Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 379876, 25 pages
doi:10.1155/2011/379876
Research Article
Global Uniqueness Results for
Fractional Order Partial Hyperbolic
Functional Differential Equations
Saı̈d Abbas,1 Mouffak Benchohra,2 and Juan J. Nieto3
1
Laboratoire de Mathématiques, Université de Saı̈da, P.O. Box 138, Saı̈da 20000, Algeria
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria
3
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela,
15782 Santiago de Compostela, Spain
2
Correspondence should be addressed to Juan J. Nieto, juanjose.nieto.roig@usc.es
Received 22 November 2010; Accepted 29 January 2011
Academic Editor: J. J. Trujillo
Copyright q 2011 Saı̈d Abbas et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We investigate the global existence and uniqueness of solutions for some classes of partial
hyperbolic differential equations involving the Caputo fractional derivative with finite and infinite
delays. The existence results are obtained by applying some suitable fixed point theorems.
1. Introduction
In this paper, we provide sufficient conditions for the global existence and uniqueness of
some classes of fractional order partial hyperbolic differential equations. As a first problem,
we discuss the global existence and uniqueness of solutions for an initial value problem IVP
for short of a system of fractional order partial differential equations given by
D0r u x, y f x, y, ux,y ; if x, y ∈ J,
u x, y φ x, y ; if x, y ∈ J,
ux, 0 ϕx, u 0, y ψ y ; x, y ∈ 0, ∞,
c
1.1
1.2
1.3
Rn , c Dr
where J 0, ∞ × 0, ∞, J : −α, ∞ × −β, ∞ \ 0, ∞ × 0, ∞; α, β > 0, φ ∈ CJ,
0
is the Caputo’s fractional derivative of order r r1 , r2 ∈ 0, 1 × 0, 1, f : J × C → Rn , is a
given function ϕ : 0, ∞ → Rn , ψ : 0, ∞ → Rn are given absolutely continuous functions
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with ϕx φx, 0, ψy φ0, y for each x, y ∈ 0, ∞, and C : C−α, 0 × −β, 0, Rn is
the space of continuous functions on −α, 0 × −β, 0.
If u ∈ C−α, ∞ × −β, ∞, Rn , then for any x, y ∈ J define ux,y by
ux,y s, t u x s, y t ,
for s, t ∈ −α, 0 × −β, 0 .
1.4
Next we consider the following initial value problem for partial neutral functional differential
equations with finite delay of the form
c
D0r u x, y − g x, y, ux,y f x, y, ux,y ; if x, y ∈ J,
u x, y φ x, y ; if x, y ∈ J,
ux, 0 ϕx, u 0, y ψ y ; x, y ∈ 0, ∞,
1.5
1.6
1.7
where f, φ, ϕ, ψ are as in problem 1.1–1.3, and g : J × C → Rn is a given function.
The third result deals with the existence of solutions to fractional order partial
hyperbolic functional differential equations with infinite delay of the form
D0r u x, y f x, y, ux,y ; if x, y ∈ J,
u x, y φ x, y ; if x, y ∈ J ,
ux, 0 ϕx, u 0, y ψ y ; x, y ∈ 0, ∞,
c
1.8
1.9
1.10
where ϕ, ψ are as in problem 1.1–1.3 and J R2 \ 0, ∞ × 0, ∞, f : J × B → Rn ,
φ ∈ CJ , Rn , and B is called a phase space that will be specified in Section 4.
We denote by ux,y the element of B defined by
ux,y s, t u x s, y t ;
s, t ∈ −∞, 0 × −∞, 0.
1.11
Finally we consider the following initial value problem for partial neutral functional
differential equations with infinite delay
c
D0r u x, y − g x, y, ux,y f x, y, ux,y ; if x, y ∈ J,
u x, y φ x, y ; if x, y ∈ J ,
ux, 0 ϕx, u 0, y ψ y ; x, y ∈ 0, ∞,
1.12
1.13
1.14
where f, φ, ϕ, ψ are as in problem 1.8–1.10 and g : J × B → Rn is a given continuous
function.
In this paper, we present global existence and uniqueness results for the above-cited
problems. We make use of the nonlinear alternative of Leray-Schauder type for contraction
maps on Fréchet spaces.
The problem of existence of solutions of Cauchy-type problems for ordinary
differential equations of fractional order without delay in spaces of integrable functions was
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studied in numerous works see 1, 2, a similar problem in spaces of continuous functions
was studied in 3. We can find numerous applications of differential equations of fractional
order in viscoelasticity, electrochemistry, control, porous media, electromagnetic, theory of
neolithic transition, and so forth, see 4–11. There has been a significant development in
ordinary and partial fractional differential equations in recent years; see the monographs of
Kilbas et al. 12, Lakshmikantham et al. 13, Miller and Ross 14, Samko et al. 15, the
papers of Abbas and Benchohra 16–18, Agarwal et al. 19, 20, Ahmad and Nieto 21–23,
Belarbi et al. 24, Benchohra et al. 25–27, Chang and Nieto 28, Diethelm et al. 4, 29,
Heinsalu et al. 30, Jumarie 31, Kilbas and Marzan 32, Luchko et al. 33, Magdziarz et
al. 34, Mainardi 9, Rossikhin and Shitikova 35, Vityuk and Golushkov 36, Yu and Gao
37, and Zhang 38 and the references therein.
For integer order derivative, various classes of hyperbolic differential equations were
considered on bounded domain; see, for instance, the book by Kamont 39, the papers
by Człapiński 40, Dawidowski and Kubiaczyk 41, Kamont, and Kropielnicka 42,
Lakshmikantham and Pandit 43, and Pandit 44.
2. Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used
throughout this paper. Let p ∈ N and J0 : 0, p × 0, p. Let CJ0 , Rn be the Banach space of
all continuous functions from J0 into Rn with the norm
z∞ sup z x, y ,
x,y∈J0
2.1
where · denotes a suitable complete norm on Rn . As usual, by ACJ0 , Rn we denote
the space of absolutely continuous functions from J0 into Rn and L1 J0 , Rn is the space of
Lebegue-integrable functions w : J0 → Rn with the norm
w1 p p
0
w x, y dy dx.
2.2
0
Let r1 , r2 > 0 and r r1 , r2 . For z ∈ L1 J0 , Rn , the expression
I0r z x, y 1
Γr1 Γr2 x y
0
r −1
x − sr1 −1 y − t 2 zs, tdt ds,
2.3
0
where Γ· is the Euler gamma function, is called the left-sided mixed Riemann-Liouville
integral of order r.
2
: ∂2 /∂x∂y, the mixed second-order partial derivative.
Denote by Dxy
Definition 2.1 see 36. For z ∈ L1 J0 , Rn , the Caputo fractional-order derivative of order
2
zx, y.
r ∈ 0, 1 × 0, 1 of z is defined by the expression c D0r zx, y I01−r Dxy
In the definition above by 1 − r we mean 1 − r1 , 1 − r2 ∈ 0, 1 × 0, 1.
If z is an absolutely continuous function, then its Caputo fractional derivative
c D0r zx, y exists for each x, y ∈ J0 .
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Let X be a Fréchet space with a family of seminorms { · n }n∈N . We assume that the
family of seminorms { · n } verifies:
x1 ≤ x2 ≤ x3 ≤ · · ·
for every x ∈ X.
2.4
Let Y ⊂ X, we say that Y is bounded if for every n ∈ N, there exists Mn > 0 such that
y ≤ Mn ,
n
2.5
∀y ∈ Y.
To X we associate a sequence of Banach spaces {X n , · n } as follows. For every n ∈ N, we
consider the equivalence relation ∼n defined by: x∼n y if and only if x − yn 0 for x, y ∈ X.
We denote X n X|∼n , · n the quotient space, the completion of X n with respect to · n . To
every Y ⊂ X, we associate a sequence {Y n } of subsets Y n ⊂ X n as follows. For every x ∈ X,
we denote xn the equivalence class of x of subset X n and we defined Y n {xn : x ∈ Y }.
We denote Y n , intn Y n and ∂n Y n , respectively, the closure, the interior and the boundary of
Y n with respect to · n in X n . For more information about this subject see 45.
Definition 2.2. Let X be a Fréchet space. A function N : X → X is said to be a contraction if
for each n ∈ N there exists kn ∈ 0, 1 such that
Nu − Nvn ≤ kn u − vn ,
∀u, v ∈ X.
2.6
Theorem 2.3 see 45. Let X be a Fréchet space and Y ⊂ X a closed subset in X. Let N : Y → X
be a contraction such that NY is bounded. Then one of the following statements holds:
a the operator N has a unique fixed point;
b there exists λ ∈ 0, 1, n ∈ N and u ∈ ∂n Y n such that u − λNun 0.
In the sequel we will make use of the following generalization of Gronwall’s lemma
for two independent variables and singular kernel.
Lemma 2.4 see 46. Let υ : J0 → 0, ∞ be a real function and ω·, · be a nonnegative, locally
integrable function on J. If there are constants c > 0 and 0 < l1 , l2 < 1 such that
υ x, y ≤ ω x, y c
x y
0
0
υs, t
l dt ds,
x − sl1 y − t 2
2.7
ωs, t
l dt ds,
x − sl1 y − t 2
2.8
then there exists a constant k kl1 , l2 such that
υ x, y ≤ ω x, y kc
x y
0
for every x, y ∈ J0 .
0
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3. Global Result for Finite Delay
Let us start by defining what we mean by a global solution of the problem 1.1–1.3.
Definition 3.1. A function u ∈ C0 : C−α, ∞ × −β, ∞, Rn such that its mixed derivative
2
exists and is integrable on J is said to be a global solution of 1.1–1.3 if u satisfies 1.1
Dxy
and 1.3 on J and the condition 1.2 on J.
Let h ∈ L1 J0 , Rn and consider the following problem
D0r u x, y h x, y ;
x, y ∈ J0 ,
ux, 0 ϕx, u 0, y ψ y ; x, y ∈ 0, p ,
c
3.1
ϕ0 ψ0.
For the existence of global solutions for the problem 1.1–1.3, we need the following known
lemma.
Lemma 3.2 see 16, 17. A function u ∈ ACJ0 , Rn is a global solution of problem 3.1 if and
only if ux, y satisfies
u x, y μ x, y I0r h x, y ,
x, y ∈ J0 ,
3.2
where
μ x, y ϕx ψ y − ϕ0.
3.3
As a consequence of Lemma 3.2, we have the following result.
Lemma 3.3. A function u ∈ ACJ0 , Rn is a global solution of problem 1.1–1.3 if and only if
ux, y φx, y, x, y ∈ J and ux, y satisfies
u x, y μ x, y I0r f x, y ,
x, y ∈ J0 ,
3.4
where
μ x, y ϕx ψ y − ϕ0.
3.5
For each p ∈ N, we consider following set:
Cp C −α, p × −β, p , Rn ,
3.6
and we define in C0 the seminorms by
up sup u x, y : −α ≤ x ≤ p, −β ≤ y ≤ p .
Then C0 is a Fréchet space with the family of seminorms {up }.
3.7
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Further, we present conditions for the existence and uniqueness of a global solution of
problem 1.1–1.3.
Theorem 3.4. Assume that
H1 the function f : J × C → Rn is continuous,
H2 for each p ∈ N, there exists lp ∈ CJ0 , Rn such that for each x, y ∈ J0
f x, y, u − f x, y, v ≤ lp x, y u − v ,
C
for each u, v ∈ C.
3.8
If
lp∗ pr1 r2
Γr1 1Γr2 1
3.9
< 1,
where
lp∗ sup lp x, y ,
3.10
x,y∈J0
then, there exists a unique solution for IVP 1.1–1.3 on −α, ∞ × −β, ∞.
Proof. Transform the problem 1.1–1.3 into a fixed point problem. Consider the operator
N : C0 → C0 defined by,
⎧ ⎪φ x, y ,
⎨
Nu x, y ⎪
⎩μ x, y 1
Γr1 Γr2 x y
0
x, y ∈ J,
r −1 x − sr1 −1 y − t 2 f s, t, us,t dt ds, x, y ∈ J.
0
3.11
Clearly, from Lemma 3.3, the fixed points of N are solutions of 1.1–1.3. Let u be a possible
solution of the problem u λNu for some 0 < λ < 1. This implies that for each x, y ∈ J0 ,
we have
u x, y λμ x, y λ
Γr1 Γr2 x y
0
r −1 x − sr1 −1 y − t 2 f s, t, us,t dt ds.
3.12
0
Introducing fs, t, 0 − fs, t, 0, it follows by H2 that
u x, y ≤ μ x, y 1
Γr1 Γr2 f ∗ pr1 r2
Γr1 1Γr2 1
x y
r −1
x − sr1 −1 y − t 2 lp s, tus,t C dt ds,
0
0
3.13
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where
f ∗ sup f x, y, 0 .
3.14
x,y∈J0
We consider the function τ defined by
τ x, y sup us, t : −α ≤ s ≤ x, −β ≤ t ≤ y; x, y ∈ 0, p .
3.15
Let x∗ , y∗ ∈ −α, x × −β, y be such that τx, y ux∗ , y∗ . If x∗ , y∗ ∈ J0 , then by the
previous inequality, we have for x, y ∈ J0 ,
u x, y ≤ μ x, y 1
Γr1 Γr2 f ∗ pr1 r2
Γr1 1Γr2 1
x y
r −1
x − sr1 −1 y − t 2 lp s, tτs, tdt ds.
0
3.16
0
then τx, y φC and the previous inequality holds.
If x∗ , y∗ ∈ J,
By 3.16 we obtain that
τ x, y ≤ μ x, y 1
Γr1 Γr2 ≤ μ x, y lp∗
Γr1 Γr2 f ∗ pr1 r2
Γr1 1Γr2 1
x y
r −1
x − sr1 −1 y − t 2 lp s, tτs, tdt ds
0
0
f ∗ pr1 r2
Γr1 1Γr2 1
x y
r −1
x − sr1 −1 y − t 2 τs, tdt ds,
0
3.17
0
and Lemma 2.4 implies that there exists a constant k kr1 , r2 such that
τ x, y ≤
μ p
f ∗ pr1 r2
Γr1 1Γr2 1
1
klp∗
Γr1 1Γr2 1
: Mp .
3.18
Then from 3.16, we have
up ≤ μp Mp lp∗
f ∗ pr1 r2
: Mp∗ .
Γr1 1Γr2 1 Γr1 1Γr2 1
3.19
Since for every x, y ∈ J0 , ux,y C ≤ τx, y, we have
up ≤ max φC , Mp∗ : Rp .
3.20
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Set
U u ∈ C0 : up ≤ Rp 1 ∀p ∈ N .
3.21
We will show that N : U → Cp is a contraction map. Indeed, consider v, w ∈ U. Then for
each x, y ∈ 0, p, we have
Nv x, y − Nw x, y x y
r −1 1
≤
x − sr1 −1 y − t 2 f s, t, vs,t − f s, t, ws,t dt ds
Γr1 Γr2 0 0
x y
r −1
1
≤
x − sr1 −1 y − t 2 lp,q s, tvs,t − ws,t C dt ds
Γr1 Γr2 0 0
≤
lp∗ pr1 r2
Γr1 1Γr2 1
3.22
v − wp .
Thus,
Nv − Nwp ≤
lp∗ pr1 r2
Γr1 1Γr2 1
3.23
v − wp .
Hence by 3.9, N : U → Cp is a contraction. By our choice of U, there is no u ∈ ∂n Un such
that u λNu, for λ ∈ 0, 1. As a consequence of Theorem 2.3, we deduce that N has a
unique fixed point u in U which is a solution to problem 1.1–1.3.
Now we present a global existence and uniqueness result for the problem 1.5–1.7.
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Definition 3.5. A function u ∈ C0 such that the mixed derivative Dxy
ux, y − gx, y, ux,y exists and is integrable on J is said to be a global solution of 1.5–1.7 if u satisfies equations
1.5 and 1.7 on J and the condition 1.6 on J.
Let f ∈ L1 J0 , Rn , g ∈ ACJ0 , Rn and consider the following linear problem
D0r u x, y − g x, y f x, y ;
x, y ∈ J0 ,
ux, 0 ϕx, u 0, y ψ y ; x, y ∈ 0, p ,
c
3.24
with ϕ0 ψ0.
For the existence of solutions for the problem 1.5–1.7, we need the following
lemma.
Lemma 3.6. A function u ∈ ACJ0 , Rn is a global solution of problem 3.24 if and only if ux, y
satisfies
u x, y μ x, y g x, y − gx, 0 − g 0, y g0, 0 I0r f x, y ;
x, y ∈ J0 .
3.25
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Proof. Let ux, y be a solution of problem 3.24. Then, taking into account the definition of
the fractional Caputo derivative, we have
2
u x, y − g x, y f x, y .
I01−r Dxy
3.26
2
u x, y − g x, y I0r f x, y ,
I0r I01−r Dxy
3.27
2
I01 Dxy
u x, y − g x, y I0r f x, y .
3.28
2
I01 Dxy
u x, y − g x, y u x, y − g x, y − ux, 0 − gx, 0
− u 0, y − g 0, y u0, 0 − g0, 0 ,
3.29
u x, y μ x, y g x, y − gx, 0 − g 0, y g0, 0 I0r f x, y .
3.30
Hence, we obtain
then,
Since
we have
Now, let ux, y satisfy 3.25. It is clear that ux, y satisfies 3.24.
As a consequence of Lemma 3.6 we have the following result.
Lemma 3.7. The function u ∈ ACJ0 , Rn is a global solution of problem 1.5–1.7 if and only if u
satisfies the equation
x y
r −1 1
u x, y x − sr1 −1 y − t 2 f s, t, us,t ds dt
Γr1 Γr2 0 0
μ x, y g x, y, ux,y − g x, 0, ux,0
− g 0, y, u0,y g 0, 0, u0,0 ,
3.31
for all x, y ∈ J0 and the condition 1.6 on J.
Theorem 3.8. Assume that H1, H2, and the following condition holds
H3 For each p 1, 2, . . ., there exists a constant cp with 0 < cp < 1/4 such that for each
x, y ∈ J0 , one has
g x, y, u − g x, y, v ≤ cp u − v ,
C
for each u, v ∈ C.
3.32
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If
4cp lp∗ pr1 r2
Γr1 1Γr2 1
3.33
< 1,
then there exists a unique solution for IVP 1.5–1.7 on −α, ∞ × −β, ∞.
Proof. Transform the problem 1.5–1.7 into a fixed point problem. Consider the operator
N1 : C0 → C0 defined by,
⎧ ⎪
φ x, y ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪μ x, y g x, y, ux,y − g x, 0, ux,0
⎨
N1 u x, y −g 0, y, u0,y g 0, 0, u0,0
⎪
⎪
⎪
⎪
x y
⎪
⎪
r −1 ⎪
1
⎪
⎩ x − sr1 −1 y − t 2 f s, t, us,t dt ds,
Γr1 Γr2 0 0
x, y ∈ J,
x, y ∈ J.
3.34
From Lemma 3.7, the fixed points of N1 are solutions to problem 1.5–1.7. In order to use
the nonlinear alternative, we will obtain a priori estimates for the solutions of the integral
equation
u x, y λ μ x, y g x, y, ux,y − g x, 0, ux,0 − g 0, y, u0,y g 0, 0, u0,0
x y
r −1 λ
x − sr1 −1 y − t 2 f s, t, us,t dt ds,
Γr1 Γr2 0 0
3.35
for some λ ∈ 0, 1. Then, using H1–H3 and 3.16 we get for each x, y ∈ J0 ,
u x, y ≤ μ x, y f ∗ pr1 r2
Γr1 1Γr2 1
g x, y, ux,y g x, 0, ux,0 g 0, y, u0,y g 0, 0, u0,0 x y
r −1
1
x − sr1 −1 y − t 2 lp s, tτs, tdt ds,
Γr1 Γr2 0 0
3.36
then, we obtain
u x, y ≤ μ x, y f ∗ pr1 r2
Γr1 1Γr2 1
4cp τ x, y g x, y, 0 gx, 0, 0 g 0, y, 0 g0, 0, 0
lp∗
Γr1 Γr2 x y
0
0
r −1
x − sr1 −1 y − t 2 τs, tdt ds.
3.37
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Replacing 3.37 in the definition of τx, y we get
f ∗ pr1 r2
1
∗
μ x, y 4g
τ x, y ≤
1 − 4cp
Γr1 1Γr2 1
l∗
p
x y
Γr1 Γr2 0
r −1
x − sr1 −1 y − t 2 τs, tdt ds,
3.38
0
where lp∗ lp∗ /1 − 4cp and gp∗ supx,y∈J0 gx, y, 0.
By Lemma 2.4, there exists a constant δ δr1 , r2 such that
τp ≤
f ∗ pr1 r2
1
∗
μ 4g
p
p
1 − 4cp
Γr1 1Γr2 1
⎤
⎡
δlp∗
⎦ : Dp .
× ⎣1 Γr1 1Γr2 1
3.39
Then, from 3.37 and 3.39, we get
up ≤ μp f ∗ pr1 r2
4gp∗
Γr1 1Γr2 1
4cp Dp Dp lp∗
Γr1 1Γr2 1
:
3.40
Dp∗ .
Since for every x, y ∈ J0 , ux,y C ≤ τx, y, we have
up ≤ max φC , Dp∗ : R∗p .
3.41
U1 u ∈ C0 : up ≤ R∗p 1 ∀p 1, 2, . . . .
3.42
Set
Clearly, U1 is a closed subset of C0 . As in Theorem 3.4, we can show that N1 : U1 → C0 is a
contraction operator. Indeed
N1 v − N1 wp ≤
4cp lp∗ pr1 r2
Γr1 1Γr2 1
v − wp
3.43
for each v, w ∈ U1 and x, y ∈ J0 . From the choice of U1 , there is no u ∈ ∂n U1n such that
u λN1 u, for some λ ∈ 0, 1. As a consequence of Theorem 2.3, we deduce that N1 has a
unique fixed point u in U1 which is a solution to problem 1.5–1.7.
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4. The Phase Space B
The notation of the phase space B plays an important role in the study of both qualitative
and quantitative theory for functional differential equations. A usual choice is a seminormed
space satisfying suitable axioms, which was introduced by Hale and Kato see 47. For
further applications see, for instance, the books 48–50 and their references.
Inspired by 47, Człapiński 40 introduced the following construction of the phase
space. For any x, y ∈ J0 denote Ex,y : 0, x×{0}∪{0}×0, y, furthermore in case x y p
we write simply E. Consider the space B, ·, ·B is a seminormed linear space of functions
mapping −∞, 0 × −∞, 0 into Rn , and satisfying the following fundamental axioms which
were adapted from those introduced by Hale and Kato for ordinary differential functional
equations.
A1 If z : −∞, p × −∞, p → Rn continuous on J0 and zx,y ∈ B, for all x, y ∈ E,
then there are constants H, K, M > 0 such that for any x, y ∈ J0 the following
conditions hold:
i zx,y is in B;
ii zx, y ≤ Hzx,y B , and
iii zx,y B ≤ K sups,t∈0,x×0,y zs, t M sups,t∈Ex,y zs,t B .
A2 For the function z·, · in A1 , zx,y is a B-valued continuous function on J0 .
A3 The space B is complete.
Now, we present some examples of phase spaces see 40.
Example 4.1. Let B be the set of all functions φ : −∞, 0 × −∞, 0 → Rn which are continuous
on −α, 0 × −β, 0, α, β ≥ 0, with the seminorm
φ B
φs, t.
sup
s,t∈−α,0×−β,0
4.1
Then, we have H K M 1. The quotient space B B/ · B is isometric to the space
C−α, 0 × −β, 0, Rn of all continuous functions from −α, 0 × −β, 0 into Rn with the
supremum norm, this means that partial differential functional equations with finite delay
are included in our axiomatic model.
Example 4.2. Let Cγ be the set of all continuous functions φ : −∞, 0 × −∞, 0 → Rn for
which a limit lims,t → ∞ eγst φs, t exists, with the norm
φ Cγ
sup
eγst φs, t.
s,t∈−∞,0×−∞,0
4.2
Then we have H K M 1.
Example 4.3. Let α, β, γ ≥ 0 and let
φ
CLγ
sup
φs, t s,t∈−α,0×−β,0
0
−∞
eγst φs, tdt ds
4.3
Advances in Difference Equations
13
be the seminorm for the space CLγ of all functions φ : −∞, 0 × −∞, 0 → Rn which are
continuous on −α, 0 × −β, 0 measurable on −∞, −α × −∞, 0 ∪ −∞, 0 × −∞, −β, and
such that φCLγ < ∞. Then,
H 1,
K
0 0
−α
−β
eγst dt ds,
M 2.
4.4
5. Global Result for Infinite Delay
In this section we present a global existence and uniqueness result for the problems 1.8–
1.10 and 1.12–1.14. Let us define the space
Ω : u : R2 −→ Rn : ux,y ∈ B for x, y ∈ E0 , u|J ∈ CJ, Rn ,
5.1
where E0 : 0, ∞ × {0} ∪ {0} × 0, ∞.
2
Definition 5.1. A function u ∈ Ω such that its mixed derivative Dxy
exists and is integrable on
J is said to be a global solutionis of 1.8–1.10 if u satisfies equations 1.8 and 1.10 on J
and the condition 1.9 on J .
For each p ∈ N, we consider following set,
Cp u : −∞, p × −∞, p −→ Rn : u ∈ B ∩ CJ0 , Rn , ux,y 0 for x, y ∈ E ,
5.2
and we define in
C0 : u : R2 −→ Rn : u ∈ B ∩ C0, ∞ × 0, ∞, Rn , ux,y 0 for x, y ∈ E0
5.3
the seminorms by
up sup ux,y B sup u x, y x,y∈E
sup u x, y ,
x,y∈J0
x,y∈J0
5.4
u ∈ Cp .
Then, C0 is a Fréchet space with the family of seminorms {up }.
Theorem 5.2. Assume that
H 1 the function f : J × B → Rn is continuous and
H 2 for each p ∈ N, there exists lp ∈ CJ0 , Rn such that for and x, y ∈ J0
f x, y, u − f x, y, v ≤ lp x, y u − v ,
B
for each u, v ∈ B.
5.5
14
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If
Klp∗ pr1 r2
Γr1 1Γr2 1
5.6
< 1,
where
lp∗ sup lp x, y ,
5.7
x,y∈J0
then, there exists a unique solution for IVP 1.8–1.10 on R2 .
Proof. Transform the problem 1.8–1.10 into a fixed point problem. Consider the operator
N : Ω → Ω defined by
⎧ ⎪
⎨φ x, y ,
N u x, y ⎪
⎩μ x, y 1
Γr1 Γr2 x y
0
x, y ∈ J ,
r −1 x − sr1 −1 y − t 2 f s, t, us,t dt ds; x, y ∈ J.
0
5.8
Let v·, · : R2 → Rn be a function defined by
⎧ ⎨φ x, y ,
v x, y ⎩μx, y,
x, y ∈ J ,
x, y ∈ J.
5.9
Then, vx,y φ for all x, y ∈ E0 . For each w ∈ CJ, Rn with wx, y 0; for all x, y ∈ E0 ,
we denote by w the function defined by
⎧
⎨0,
x, y ∈ J ,
w x, y ⎩wx, y, x, y ∈ J.
5.10
If u·, · satisfies the integral equation,
u x, y μ x, y 1
Γr1 Γr2 x y
0
r −1 x − sr1 −1 y − t 2 f s, t, us,t dt ds,
5.11
0
we can decompose u·, · as ux, y wx, y vx, y; x, y ≥ 0, which implies that ux,y wx,y vx,y , for every x, y ≥ 0, and the function w·, · satisfies
w x, y 1
Γr1 Γr2 x y
0
0
r −1 x − sr1 −1 y − t 2 f s, t, ws,t vs,t dt ds.
5.12
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15
Let the operator P : C0 → C0 be defined by
x y
r −1
1
x − sr1 −1 y − t 2
Γr1 Γr2 0 0
× f s, t, ws,t vs,t dt ds;
x, y ∈ J.
P w x, y 5.13
Obviously, the operator N has a fixed point is equivalent to P having a fixed point, and so
we turn to prove that P has a fixed point. We will use the alternative to prove that P has a
fixed point. Let w be a possible solution of the problem w P w for some 0 < λ < 1. This
implies that for each x, y ∈ J0 , we have
w x, y λ
Γr1 Γr2 x y
0
r −1 x − sr1 −1 y − t 2 f s, t, ws,t vs,t dt ds.
5.14
0
This implies by H 1 that
w x, y ≤
fp∗ pr1 r2
Γr1 1Γr2 1
x y
r −1
1
x − sr1 −1 y − t 2 lp s, tws,t vs,t B dt ds,
Γr1 Γr2 0 0
5.15
where
fp∗ sup f x, y, 0 : x, y ∈ J0 .
5.16
ws,t vs,t ≤ ws,t vs,t B
B
B
≤ K sup u s, t : s, t ∈ 0, s × 0, t
5.17
But
MφB K φ0, 0.
If we name zs, t the right-hand side of 5.17, then we have
ws,t vs,t ≤ zs, t.
B
5.18
Therefore, from 5.15 and 5.18 we get
w x, y ≤
fp∗ pr1 r2
Γr1 1Γr2 1
x y
r −1
1
x − sr1 −1 y − t 2 lp s, tzs, tdt ds.
Γr1 Γr2 0 0
5.19
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Advances in Difference Equations
Replacing 5.19 in the definition of w, we have that
z x, y ≤
Kfp∗ pr1 r2
MφB
Γr1 1Γr2 1
x y
Klp∗
r −1
x − sr1 −1 y − t 2 zs, tdt ds.
Γr1 Γr2 0 0
5.20
By Lemma 2.4, there exists a constant δ δr1 , r2 such that
zp ≤
Kfp∗ pr1 r2
MφB
Γr1 1Γr2 1
δKlp∗
× 1
Γr1 1Γr2 1
5.21
: M.
Then, from 5.19, we have
wp ≤ M
lp∗ pr1 r2
Γr1 1Γr2 1
fp∗ pr1 r2
Γr1 1Γr2 1
∗ .
: M
5.22
Since for every x, y ∈ J0 , wx,y B ≤ zx, y, we have
∗.
∗ : R
wp ≤ max φB , M
5.23
∗ 1 ∀p ∈ N .
U w ∈ C0 : wp ≤ R
5.24
Set
We will show that P : U → Cp is a contraction operator. Indeed, consider w, w∗ ∈ U . Then
for each x, y ∈ J0 , we have
P w x, y − P w∗ x, y x y
r −1
1
≤
x − sr1 −1 y − t 2
Γr1 Γr2 0 0
× f s, t, ws,t vs,t − f s, t, w∗ s,t vs,t dt ds
≤
1
Γr1 Γr2 ≤K
x y
0
0
r −1
x − sr1 −1 y − t 2 lp s, tws,t − w∗ s,t dt ds
lp∗ pr1 r2
Γr1 1Γr2 1
B
w − w∗ p .
5.25
Advances in Difference Equations
17
Thus,
P w − P w∗ ≤
p
Klp∗ pr1 r2
Γr1 1Γr2 1
w − w∗ p .
5.26
Hence by 5.6, P : U → Cp is a contraction. By our choice of U , there is no w ∈ ∂n U n
such that w λP w, for λ ∈ 0, 1. As a consequence of Theorem 2.3, we deduce that N has
a unique fixed point which is a solution to problem 1.8–1.10.
Now, we present an existence result for the problem 1.12–1.14.
2
Definition 5.3. A function u ∈ Ω such that the mixed derivative Dxy
ux, y − gx, y, ux,y exists and is integrable on J is said to be a global solutionis of 1.12–1.14 if u satisfies
equations 1.12 and 1.14 on J and the condition 1.13 on J .
Theorem 5.4. Let f, g : J × B → Rn be continuous functions. Assume that H 1, H 2, and the
following condition hold.
H 3 For each p 1, 2, . . ., there exists a constant cp with 0 < Kcp < 1/4 such that for any
x, y ∈ J0 , one has
g x, y, u − g x, y, v ≤ cp u − v ,
B
for any u, v ∈ B.
5.27
If
4cp Klp∗ pr1 r2
Γr1 1Γr2 1
< 1,
for each p ∈ N,
5.28
then, there exists a unique solution for IVP 1.12–1.14 on R2 .
Proof. Consider the operator N1 : Ω → Ω defined by
⎧ ⎪
φ x, y ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪μ x, y g x, y, ux,y − g x, 0, ux,0
⎪
⎪
⎪
⎨
−g 0, y, u0,y g 0, 0, u0,0
N1 u x, y ⎪
x y
⎪
⎪
⎪
r −1
1
⎪
⎪
x − sr1 −1 y − t 2
⎪
⎪ Γr1 Γr2 ⎪
⎪
0 0
⎪
⎩ ×f s, t, u dt ds,
s,t
x, y ∈ J ,
5.29
x, y ∈ J.
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Advances in Difference Equations
In analogy to Theorem 5.2, we consider the operator P1 : C0 → C0 defined by
P1 w x, y g x, y, wx,y vx,y − g x, 0, wx,0 vx,0
− g 0, y, w0,y v0,y g 0, 0, w0,0 v0,0
x y
r −1
1
x − sr1 −1 y − t 2
Γr1 Γr2 0 0
x, y ∈ J.
× f s, t, ws,t vs,t dt ds,
5.30
In order to use the nonlinear alternative, we will obtain a priori estimates for the solutions of
the integral equation
w x, y λ g x, y, wx,y vx,y − g x, 0, wx,0 vx,0
−g 0, y, w0,y v0,y g 0, 0, w0,0 v0,0
x y
r −1 λ
x − sr1 −1 y − t 2 f s, t, ws,t vs,t dt ds,
Γr1 Γr2 0 0
5.31
for some λ ∈ 0, 1. Then from H1 –H3 , 5.15, and 5.18 we get for each x, y ∈ J0 ,
w x, y ≤
fp∗ pr1 r2
4cp z x, y
Γr1 1Γr2 1
g x, y, 0 gx, 0, 0 g 0, y, 0 g0, 0, 0
x y
r −1
1
x − sr1 −1 y − t 2 lp s, tzs, tdt ds.
Γr1 Γr2 0 0
5.32
Replacing 5.32 in the definition of zx, y, we get
z x, y ≤
1
M φ B 4K φ0, 0 4K g 0, 0, φ0, 0
1 − 4Kcp
!
Kfp∗ pr1 r2
∗
4Kgp Γr1 1Γr2 1
l∗
p
Γr1 Γr2 x y
0
r −1
x − sr1 −1 y − t 2 zs, tdt ds,
0
where lp∗ x, y lp∗ /1 − 4Kcp and gp∗ sup{gx, y, 0 : x, y ∈ J0 }.
5.33
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19
By 5.32 and Lemma 2.4, there exists a constant δ δr1 , r2 such that
z x, y ≤
1
M φ B 4K φ0, 0 4K g 0, 0, φ0, 0
1 − 4Kcp
!
Kfp∗ pr1 r2
∗
4Kgp Γr1 1Γr2 1
⎤
⎡
δlp∗
⎦ : D .
× ⎣1 Γr1 1Γr2 1
5.34
Then, from 5.32 and 5.34, we get
wp ≤
D lp∗ fp∗ pr1 r2
Γr1 1Γr2 1
∗
4cp D 4gp∗ : D .
5.35
Since for every x, y ∈ J0 , wx,y B ≤ zx, y, we have
wp ≤ max φB , D∗ : R∗ .
5.36
U1 w ∈ C0 : wp ≤ R∗ 1 .
5.37
Set
Clearly, U1 is a closed subset of C0 . As in Theorem 5.2, we can show that P1 : U1 → C0 is a
contraction operator. Indeed
N1 v − N1 wp ≤
4cp
Klp∗ pr1 r2
Γr1 1Γr2 1
v − wp ,
5.38
for each v, w ∈ U1 , andx, y ∈ J0 . From the choice of U1 , there is no w ∈ ∂n U1 n such that
w λP1 w, for some λ ∈ 0, 1. As a consequence of Theorem 2.3, we deduce that N1 has a
unique fixed point which is a solution to problem 1.12–1.14.
20
Advances in Difference Equations
6. Examples
Example 6.1. As an application of our results we consider the following partial hyperbolic
functional differential equations with finite delay of the form
c
cp
; if x, y ∈ 0, ∞ × 0, ∞,
e
1 u x − 1, y − 2 ux, 0 x, u 0, y y2 ; x, y ∈ 0, ∞,
u x, y x y2 ;
x, y ∈ −1, ∞ × −2, ∞ \ 0, ∞ × 0, ∞,
D0r u x, y xy2
6.1
where
cp Γr1 1Γr2 1
;
pr1 r2
p ∈ N∗ .
6.2
Set
f x, y, ux,y cp
;
exy2 1 u x − 1, y − 2 x, y ∈ 0, ∞ × 0, ∞.
6.3
For each p ∈ N∗ and x, y ∈ 0, p × 0, p, we have
c
f x, y, ux,y − f x, y, ux,y ≤ p u − u .
C
e2
6.4
Hence conditions H1 and H2 are satisfied with lp∗ cp /e2 . We will show that condition
3.9 holds for all p ∈ N∗ . Indeed
lp∗ pr1 r2
Γr1 1Γr2 1
1
< 1,
e2
6.5
which is satisfied for each r1 , r2 ∈ 0, 1 × 0, 1. Consequently Theorem 3.4 implies that
problem 6.1 has a unique global solution defined on −1, ∞ × −2, ∞.
Example 6.2. We consider now the following partial hyperbolic functional differential
equations with infinite delay of the form
c
D0r u x, y 4exy
cp π 2 exy e−x−y −x −y γθη e
u x θ, y η dηdθ
×
; if x, y ∈ 0, ∞ × 0, ∞,
2
−∞ −∞ 1 x θ2
1 yη
2
u x, y x y ;
x, y ∈ R2 \ 0, ∞ × 0, ∞,
ux, 0 x, u 0, y y2 ; x, y ∈ 0, ∞,
6.6
where cp 3pr1 r2 /Γr1 1Γr2 1, p ∈ N∗ and γ a positive real constant.
Advances in Difference Equations
21
Let
Bγ "
u ∈ C−∞, 0 × −∞, 0, R :
lim
θ,η → ∞
#
eγθη u θ, η exists in R .
6.7
The norm of Bγ is given by
uγ sup eγθη u θ, η .
6.8
−∞<θ, η≤0
Let u : R2 → R such that ux,y ∈ Bγ , x, y ∈ E : 0, p × {0} ∪ {0} × 0, p, then
lim
θ,η → ∞
eγθη ux,y θ, η eγθ−xη−y u θ, η
lim
θ,η → ∞
e−γxy
eγθη u θ, η < ∞.
lim
6.9
θ,η → ∞
Hence, ux,y ∈ Bγ . Finally we prove that
ux,y K
γ
|us, t| M sup us,t γ ,
sup
s,t∈0,x×0,y
s,t∈Ex,y
6.10
where K M 1 and H 1.
If x θ ≤ 0, y η ≤ 0 we get
ux,y sup{|us, t| : s, t ∈ −∞, 0 × −∞, 0},
γ
6.11
and if x θ ≥ 0, y η ≥ 0 then we have
ux,y sup |us, t| : s, t ∈ 0, x × 0, y .
γ
6.12
Thus, for all x θ, y η ∈ 0, p, we get
ux,y γ
|us, t| sup
s,t∈−∞,0×−∞,0
sup
|us, t|.
s,t∈0,x×0,y
6.13
Then,
ux,y sup us,t γ
γ
s,t∈E
sup
|us, t|.
s,t∈0,x×0,y
6.14
Bγ , · γ is a Banach space. We conclude that Bγ is a phase space.
Let
4exy
f x, y, u cp π 2 exy e−x−y −x −y
−∞
−∞
eγθη u x θ, y η
2 dη dθ;
1 x θ2 1 y η
6.15
22
Advances in Difference Equations
for each x, y, u ∈ J × Bγ . Then for each p ∈ N∗ , x, y ∈ 0, p × 0, p and u, v ∈ Bγ , we have
f x, y, u − f x, y, v −x −y
eγθη u x θ, y η
4exy
dη dθ
cp π 2 exy e−x−y −∞ −∞ 1 x θ2 1 y η2
−
4e
cp
−x −y
xy
π 2 exy
e−x−y 4exy
≤
cp π 2 exy e−x−y 4exy
≤
cp π 2 exy e−x−y 4exy
≤
cp π 2 exy e−x−y ≤
e
cp
xy
exy
e−x−y −∞
−x −y
−∞
−∞
−∞
e
v x θ, y η
dη
dθ
2
2
1 x θ
1 yη
γθη
eγθη u x θ, y η − v x θ, y η dη dθ
2 1 x θ2 1 y η
6.16
eγθη u θ, η − v θ, η dη dθ
1 θ2 1 η2
−∞
0
∞
0
1
dη dθu − vγ
1 θ2 1 η2
u − vγ .
Hence, condition H 2 is satisfied with lp x, y exy /cp exy e−x−y . Since
$
lp∗
exy
: x, y ∈ 0, ∞ × 0, ∞
sup
xy
−x−y
cp e
e
%
≤
1
cp
6.17
and K 1, we have
Klp∗ pr1 r2
Γr1 1Γr2 1
1
< 1.
3
6.18
Hence, condition 5.6 holds for each r1 , r2 ∈ 0, 1 × 0, 1 and all p ∈ N∗ . Consequently
Theorem 5.2 implies that problem 6.6 has a unique global solution defined on R2 .
Acknowledgments
The authors are grateful to the referees for their remarks. The research of J. J. Nieto has been
partially supported by Ministerio de Ciencia e Innovacion and FEDER, project MTM200761724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PN.
Advances in Difference Equations
23
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