x))(t)

i.e., T(.) is continuous on Br(0).
Repeating the arguments in the proof of Theorem 3.1 with corresponding modifications it follows

that T(.) is compact on Br(0). We apply Corollary 2.2 and we find that either i) the equation x = T(x)

has a solution in Br(0), or ii) there exists x ∈ X with |x|C = r and x = λT(x) for some λ < 1.
As in the proof of Theorem 3.1 if the statement ii) holds true, then we obtain a contradiction

to 3.1. Thus only the statement i) is true and problem 1.1 has a solution x(.) ∈ W4,1(I,R) with
|x(.)|C < r �

4. A Filippov type existence result

In this section we consider the, even, more general problem

x(4) ∈ F(t,x,V (x)(t)), a.e. ([0, 1]), x(0) = x′(0) = x′′(1) = x′′′(1) = 0, (4.1)

where F : [0, 1] × R × R → P(R) is a set-valued map, V : C([0, 1],R) → C([0, 1],R) is a nonlinear
Volterra integral operator defined by V (x)(t) =

∫ t
0
k(t,s,x(s))ds with k(., ., .) : [0, 1] × R × R → R a

given function. We show that Filippov’s ideas ( [6]) can be suitably adapted in order to obtain the
existence of solutions for problem 4.1.

In order to prove our results we need the following hypotheses.

Hypothesis H3. i) F(., .) : I ×R×R →P(R) has nonempty closed values and is L(I) ⊗B(R×R)
measurable.

ii) There exists L(.) ∈ L1(I, (0,∞)) such that, for almost all t ∈ I,F(t, ., .) is L(t)-Lipschitz in the
sense that

dH (F(t,x1,y1),F(t,x2,y2)) ≤ L(t)(|x1 −x2| + |y1 −y2|) ∀ x1,x2,y1,y2 ∈ R.

iii) k(., ., .) : I ×R×R → R is a function such that ∀x ∈ R, (t,s) → k(t,s,x) is measurable.
iv) |k(t,s,x) −k(t,s,y)| ≤ L(t)|x−y| a.e. (t,s) ∈ I × I, ∀x,y ∈ R.

We use next the following notations

M(t) := L(t)(1 +

∫ t
0

L(u)du), t ∈ I, M0 =
∫ 1
0

M(t)dt.



32 CERNEA

Theorem 4.1. Assume that Hypothesis H3 is satisfied and M0 < 2. Let y(.) ∈ C(I,R) be such that
y(0) = y′(0) = y′′(1) = y′′′(1) = 0 and there exists p(.) ∈ L1(I,R+) with d(y(4)(t),F(t,y(t),V (y)(t)))
≤ p(t) a.e. (I).

Then there exists x(.) a solution of problem 4.1 satisfying for all t ∈ I

|x(t) −y(t)| ≤
1

2 −M0

∫ 1
0

p(t)dt.

Proof. The set-valued map t → F(t,y(t),V (y)(t)) is measurable with closed values and F(t,y(t),
V (y)(t)) ∩{y(4)(t) + p(t)[−1, 1]} 6= ∅ a.e. (I).

It follows from Lemma 2.1 that there exists a measurable selection f1(t) ∈ F(t,y(t),V (y)(t)) a.e.
(I) such that

|f1(t) −y(4)(t)| ≤ p(t) a.e. (I) (4.2)

Define x1(t) =
∫ 1
0
G(t,s)f1(s)ds and one has |x1(t) −y(t)| ≤ 12

∫ 1
0
p(t)dt.

We claim that it is enough to construct the sequences xn(.) ∈ C(I,R), fn(.) ∈ L1(I,R), n ≥ 1 with
the following properties

xn(t) =

∫ 1
0

G(t,s)fn(s)ds, t ∈ I, (4.3)

fn(t) ∈ F(t,xn−1(t),V (xn−1)(t)) a.e. (I), (4.4)

|fn+1(t) −fn(t)| ≤ L(t)(|xn(t) −xn−1(t)| +
∫ t
0

L(s)|xn(s) −xn−1(s)|ds) a.e. (I) (4.5)

If this construction is realized then from 4.2-4.5 we have for almost all t ∈ I

|xn+1(t) −xn(t)| ≤
1

2
(
M0
2

)n
∫ 1
0

p(t)dt ∀n ∈ N.

Indeed, assume that the last inequality is true for n− 1 and we prove it for n. One has

|xn+1(t) −xn(t)| ≤
∫ 1
0

|G(t,t1)|.|fn+1(t1) −fn(t1)|dt1 ≤

1

2

∫ 1
0

L(t1)[|xn(t1) −xn−1(t1)| +
∫ t1
0

L(s)|xn(s) −xn−1(s)|ds]dt1 ≤
1

2∫ 1
0

L(t1)(1 +

∫ t1
0

L(s)ds)dt1.(
1

2
)nMn−10

∫ 1
0

p(t)dt =
1

2
(
M0
2

)n
∫ 1
0

p(t)dt

Therefore {xn(.)} is a Cauchy sequence in the Banach space C(I,R), hence converging uniformly
to some x(.) ∈ C(I,R). Therefore, by 4.5, for almost all t ∈ I, the sequence {fn(t)} is Cauchy in R.
Let f(.) be the pointwise limit of fn(.). Moreover, one has

|xn(t) −y(t)| ≤ |x1(t) −y(t)| +
∑n−1

i=1 |xi+1(t) −xi(t)| ≤
1
2

∫ 1
0
p(t)dt +

∑n−1
i=1 (

1
2

∫ 1
0
p(t)dt)( M0

2
)i =

1
2

∫
1
0
p(t)dt

1−M0
2

.
(4.6)

On the other hand, from 4.2, 4.5 and 4.6 we obtain for almost all t ∈ I

|fn(t) −y(4)(t)| ≤
n−1∑
i=1

|fi+1(t) −fi(t)| + |f1(t) −D
q
Cy(t)| ≤ L(t)

∫ 1
0
p(t)dt

2 −M0
+ p(t).

Hence the sequence fn(.) is integrably bounded and therefore f(.) ∈ L1(I,R).
Using Lebesgue’s dominated convergence theorem and taking the limit in 4.2, 4.4 we deduce that

x(.) is a solution of 1.1. Finally, passing to the limit in 4.6 we obtained the desired estimate on x(.).
It remains to construct the sequences xn(.),fn(.) with the properties in 4.2-4.5. The construction

will be done by induction.
Since the first step is already realized, assume that for some N ≥ 1 we already constructed xn(.) ∈

C(I,R) and fn(.) ∈ L1(I,R), n = 1, 2, ...N satisfying 4.2, 4.5 for n = 1, 2, ...N and 4.4 for n =
1, 2, ...N − 1. The set-valued map t → F(t,xN (t),V (xN )(t)) is measurable. Moreover, the map



A FOURTH ORDER DIFFERENTIAL INCLUSION 33

t → L(t)(|xN (t) − xN−1(t)| +
∫ t
0
L(s)|xN (s) − xN−1(s)|ds) is measurable. By the lipschitzianity of

F(t, .) we have that for almost all t ∈ I

F(t,xN (t)) ∩{fN (t) + L(t)(|xN (t) −xN−1(t)| +
∫ t
0

L(s)|xN (s) −xN−1(s)|ds)[−1, 1]} 6= ∅.

Lemma 2.1 yields that there exist a measurable selection fN+1(.) of F(.,xN (.),V (xN )(.)) such that
for almost all t ∈ I

|fN+1(t) −fN (t)| ≤ L(t)(|xN (t) −xN−1(t)| +
∫ t
0

L(s)|xN (s) −xN−1(s)|ds).

We define xN+1(.) as in 4.2 with n = N + 1. Thus fN+1(.) satisfies 4.4 and 4.5 and the proof is
complete. �

The assumptions in Theorem 4.1 are satisfied, in particular, for y(.) = 0 and therefore with p(.) =
L(.). We obtain the following consequence of Theorem 4.1.

Corollary 4.1. Assume that Hypothesis H3 is satisfied, M0 < 2 and d(0,F(t, 0,V (0)(t)) ≤ L(t) a.e.
(I). Then there exists x(.) a solution of problem 4.1 satisfying for all t ∈ I, |x(t)| ≤ 1

2−M0

∫ 1
0
L(t)dt.

If F does not depend on the last variable, Hypothesis H3 becames

Hypothesis H4. i) F(., .) : I×R →P(R) has nonempty closed values and is L(I)⊗B(R) measurable.
ii) There exists L(.) ∈ L1(I, (0,∞)) such that, for almost all t ∈ I, F(t, .) is L(t)-Lipschitz in the

sense that
dH (F(t,x1),F(t,x2)) ≤ L(t)|x1 −x2| ∀ x1,x2 ∈ R.

Denote L0 =
∫ 1
0
L(t)dt.

Corollary 4.2. Assume that Hypothesis H4 is satisfied, M0 < 2 and d(0,F(t, 0) ≤ L(t) a.e. (I).
Then there exists x(.) a solution of problem 1.1 satisfying for all t ∈ I

|x(t)| ≤
L0

2 −L0
.

References

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1Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014 Bucharest,
Romania

2Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest, Romania

∗acernea@fmil.unibuc.ro


	1. Introduction
	2. Preliminaries
	3. Existence via fixed points
	4. A Filippov type existence result
	References