International Journal of Analysis and Applications

Volume 17, Number 6 (2019), 904-916

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-904

ON THE SOLUTIONS OF A CLASS OF FRACTIONAL HYPERBOLIC

INTEGRO-DIFFERENTIAL INCLUSIONS

AURELIAN CERNEA1,2,∗

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

Romania

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

∗Corresponding author: acernea@fmil.unibuc.ro

Abstract. We study a Darboux problem associated to a fractional hyperbolic integro-differential inclusion

defined by Caputo-Katugampola fractional derivative and we prove several existence results for this problem.

1. Introduction

In the last years one may see a strong development of the theory of differential equations and inclusions of

fractional order ( [2, 7, 11–13] etc.). The main reason is that fractional differential equations are very useful

tools in order to model many physical phenomena.

Recently, a generalized Caputo-Katugampola fractional derivative was proposed in [10] by Katugampola

and afterwards he provided the existence of solutions for fractional differential equations defined by this

derivative. This Caputo-Katugampola fractional derivative extends the well known Caputo and Caputo-

Hadamard fractional derivatives into a single form. Even if Katugampola fractional integral operator is an

Erdélyi-Kober type operator ( [8]) it is argued ( [10]) that is not possible to derive Hadamard equivalence op-

erators from Erdélyi-Kober type operators. Also, in some recent papers [1, 15], several qualitative properties

of solutions of fractional differential equations defined by Caputo-Katugampola derivative were obtained.

Received 2019-04-04; accepted 2019-04-29; published 2019-11-01.

2010 Mathematics Subject Classification. 34A60, 26A42, 26A33.

Key words and phrases. fractional derivative; hyperbolic differential inclusion; decomposable set.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

904

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-904


Int. J. Anal. Appl. 17 (6) (2019) 905

The set-valued framework was studied in [5, 6] and existence results are obtained in the situation when the

values of the set-valued map are not necesarily convex provided the set-valued map is Lipschitz in the state

variable.

In the present paper we study fractional hyperbolic integro-differential inclusions of the form

Dα,ρc u(x,y) ∈ F(x,y,u(x,y), (I
α,ρ
0 u)(x,y)) a.e. (x,y) ∈ Π, (1.1)

u(x, 0) = ϕ(x), u(0,y) = ψ(y) (x,y) ∈ Π, (1.2)

where Π = [0,T1] × [0,T2], ϕ(.) : [0,T1] → R, ψ(.) : [0,T2] → R with ϕ(0) = ψ(0), F(., .) : Π × R × R →

P(R) is a set-valued map, Iα,ρ0 is the generalized left-sided mixed integral and D
α,ρ
c is the mixed Caputo-

Katugampola fractional derivative, α = (α1,α2) ∈ [0, 1) × [0, 1) and ρ = (ρ1,ρ2), ρ1,ρ2 > 0.

The goal of the present paper is twofold. First, we show that Filippov’s ideas ( [9]) can be suitably

adapted in order to obtain the existence of a solution of problem 1.1-1.2. We recall that for an ”ordinary”

differential inclusion defined by a lipschitzian set-valued map with nonconvex values Filippov’s theorem ( [9])

provides the existence of a solution starting from a given ”quasi” solution. At the same time, the result

gives an estimate between the ”quasi” solution and the solution of the differential inclusion. Secondly, we

obtain the existence of solutions continuously depending on a parameter for problem 1.1-1.2. This result is,

in fact, a continuous version of our first result. In the proof of this second theorem we essentially use a result

of Bressan and Colombo ( [3]) concerning the existence of continuous selections of lower semicontinuous

multifunctions with decomposable values. Our second theorem allows to deduce a continuous selection of

the solution set of the problem considered.

The results in the present paper may be regarded as extensions of the results in [5, 6] to the hyperbolic

framework and generalizations of the results in [4] obtained for problems defined by Caputo’s derivative to

the more general problem 1.1-1.2.

The paper is organized as follows: in Section 2 we briefly recall some preliminary results that we will use

in the sequel and in Section 3 we prove the main results of the paper.

2. Preliminaries

In [10] the following notions were introduced. Let ρ > 0.

Definition 2.1. a) The generalized left-sided fractional integral of order α > 0 of a Lebesgue integrable

function f : [0,∞) → R is defined by

Iα,ρf(t) =
ρ1−α

Γ(α)

∫ t
0

(tρ −sρ)α−1sρ−1f(s)ds,



Int. J. Anal. Appl. 17 (6) (2019) 906

provided the right-hand side is pointwise defined on (0,∞) and Γ(.) is (Euler’s) Gamma function defined by

Γ(α) =
∫∞

0
tα−1e−tdt.

b) The generalized fractional derivative, corresponding to the generalized left-sided fractional integral of a

function f : [0,∞) → R is defined by

Dα,ρf(t) = (t1−ρ
d

dt
)n(In−α,ρ)(t) =

ρα−n+1

Γ(n−α)
(t1−ρ

d

dt
)n

∫ t
0

sρ−1f(s)

(tρ −sρ)α−n+1
ds

if the integral exists and n = [α] + 1.

c) The Caputo-Katugampola generalized fractional derivative is defined by

Dα,ρc f(t) = (D
α,ρ[f(s) −

n−1∑
k=0

f(k)(0)

k!
sk])(t)

If n = 1 (i.e., α ∈ [0, 1)), the Caputo-Katugampola fractional derivative is

Dα,ρc f(t) =
ρα

Γ(1 −α)

∫ t
0

f′(s)

(tρ −sρ)α
ds.

We note that if ρ = 1, the Caputo-Katugampola fractional derivative becames the well known Caputo

fractional derivative. On the other hand, passing to the limit with ρ → 0+, the above definition yields the

Hadamard fractional derivative.

Consider I1 = [0,T1], I2 = [0,T2] and Π = [0,T1]× [0,T2]. Denote by L(Π) the σ- algebra of the Lebesgue

measurable subsets of Π and by B(R) the family of all Borel subsets of R.

Let C(Π, R) be the Banach space of all continuous functions from Π to R with the norm ||u||C =

sup{|u(x,y)|; (x,y) ∈ Π} and L1(Π, R) be the Banach space of functions u(·, ·) : Π → R which are integrable,

normed by ‖u‖L1 =
∫ T1

0

∫ T2
0
|u(x,y)|dxdy.

Let ρ1,ρ2 > 0. The corresponding versions of the above definition in the case of function with two

variables are as follows.

Definition 2.2. a) The generalized left-sided mixed integral of order α = (α1,α2) ∈ [0, 1)× [0, 1) of f(., .) ∈

L1(Π, R) is defined by

(I
α,ρ
0 f)(x,y) =

ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1·

f(s,t)dsdt.

b) The mixed Caputo-Katugampola fractional-order derivative of order α of

f(., .) ∈ L1(Π, R) is defined by

(Dα,ρc f)(x,y) = (I
1−α,ρ
0

∂2f
∂x∂y

)(x,y) =

ρ
α1
1 ρ

α2
2

Γ(1−α1)Γ(1−α2)
∫ x

0

∫ y
0

(xρ1 −sρ1 )−α1 (yρ2 − tρ2 )−α2 ∂
2f

∂s∂t
(s,t)dsdt.

In the definition above by 1 −α we mean (1 −α1, 1 −α2) ∈ (0, 1] × (0, 1].



Int. J. Anal. Appl. 17 (6) (2019) 907

Definition 2.3. A function u(., .) ∈ C(Π, R) is said to be a solution of problem 1.1-1.2 if there exists

f(., .) ∈ L1(Π, R) such that

f(x,y) ∈ F(x,y,u(x,y), (Iα,ρ0 u)(x,y)) a.e. (Π), (2.1)

u(x,y) = ν(x,y) + (I
α,ρ
0 f)(x,y), (x,y) ∈ Π, (2.2)

where ν(x,y) = ϕ(x) + ψ(y) −ϕ(0).

The pair (u(., .),f(., .)) is called a trajectory-selection pair of problem 1.1-1.2.

The previous definition is justified by the fact that a simple computation shows that u(., .) satisfies

Dα,ρc u(x,y) ≡ f(x,y), u(x, 0) ≡ ϕ(x), u(0,y) ≡ ψ(y), (x,y) ∈ Π if and only if 2.2 is verified.

Let (X,d) be a metric space. 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) = inf{d(x,y); y ∈ B}.

With cl(A) we denote the closure of the set A ⊂ X.

Recall that a subset D ⊂ L1(Π, R) is said to be decomposable if for any u(·),v(·) ∈ D and any subset

A ∈L(Π) one has uχA + vχB ∈ D, where B = I\A. We denote by D the family of all decomposable closed

subsets of L1(Π, R).

Let G(., .) : Π×Rm →P(Rn) be a set-valued map. Recall that G(., .) is called L(Π)⊗B(Rm) measurable

if for any closed subset C ⊂ Rn we have {(x,y,z) ∈ Π × Rm; F(x,y,z) ∩C} 6= ∅}∈L(Π) ⊗B(Rm).

Consider the Banach space S := {(ϕ,ψ) ∈ C(I1, R) × C(I2, R); ϕ(0) = ψ(0)} endowed with the norm

||(ϕ,ψ)|| = ||ϕ||C + ||ψ||C and for (ϕ,ψ) ∈ S denote S(ϕ,ψ) the set of all solutions of problem 1.1-1.2.

We recall now some results that we are going to use in the next section.

Lemma 2.1. ( [14]) Let G(·, ·) : Π → P(Rn) be a compact valued measurable multifunction and h(·, ·) :

Π → Rn a measurable function.

Then there exists a measurable selection g(·, ·) of G(·, ·) such that

‖g(x,y) −h(x,y)‖ = d(h(x,y),G(x,y)), a.e. (Π).

Next (S, d) is a separable metric space and X is a Banach space. We recall that a multifunction G(·) : S →

P(X) is said to be lower semicontinuous (l.s.c.) if for any closed subset C ⊂ X, the subset {s ∈ S; G(s) ⊂ C}

is closed in S.

Lemma 2.2. ( [3]) Let G∗(., .) : Π ×S →P(Rn) be a closed valued L(Π) ⊗B(S) measurable multifunction

such that G∗((x,y), .) is l.s.c. for any (x,y) ∈ Π.



Int. J. Anal. Appl. 17 (6) (2019) 908

Then the set-valued map H(.) defined by

H(s) = {g ∈ L1(Π, Rn); g(x,y) ∈ G∗(x,y,s) a.e. (Π)}

is l.s.c. with nonempty decomposable closed values if and only if there exists a continuous mapping q(.) :

S → L1(Π, R) such that

d(0,G∗(x,y,s)) ≤ q(s)(x,y) a.e. (Π), ∀s ∈ S.

Lemma 2.3. ( [3]) Let H(.) : S → D be a l.s.c. set-valued map with closed decomposable values and let

f(.) : S → L1(Π, Rn), p(.) : S → L1(Π, R) be continuous such that the multifunction G(.) : S → D defined

by

G(s) = cl{h ∈ H(s); ||h(x,y) −f(s)(x,y)|| < p(s)(x,y) a.e. (Π)}

has nonempty values.

Then G(.) has a continuous selection.

3. The main results

In order to obtain an existence result for problem 1.1-1.2 one need the following assumptions on F(., .).

Hypothesis H1. F(., .) : Π × R × R → P(R) is a set-valued map with non-empty, compact values that

verifies:

i) For all u,v ∈ R, F(., .,u,v) is measurable.

ii) There exists K1,K2 > 0 such that for almost all (x,y) ∈ Π,

dH(F(x,y,u1,v1),F(x,y,u2,v2)) ≤ K1|u1 −u2| + K2|v1 −v2|,

∀u1,v1,u2,v2 ∈ R.

In what follows g(., .) ∈ L1(Π, R) is given and there exists ξ(., .) ∈ L1(Π, R+) with Ξ :=

sup(x,y)∈Π(I
α,ρ
0 ξ)(x,y) < +∞ which satisfies

d(g(x,y),F(x,y,w(x,y), (I
α,ρ
0 w)(x,y))) ≤ ξ(x,y) a.e. (Π),

where w(., .) is a solution of the fractional hyperbolic differential equation

Dα,ρc w(x,y) = g(x,y) (x,y) ∈ Π, (3.1)

w(x, 0) = ϕ1(x), w(0,y) = ψ1(y) (x,y) ∈ Π, (3.2)

with (ϕ1,ψ1) ∈ S.

Set ν1(x,y) = ϕ1(x) + ψ1(y) −ϕ1(0), (x,y) ∈ Π, K3 =
T
ρ1
1 T

ρ2
2 ρ

1−α1
1 ρ

1−α2
2

Γ(1+α1)Γ(1+α2)
and K = K3(K1 + K2K3).



Int. J. Anal. Appl. 17 (6) (2019) 909

Theorem 3.1. Let Hypothesis H1 be satisfied, K < 1 and consider g(., .), ξ(., .), w(., .) as above, (ϕ,ψ) ∈ S

and ν(x,y) = ϕ(x) + ψ(y) −ϕ(0), (x,y) ∈ Π.

Then there exists (v(., .),f(., .)) a trajectory-selection pair of problem 1.1-1.2 such that

|v(x,y) −w(x,y)| ≤
||ν −ν1||C + Ξ

1 −K
, ∀(x,y) ∈ Π, (3.3)

|f(x,y) −g(x,y)| ≤
(K1 + K2K3)(||ν −ν1||C + Ξ)

1 −K
+ ξ(x,y), a.e. (Π). (3.4)

Proof. We define f0(., .) = g(., .), v0(., .) = w(., .). It follows from Lemma 2.1 that there exists a measurable

function f1(., .) such that f1(x,y) ∈ F(x,y,v0(x,y),

(I
α,ρ
0 v0)(x,y)) a.e. (Π) and for almost all (x,y) ∈ Π

|f0(x,y) −f1(x,y)| = d(g(x,y),F(x,y,v0(x,y), (I
α,ρ
0 v0)(x,y))) ≤ ξ(x,y).

Define, for (x,y) ∈ Π

v1(x,y) = ν(x,y) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1·

tρ2−1f1(s,t)dsdt.

Since

w(x,y) = ν1(x,y) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1·

tρ2−1f0(s,t)dsdt.

one has

|v1(x,y) −v0(x,y)| ≤ |ν(x,y) −ν1(x,y)| +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1

(yρ2 − tρ2 )α2−1sρ1−1tρ2−1||f1(s,t) −f0(s,t)||dsdt ≤ ||ν −ν1||C +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1ξ(s,t)dsdt ≤ ||ν −ν1||C + Ξ.

From Lemma 2.1 we deduce the existence of a measurable function f2(., .) such that f2(x,y) ∈

F(x,y,v1(x,y), (I
r
0v1)(x,y)) a.e. (Π) and for almost all (x,y) ∈ Π

|f2(x,y) −f1(x,y)| ≤ d(f1(x,y),F(x,y,v1(x,y), (I
α,ρ
0 v1)(x,y))) ≤

dH(F(x,y,v0(x,y), (I
α,ρ
0 v0)(x,y)),F(x,y,v1(x,y), (I

α,ρ
0 v1)(x,y))) ≤

K1|v1(x,y) −v0(x,y)| + K2|(I
α,ρ
0 v0)(x,y) − (I

α,ρ
0 v1)(x,y)| ≤

K1(||ν −ν1||C + Ξ) + K2
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1·

sρ1−1tρ2−1(||ν −ν1||C + Ξ)dsdt = (K1 + K2K3)(||ν −ν1||C + Ξ).

Define, for (x,y) ∈ Π

v2(x,y) = ν(x,y) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1·

tρ2−1f2(s,t)dsdt



Int. J. Anal. Appl. 17 (6) (2019) 910

and one has

|v2(x,y) −v1(x,y)| ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1

sρ1−1tρ2−1|f2(s,t) −f1(s,t)|dsdt ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2−

tρ2 )α2−1sρ1−1tρ2−1(K1 + K2K3)(||ν −ν1||C + Ξ)dsdt = K(||ν −ν1||C + Ξ).

Assuming that for some p ≥ 2 we have already constructed the sequences (vi(., .))
p
i=1, (fi(., .))

p
i=1 satisfying

|vp(x,y) −vp−1(x,y)| ≤ Kp−1(||ν −ν1||C + Ξ) (x,y) ∈ Π, (3.5)

|fp(x,y) −fp−1(x,y)| ≤ (K1 + K2K3)Kp−2(||ν −ν1||C + Ξ) a.e. (Π). (3.6)

We apply Lemma 2.1 and we find a measurable function fp+1(., .) such that

fp+1(x,y) ∈ F(x,y,vp(x,y), (I
α,ρ
0 vp)(x,y)) a.e. (Π) and for almost all (x,y) ∈ Π

|fp+1(x,y) −fp(x,y)| ≤ d(fp+1(x,y),F(x,y,vp−1(x,y), (I
α,ρ
0 vp−1)(x,y)))

≤ dH(F(x,y,vp(x,y), (I
α,ρ
0 vp)(x,y)),F(x,y,vp−1(x,y), (I

α,ρ
0 vp−1)(x,y)))

≤ L1|vp(x,y) −vp−1(x,y)| + L2|(I
α,ρ
0 vp)(x,y) − (I

α,ρ
0 vp−1)(x,y)| ≤

K1[K
p−2(||ν −ν1||C + Ξ)] + K2K3Kp−2(||ν −ν1||C + Ξ) = Kp−1(||ν −ν1||C

+Ξ)(K1 + K2K3).

Define, for (x,y) ∈ Π

vp+1(x,y) = ν(x,y)+

ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1fp+1(s,t)dsdt.
(3.7)

We have

|vp+1(x,y) −vp(x,y)| ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1

sρ1−1tρ2−1|fp+1(s,t) −fp(s,t)|dsdt ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2−

tρ2 )α2−1sρ1−1tρ2−1Kp−1(||ν −ν1||C + Ξ)(K1 + K2K3)dsdt = Kp−1(||ν −ν1||C

+Ξ)K3(K1 + K2K3) = K
p(||ν −ν1||C + Ξ).

Taking into account 3.5 we deduce that the sequence (vp(., .))p≥0 is Cauchy in C(Π, R), so it converges to

v(., .) ∈ C(Π, R). From 3.6 we infer that the sequence (fp(., .))p≥0 is Cauchy in L1(Π, R), thus it converges

to f(., .) ∈ L1(Π, R).

Using the fact that the values of F(., .) are closed we get that f(x,y) ∈ F(x,y,

v(x,y), (I
α,ρ
0 v)(x,y)) a.e. (Π).



Int. J. Anal. Appl. 17 (6) (2019) 911

One may write successively,

|ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1fp(s,t)dsdt−
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1f(s,t)dsdt| ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1|fp(s,t)−

f(s,t)|dsdt ≤ ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1·

(K1 + K2K3)|up−1(s,t) −u(s,t)|dsdt ≤ K||up−1(., .) −u(., .)||C.

Thus, we pass to the limit in 3.2 and we get that v(., .) is a solution of problem 1.1-1.2. At the same time,

by adding inequalities 3.5 for any (x,y) ∈ Π we have

|vp(x,y) −w(x,y)| ≤ |vp(x,y) −vp−1(x,y)| + |vp−1(x,y) −vp−2(x,y)|

+ . . . + |v2(x,y) −v1(x,y)| + |v1(x,y) −v0(x,y)| ≤

(Kp−1 + Kp−2 + ... + K + 1)(||ν −ν1||C + Ξ) ≤
||ν−ν1||C+Ξ

1−K .

(3.8)

Similarly, by adding inequalities 3.6 for almost all (x,y) ∈ Π we have

|fp(x,y) −g(x,y)| ≤ |fp(x,y) −fp−1(x,y)| + |fp−1(x,y)−

fp−2(x,y)| + . . . + |f2(x,y) −f1(x,y)| + |f1(x,y) −f0(x,y)| ≤

(K1 + K2K3)(K
p−2 + ... + K + 1)(||ν −ν1||C + Ξ) + ξ(x,y) ≤

(K1 + K2K3)
||ν−ν1||C+Ξ

1−K + ξ(x,y).

(3.9)

Finally we pass to the limit with p →∞ in (3.8) and (3.9) and we get (3.3) and (3.4), respectively, which

completes the proof. �

If in Theorem 3.1 we take g = 0, w = 0, ϕ1 = 0, ψ1 = 0 then we obtain the following existence result for

solutions of problem 1.1-1.2.

Corollary 3.1. Let Hypothesis H1 be satisfied, K < 1 and assume that there exists ξ(., .) ∈ L1(Π, R+) with

Ξ := sup(x,y)∈Π(I
α,ρ
0 ξ)(x,y) < +∞ such that d(0,F(x,y, 0,

0)) ≤ ξ(x,y) ∀(x,y) ∈ Π.

Then there exists v(., .) ∈ C(Π, R) a solution of problem 1.1-1.2 such that

|v(x,y)| ≤
||ν||C + Ξ

1 −K
, ∀(x,y) ∈ Π.

Next we obtain a continuous version of Theorem 3.1.

Hypothesis H2. i) S is a separable metric space, ϕ(.) → C(I1, R),ψ(.) : S → C(I2, R) and ε(.) : S →

(0,∞) are continuous mappings and ϕ(s)(0) ≡ ψ(s)(0).



Int. J. Anal. Appl. 17 (6) (2019) 912

ii) There exists the continuous mappings ϕ1(.) → C(I1, R),ψ1(.) : S → C(I2, R) g(.) : S → L1(Π, R),

ξ(.) : S → L1(Π, R) and w(.) : S → C(Π, R) such that ϕ1(s)(0) ≡ ψ1(s)(0),

(Dw(s))α,ρc (x,y) = g(s)(x,y) a.e. (Π), ∀s ∈ S,

w(s)(x, 0) = ϕ1(s)(x), w(s)(0,y) = ψ1(s)(y) (x,y) ∈ Π, ∀s ∈ S,

d(g(s)(x,y),F(x,y,w(s)(x,y), (I
α,ρ
0 w(s))(x,y))) ≤ ξ(s)(x,y) a.e. (Π),∀s ∈ S

and the mapping s → Ξ(s) := sup(x,y)∈Π(I
α,ρ
0 ξ(s))(x,y) is continuous.

We use next the following notations ν(s)(x,y) = ϕ(s)(x) + ψ(s)(y) − ϕ(s)(0), ν1(s)(x,y) = ϕ1(s)(x) +

ψ1(s)(y) −ϕ1(s)(0) (x,y) ∈ Π, a(s) = sup(x,y)∈Π |ν(s)(x,y) −ν1(s)(x,y)| s ∈ S.

Theorem 3.2. Assume that Hypotheses H1 and H2 are satisfied and K < 1.

Then there exist a continuous mapping v(.) : S → C(Π, R) such that for any s ∈ S, v(s)(., .) is a solution

of problem 1.1 which satisfies v(s)(x, 0) = ϕ(s)(x), v(s)(0,y) = ψ(s)(y) (x,y) ∈ Π,s ∈ S and

|v(s)(x,y) −w(s)(x,y)| ≤
a(s) + ε(s) + Ξ(s)

1 −K
∀(x,y) ∈ Π,∀s ∈ S.

Proof. We make the following notations

v0(., .) = w(., .), ξp(s) := K
p−1(a(s) + ε(s) + Ξ(s)), p ≥ 1.

We consider the set-valued maps G0(.),H0(.) defined, respectively, by

G0(s) = {h ∈ L1(Π, R); h(x,y) ∈ F(x,y,w(s)(x,y), (I
α,ρ
0 w(s))(x,y))a.e.(Π)}

H0(s) = cl{h ∈ G0(s); |h(x,y) −g(s)(x,y)| < ξ(s)(x,y) +
1

K3
ε(s)}.

Taking into account that d(g(s)(x,y),F(x,y,w(s)(x,y), (I
α,ρ
0 w(s))(x,y)) ≤

ξ(s)(x,y) < ξ(s)(x,y) + 1
K3
ε(s) the set H0(s) is not empty.

Set F∗0 (x,y,s) = F(x,y,w(s)(x,y), (I
α,ρ
0 w(s))(x,y)) and note that

d(0,F∗0 (x,y,s)) ≤ |g(s)(x,y)| + ξ(s)(x,y) =: ξ
∗(s)(x,y)

and ξ∗(.) : S → L1(I, R) is continuous.

Applying now Lemma 2.2 and Lemma 2.3 we obtain the existence of a continuous selection f0 of H0 such

that ∀s ∈ S, (x,y) ∈ Π,

f0(s)(x,y) ∈ F(x,y,w(s)(x,y), (I
α,ρ
0 w(s))(x,y)) a.e. (Π), ∀s ∈ S,

|f0(s)(x,y) −g(s)(x,y)| ≤ ξ0(s)(x,y) = ξ(s)(x,y) +
1

K3
ε(s).



Int. J. Anal. Appl. 17 (6) (2019) 913

We define

v1(s)(x,y) = ν(s)(x,y) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1·

zρ1−1tρ2−1f0(s)(z,t)dzdt

and one has

|v1(s)(x,y) −v0(s)(x,y)| ≤ a(s) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1·

(yρ2 − tρ2 )α2−1zρ1−1tρ2−1|f0(s)(z,t) −g(s)(z,t)|dzdt ≤ a(s)+
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1(ξ(s)(z,t)

+ 1
K3
ε(s))dzdt ≤ a(s) + Ξ(s) + ε(s) =: ξ1(s), (x,y) ∈ Π,s ∈ S.

We construct the sequences of approximations fp(., .) : S → L1(Π, R), vp(., .) : S → C(Π, R) with the

following properties:

a) fp(., .) : S → L1(Π, R), vp(., .) : S → C(Π, R) are continuous,

b) fp(s)(x,y) ∈ F(x,y,vp(s)(x,y), (I
α,ρ
0 vp(s))(x,y)), a.e. (Π), s ∈ S,

c) |fp(s)(x,y) −fp−1(s)(x,y)| ≤ (K1 + K2K3)ξp(s), a.e. (Π), s ∈ S.

d) vp+1(s)(x,y) = ν(s)(x,y) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1 ·

zρ1−1tρ2−1fp(s)(z,t)dzdt, (x,y) ∈ Π,s ∈ S.

Assume that we have already constructed fi(.),vi(.) satisfying a)-c) and define vp+1(.) as in d). From c)

and d) one has

|vp+1(s)(x,y) −vp(s)(x,y)| ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1·

(yρ2 − tρ2 )α2−1zρ1−1tρ2−1|fp(s)(z,t) −fp−1(s)(z,t)|dzdt ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1yρ2 − tρ2 )α2−1zρ1−1tρ2−1(K1+

K2K3)ξp(s)dzdt = K3(K1 + K2K3)ξp(s) = ξp+1(s).

(3.10)

On the other hand,

d(fp(s)(x,y),F(x,y,vp+1(s)(x,y), (I
α,ρ
0 vp+1(s))(x,y))) ≤

K1|vp+1(s)(x,y) −vp(s)(x,y)| + K2
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1·

(yρ2 − tρ2 )α2−1zρ1−1tρ2−1|vp+1(s)(z,t) −vp(s)(z,t)|dzdt ≤

(K1 + K2K3)ξp+1(s).

(3.11)

For any s ∈ S we define the set-valued maps Gp+1(s) = {u ∈ L1(Π, R); u(x,y) ∈

F(x,y,vp+1(s)(x,y), (I
α,ρ
0 vp+1(s))(x,y)) a.e. (Π)} and

Hp+1(s) = cl{u ∈ Gp+1(s); |u(x,y) −fp(s)(x,y)| < (K1 + K2K3)ξp+1(s)}.

We note that from 3.11 the set Hp+1(s) is not empty.

Set F∗p+1(x,y,s) = F(x,y,vp+1(s)(x,y), (I
α,ρ
0 vp+1(s))(x,y)) and note that

d(0,F∗p+1(x,y,s)) ≤ |fp(s)(x,y)| + (K1 + K2K3)ξp+1(s) =: ξ
∗
p+1(s)(x,y)



Int. J. Anal. Appl. 17 (6) (2019) 914

and ξ∗p+1(.) : S → L1(I, R) is continuous.

By Lemma 2.2 and Lemma 2.3 we obtain the existence of a continuous function fp+1(.) : S → L1(Π, R)

such that

fp+1(s)(x,y) ∈ F(x,y,vp+1(s)(x,y), (I
α,ρ
0 vp+1(s))(x,y)) a.e. (Π), ∀s ∈ S,

|fp+1(s)(x,y) −fp(s)(x,y)| ≤ (K1 + K2K3)ξp+1(s) ∀s ∈ S, (x,y) ∈ Π.

From 3.10, c) and d) we obtain

|vp+1(s)(., .) −vp(s)(., .)|C ≤ ξp+1(s) = Kp(a(s) + ε(s) + Ξ(s)), (3.12)

|fp+1(s)(., .) −fp(s)(., .)|1 ≤ Kp−1(K1 + K2K3)T1T2(a(s) + ε(s) + Ξ(s)). (3.13)

Thus, fp(s)(., .), vp(s)(., .) are Cauchy sequences in the Banach spaces L
1(Π, R) and C(Π, R), respectively.

Consider f(.) : S → L1(Π, R), v(.) : S → C(Π, R) their limits. The function s → a(s) + ε(s) + Ξ(s) is

continuous, hence locally bounded. Therefore 3.13 implies that for every s′ ∈ S the sequence fp(s′)(., .)

satisfies the Cauchy condition uniformly with respect to s′ on some neighborhood of s. Therefore, s →

f(s)(., .) is continuous from S into L1(Π, R).

As before, from 3.12, vp(s)(., .) is Cauchy in C(Π, R) locally uniformly with respect to s. Hence s →

v(s)(., .) is continuous from S into C(Π, R). At the same time, since vp(s)(., .) converges uniformly to

v(s)(., .) and

d(fp(s)(x,y),F(x,y,v(s)(x,y), (I
α,ρ
0 v(s))(x,y)) ≤

(K1 + K2K3)|vp(s)(x,y) −v(s)(x,y)| a.e. (Π), ∀s ∈ S

passing to the limit along a subsequence of fp(s)(., .) converging pointwise to f(s)(., .) we obtain

f(s)(x,y) ∈ F(x,y,v(s)(x,y), (Iα,ρ0 v(s))(x,y)) a.e. (Π), ∀s ∈ S.

One may write successively,

|ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1fp(s)(z,t)dzdt−
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1f(s)(z,t)dzdt| ≤
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1|fp(s)(z,t)−

f(s)(z,t)|dzdt ≤ ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1

(K1 + K2K3)|vp−1(s)(z,t) −v(s)(z,t)|dzdt ≤ K||vp−1(s)(., .) −v(s)(., .)||C.

So, we pass to the limit in d) and we get ∀(x,y) ∈ Π,s ∈ S

v(s)(x,y) = ν(s)(x,y) +
ρ
1−α1
1 ρ

1−α2
2

Γ(α1)Γ(α2)

∫ x
0

∫ y
0

(xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1·

zρ1−1tρ2−1f(s)(z,t)dzdt,

i.e., v(s)(., .) is the required solution.



Int. J. Anal. Appl. 17 (6) (2019) 915

Finally, by adding inequalities 3.10 for all p ≥ 1 we get

|vp+1(s)(x,y) −w(s)(x,y)| ≤
p+1∑
l=1

ξl(s) ≤
a(s) + ε(s) + Ξ(s)

1 −K
. (3.14)

Passing to the limit in 3.14 we obtain the conclusion of the theorem. �

Theorem 3.2 allows to provide a continuous selection of the solution set of problem 1.1-1.2.

Hypothesis H3. Hypothesis H1 is satisfied, K < 1 and there exists q(., .) ∈ L1(Π, R+) with

sup(x,y)∈Π(I
α,ρ
0 q)(x,y) < ∞ such that d(0,F(x,y, 0, 0)) ≤ q(x,y) a.e. (Π).

Corollary 3.2. Assume that Hypothesis H3 is satisfied.

Then there exists a function v(., .) : Π × S → R such that

a) v(., (ξ,η)) ∈S(ξ,η), ∀(ξ,η) ∈ S.

b) (ξ,η) → v(., (ξ,η)) is continuous from S into C(Π, R).

Proof. We take S = S, ϕ(µ,η) = µ, ψ(µ,η) = η ∀(µ,η) ∈ S, ε(.) : S → (0,∞) an arbitrary continuous

function, g(.) = 0, w(.) = 0, ξ(s)(x,y) ≡ q(x,y) ∀s = (µ,η) ∈ S, (x,y) ∈ Π and we apply Theorem 3.2 in

order to obtain the conclusion of the corollary. �

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	1. Introduction
	2. Preliminaries
	3. The main results
	References