Fractional Order Partial Hyperbolic Functional Differential Equations

doi:10.1155/2011/379876

Research Article

Global Uniqueness Results for

Fractional Order Partial Hyperbolic Functional Differential Equations

Sa¨ıd Abbas,

Mouffak Benchohra,

and Juan J. Nieto

1Laboratoire de Math´ematiques, Universit´e de Sa¨ıda, P.O. Box 138, Sa¨ıda 20000, Algeria

2Laboratoire de Math´ematiques, Universit´e de Sidi Bel-Abb`es, P.O. Box 89, Sidi Bel-Abb`es 22000, Algeria

3Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

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

Copyrightq2011 Sa¨ıd 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 problemIVP for shortof a system of fractional order partial differential equations given by

_c D^r₀u

x, y f

x, y, u_x,y

; if x, y

∈J, 1.1

u x, y

φ x, y

; if x, y

∈J, 1.2

ux,0 ϕx, u 0, y

ψ y

; x, y∈0,∞, 1.3

whereJ 0,∞×0,∞,J: −α,∞×−β,∞\0,∞×0,∞;α, β >0,φ∈CJ,Rⁿ,^cD^r₀ is the Caputo’s fractional derivative of orderr r1, r2∈0,1×0,1, f :J× C → Rⁿ, is a given functionϕ:0,∞ → Rⁿ,ψ :0,∞ → Rⁿare given absolutely continuous functions

withϕx φx,0,ψy φ0, yfor eachx, y ∈0,∞, andC:C−α,0×−β,0,Rⁿis the space of continuous functions on−α,0×−β,0.

Ifu∈C−α,∞×−β,∞,Rⁿ, then for anyx, y∈Jdefineu_x,yby u_x,ys, t u

x s, y t

, fors, 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

cD^r₀ u

x, y

−g

x, y, u_x,y f

x, y, u_x,y

; if x, y

∈J, 1.5 u

x, y φ

x, y

; if x, y

∈J, 1.6

ux,0 ϕx, u 0, y

ψ y

; x, y∈0,∞, 1.7

wheref,φ,ϕ,ψare as in problem1.1–1.3, andg:J× C → Rⁿis 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

_c D^r₀u

x, y f

x, y, u_x,y

; if x, y

∈J, 1.8

u x, y

φ x, y

; if x, y

∈J, 1.9

ux,0 ϕx, u 0, y

ψ y

; x, y∈0,∞, 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, andBis called a phase space that will be specified inSection 4.

We denote byu_x,ythe element ofBdefined by u_x,ys, 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

cD₀^r u

x, y

−g

x, y, u_x,y f

x, y, u_x,y

; if x, y

∈J, 1.12 u

x, y φ

x, y

; if x, y

∈J, 1.13

ux,0 ϕx, u 0, y

ψ y

; x, y∈0,∞, 1.14

wheref,φ,ϕ,ψ are as in problem1.8–1.10and 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 Fr´echet 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

studied in numerous workssee1,2, a similar problem in spaces of continuous functions was studied in3. 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,see4–11. 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 Ross14, Samko et al.15, the papers of Abbas and Benchohra16–18, Agarwal et al.19,20, Ahmad and Nieto21–23, Belarbi et al.24, Benchohra et al. 25–27, Chang and Nieto28, Diethelm et al.4,29, Heinsalu et al.30, Jumarie31, Kilbas and Marzan32, Luchko et al.33, Magdziarz et al.34, Mainardi9, Rossikhin and Shitikova35, Vityuk and Golushkov36, Yu and Gao 37, and Zhang38and 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 Człapi ´nski 40, Dawidowski and Kubiaczyk 41, Kamont, and Kropielnicka 42, Lakshmikantham and Pandit43, and Pandit44.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Letp∈NandJ0: 0, p×0, p. LetCJ0,Rⁿbe the Banach space of all continuous functions fromJ0intoRⁿwith the norm

z_∞ sup

x,y∈J0

z

x, y, 2.1

where · denotes a suitable complete norm on Rⁿ. As usual, by ACJ0,Rⁿ we denote the space of absolutely continuous functions fromJ0 intoRⁿ andL¹J0,Rⁿis the space of Lebegue-integrable functionsw:J₀ → Rⁿwith the norm

w_{1 p}

0

_p

0

w

x, ydy dx. 2.2

Letr₁, r₂>0 andr r1, r₂. Forz∈L¹J0,Rⁿ, the expression I₀^rz

x, y

1 Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

zs, tdt ds, 2.3

where Γ· is the Euler gamma function, is called the left-sided mixed Riemann-Liouville integral of orderr.

Denote byD²_xy:∂²/∂x∂y, the mixed second-order partial derivative.

Definition 2.1see36. Forz ∈ L¹J0,Rⁿ, the Caputo fractional-order derivative of order r∈0,1×0,1ofzis defined by the expression^cD₀^rzx, y I₀^1−rD²_xyzx, y.

In the definition above by 1−rwe mean1−r₁,1−r₂∈0,1×0,1.

If z is an absolutely continuous function, then its Caputo fractional derivative ^cD^r₀zx, yexists for eachx, y∈J₀.

LetXbe a Fr´echet space with a family of seminorms{ · n}_n∈N. We assume that the family of seminorms{ · n}verifies:

x₁≤ x₂ ≤ x₃≤ · · · for everyx∈X. 2.4

LetY ⊂X, we say thatYis bounded if for everyn∈N, there existsMn>0 such that y

n≤Mn, ∀y∈Y. 2.5

ToXwe associate a sequence of Banach spaces{Xⁿ, · n}as follows. For everyn∈N, we consider the equivalence relation∼ndefined by:x∼nyif and only ifx−yn0 forx, y∈X.

We denoteXⁿ X|∼n, · nthe quotient space, the completion ofXⁿwith respect to · n. To everyY ⊂ X, we associate a sequence{Yⁿ}of subsetsYⁿ ⊂Xⁿas follows. For everyx∈X, we denotex_nthe equivalence class ofxof subsetXⁿand we definedYⁿ {x_n :x∈Y}.

We denoteYⁿ, int_nYⁿand∂_nYⁿ, respectively, the closure, the interior and the boundary of Yⁿwith respect to · ninXⁿ. For more information about this subject see45.

Definition 2.2. LetXbe a Fr´echet space. A functionN :X → Xis said to be a contraction if for eachn∈Nthere existsk_n∈0,1such that

Nu−Nv_n≤k_nu−v_n, ∀u, v∈X. 2.6 Theorem 2.3see45. Let X be a Fr´echet space andY ⊂Xa closed subset inX. LetN:Y → X be a contraction such thatNYis bounded. Then one of the following statements holds:

athe operatorNhas a unique fixed point;

bthere existsλ∈0,1, n∈Nandu∈∂_nYⁿsuch thatu−λNun0.

In the sequel we will make use of the following generalization of Gronwall’s lemma for two independent variables and singular kernel.

Lemma 2.4see46. Letυ:J₀ → 0,∞be a real function andω·,·be a nonnegative, locally integrable function onJ. If there are constantsc >0 and 0< l₁,l₂<1 such that

υ x, y

≤ω x, y

c _x

0

_y

0

υs, t x−s^l1

y−t_l2dt ds, 2.7

then there exists a constantkkl1, l2such that

υ x, y

≤ω x, y

kc _x

0

_y

0

ωs, t x−s^l1

y−tl2dt ds, 2.8

for everyx, y∈J₀.

3. Global Result for Finite Delay

Let us start by defining what we mean by a global solution of the problem1.1–1.3.

Definition 3.1. A functionu ∈C₀ : C−α,∞×−β,∞,Rⁿsuch that its mixed derivative D_xy² exists and is integrable onJis said to be a global solution of1.1–1.3ifusatisfies1.1 and1.3onJand the condition1.2onJ.

Leth∈L¹J0,Rⁿand consider the following problem _c

D₀^ru x, y

h x, y

;

x, y

∈J₀, ux,0 ϕx, u

0, y ψ

y

; x, y∈ 0, p

, ϕ0 ψ0.

3.1

For the existence of global solutions for the problem1.1–1.3, we need the following known lemma.

Lemma 3.2see16,17. A functionu∈ ACJ0,Rⁿis a global solution of problem3.1if and only ifux, ysatisfies

u x, y

μ

x, y I₀^rh

x, y

,

x, y

∈J0, 3.2

where

μ x, y

ϕx ψ y

−ϕ0. 3.3

As a consequence ofLemma 3.2, we have the following result.

Lemma 3.3. A functionu ∈ ACJ0,Rⁿis a global solution of problem1.1–1.3if and only if ux, y φx, y, x, y∈Jandux, ysatisfies

u x, y

μ

x, y I₀^rf

x, y

,

x, y

∈J0, 3.4

where

μ x, y

ϕx ψ y

−ϕ0. 3.5

For eachp∈N, we consider following set:

C_pC

−α, p

×

−β, p ,Rⁿ

, 3.6

and we define inC0the seminorms by u_psup u

x, y:−α≤x≤p, −β≤y≤p

. 3.7

ThenC₀is a Fr´echet space with the family of seminorms{up}.

Further, we present conditions for the existence and uniqueness of a global solution of problem1.1–1.3.

Theorem 3.4. Assume that

H1the functionf:J× C → Rⁿis continuous,

H2for eachp∈N, there existsl_p∈CJ0,Rⁿsuch that for eachx, y∈J₀ f

x, y, u

−f

x, y, v≤lp

x, y

u−v_C, for eachu, v∈ C. 3.8

If

l^∗_pp^{r1 r2}

Γr1 1Γr2 1 <1, 3.9 where

l^∗_p sup

x,y∈J0

lp

x, y

, 3.10

then, there exists a unique solution for IVP1.1–1.3on−α,∞×−β,∞.

Proof. Transform the problem1.1–1.3into a fixed point problem. Consider the operator N:C0 → C0defined by,

Nu

x, y

⎧⎪

⎨

⎪⎩ φ

x, y

,

x, y

∈J, μ

x, y 1

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, u_s,t

dt ds, x, y

∈J.

3.11

Clearly, fromLemma 3.3, the fixed points ofNare solutions of1.1–1.3. Letube a possible solution of the problemuλNufor some 0< λ <1. This implies that for eachx, y∈J₀, we have

u x, y

λμ

x, y λ

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, u_s,t

dt ds. 3.12

Introducingfs, t,0−fs, t,0, it follows byH2that u

x, y≤μ

x, y f^∗p^{r1 r2} Γr₁ 1Γr2 1 1

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tu_s,t

Cdt ds,

3.13

where

f^∗ sup

x,y∈J0

f

x, y,0. 3.14

We consider the functionτdefined by τ

x, y

sup us, t:−α≤s≤x, −β≤t≤y; x, y∈ 0, p

. 3.15

Letx^∗, y^∗ ∈−α, x×−β, ybe such thatτx, y ux^∗, y^∗. Ifx^∗, y^∗∈J₀, then by the previous inequality, we have forx, y∈J₀,

u

x, y≤μ

x, y f^∗p^{r1 r2} Γr₁ 1Γr2 1 1

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tτs, tdt ds.

3.16

Ifx^∗, y^∗∈J, then τx, y φ_Cand the previous inequality holds.

By3.16we obtain that τ

x, y

≤μ

x, y f^∗p^{r1 r2} Γr₁ 1Γr2 1 1

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tτs, tdt ds

≤μ

x, y f^∗p^{r1 r2} Γr1 1Γr2 1 l^∗_p

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

τs, tdt ds,

3.17

andLemma 2.4implies that there exists a constantkkr1, r₂such that

τ x, y

≤μ

p

f^∗p^{r1 r2} Γr₁ 1Γr2 1

1 kl^∗_p

Γr₁ 1Γr2 1

:M_p. 3.18

Then from3.16, we have u_p≤μ

p

f^∗p^{r1 r2} Γr₁ 1Γr2 1

Mpl^∗_p

Γr₁ 1Γr2 1 :M_p^∗. 3.19

Since for everyx, y∈J0, ux,yC≤τx, y, we have u_p≤maxφ

C, M_p^∗

:R_p. 3.20

Set

U

u∈C₀:u_p≤R_p 1∀p∈N

. 3.21

We will show thatN : U → C_pis a contraction map. Indeed, considerv, w∈ U. Then for eachx, y∈0, p, we have

Nv x, y

−Nw x, y

≤ 1 Γr1Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1f

s, t, v_s,t

−f

s, t, w_s,tdt ds

≤ 1 Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

l_p,qs, tv_s,t−w_s,t

Cdt ds

≤ l^∗_pp^{r1 r2}

Γr1 1Γr2 1v−w_p.

3.22

Thus,

Nv−Nw_p≤ l^∗_pp^{r1 r2}

Γr₁ 1Γr2 1v−w_p. 3.23 Hence by3.9,N:U → C_pis a contraction. By our choice ofU, there is nou∈∂_nUⁿsuch thatu λNu, forλ ∈ 0,1. As a consequence ofTheorem 2.3, we deduce thatN has a unique fixed pointuinUwhich is a solution to problem1.1–1.3.

Now we present a global existence and uniqueness result for the problem1.5–1.7.

Definition 3.5. A functionu∈C₀ such that the mixed derivativeD²_xyux, y−gx, y, ux,y exists and is integrable onJis said to be a global solution of1.5–1.7ifusatisfies equations 1.5and1.7onJand the condition1.6onJ.

Letf∈L¹J0,Rⁿ, g ∈ACJ0,Rⁿand consider the following linear problem

cD^r₀ u

x, y

−g x, y

f x, y

;

x, y

∈J₀, ux,0 ϕx, u

0, y ψ

y

; x, y∈ 0, p

, 3.24

withϕ0 ψ0.

For the existence of solutions for the problem 1.5–1.7, we need the following lemma.

Lemma 3.6. A functionu∈ACJ0,Rⁿis a global solution of problem3.24if and only ifux, y satisfies

u x, y

μ x, y

g x, y

−gx,0−g 0, y

g0,0 I₀^r f

x, y

;

x, y

∈J0. 3.25

Proof. Letux, ybe a solution of problem3.24. Then, taking into account the definition of the fractional Caputo derivative, we have

I₀^1−rD²_xy u

x, y

−g x, y

f x, y

. 3.26

Hence, we obtain

I₀^rI₀^1−rD_xy² u

x, y

−g x, y

I₀^rf

x, y

, 3.27

then,

I₀¹D_xy² u

x, y

−g x, y

I₀^rf

x, y

. 3.28

Since

I₀¹D²_xy 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

we have u

x, y μ

x, y g

x, y

−gx,0−g 0, y

g0,0 I₀^r f

x, y

. 3.30

Now, letux, ysatisfy3.25. It is clear thatux, ysatisfies3.24.

As a consequence ofLemma 3.6we have the following result.

Lemma 3.7. The functionu∈ACJ0,Rⁿis a global solution of problem1.5–1.7if and only ifu satisfies the equation

u x, y

1 Γr1Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, u_s,t ds dt μ

x, y g

x, y, u_x,y

−g

x,0, u_x,0

−g

0, y, u_0,y g

0,0, u_0,0 ,

3.31

for allx, y∈J0and the condition1.6onJ.

Theorem 3.8. Assume thatH1,H2, and the following condition holds

H3For each p 1,2, . . ., there exists a constant c_p with 0 < c_p < 1/4 such that for each x, y∈J0, one has

g x, y, u

−g

x, y, v≤cpu−v_C, for eachu, v∈ C. 3.32

If

4cp

l^∗_pp^{r1 r2}

Γr₁ 1Γr2 1<1, 3.33 then there exists a unique solution for IVP1.5–1.7on−α,∞×−β,∞.

Proof. Transform the problem1.5–1.7into a fixed point problem. Consider the operator N1:C0 → C0defined by,

N₁u x, y

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩ φ

x, y

,

x, y

∈J, μ

x, y g

x, y, u_x,y

−g

x,0, u_x,0

−g

0, y, u_0,y g

0,0, u_0,0 1

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, u_s,t

dt ds, x, y

∈J.

3.34 FromLemma 3.7, the fixed points ofN₁are solutions to problem1.5–1.7. 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, u_x,y

−g

x,0, u_x,0

−g

0, y, u_0,y g

0,0, u_0,0 λ

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−tr2−1f

s, t, u_s,t

dt ds, 3.35

for someλ∈0,1. Then, usingH1–H3and3.16we get for eachx, y∈J0, u

x, y≤μ

x, y f^∗p^{r1 r2} Γr₁ 1Γr2 1 g

x, y, u_x,y g

x,0, u_x,0 g

0, y, u_0,y g

0,0, u_0,0 1

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tτs, tdt ds,

3.36 then, we obtain

u

x, y≤μ

x, y f^∗p^{r1 r2} Γr₁ 1Γr2 1 4c_pτ

x, y g

x, y,0 gx,0,0 g

0, y,0 g0,0,0 l_p^∗

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

τs, tdt ds.

3.37

Replacing3.37in the definition ofτx, ywe get

τ x, y

≤ 1 1−4c_p

μ

x, y f^∗p^{r1 r2}

Γr₁ 1Γr2 1 4g^∗

l^∗_p Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

τs, tdt ds,

3.38

wherel^∗_pl^∗_p/1−4c_pandg_p^∗sup_x,y∈J

0gx, y,0.

ByLemma 2.4, there exists a constantδδr1, r₂such that τ_p≤ 1

1−4c_p μ

p

f^∗p^{r1 r2}

Γr₁ 1Γr2 1 4g_p^∗

×

⎡

⎣1 δl^∗_p Γr1 1Γr2 1

⎤

⎦:Dp.

3.39

Then, from3.37and3.39, we get u_p≤μ

p

f^∗p^{r1 r2}

Γr₁ 1Γr2 1 4g^∗_p 4c_pD_p Dpl^∗_p

Γr₁ 1Γr2 1 :D_p^∗.

3.40

Since for everyx, y∈J₀, u_x,yC ≤τx, y, we have u_p≤max

φC, D_p^∗

:R^∗_p. 3.41

Set

U1

u∈C0:u_p≤R^∗_p 1 ∀p1,2, . . .

. 3.42

Clearly,U₁is a closed subset ofC₀. As inTheorem 3.4, we can show thatN₁ :U₁ → C₀is a contraction operator. Indeed

N1v−N₁wp≤

4c_p l^∗_pp^{r1 r2} Γr₁ 1Γr2 1

v−w_p 3.43

for eachv, w ∈ U₁ andx, y ∈ J₀. From the choice ofU₁, there is nou ∈ ∂_nUⁿ₁ such that uλN1u, for someλ ∈0,1. As a consequence ofTheorem 2.3, we deduce thatN1has a unique fixed pointuinU₁which is a solution to problem1.5–1.7.

4. The Phase Space B

The notation of the phase spaceBplays 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 Katosee 47. For further applications see, for instance, the books48–50and their references.

Inspired by47, Człapi ´nski40introduced the following construction of the phase space. For anyx, y∈J0denoteE_x,y: 0, x×{0}∪{0}×0, y, furthermore in casexyp we write simplyE. Consider the spaceB,·,·Bis a seminormed linear space of functions mapping−∞,0×−∞,0intoRⁿ, and satisfying the following fundamental axioms which were adapted from those introduced by Hale and Kato for ordinary differential functional equations.

A1Ifz : −∞, p×−∞, p → Rⁿcontinuous onJ0and z_x,y ∈ B, for allx, y ∈ E, then there are constants H, K, M > 0 such that for anyx, y ∈ J0 the following conditions hold:

iz_x,yis inB;

iizx, y ≤Hzx,yB, and

iiizx,yB≤Ksups,t∈0,x×0,yzs, t Msup_s,t∈Ex,yzs,tB.

A2For the functionz·,·inA1, z_x,yis aB-valued continuous function onJ₀. A3The spaceBis complete.

Now, we present some examples of phase spacessee40.

Example 4.1. LetBbe the set of all functionsφ:−∞,0×−∞,0 → Rⁿwhich are continuous on−α,0×−β,0,α, β≥0, with the seminorm

φ

B sup

s,t∈−α,0×−β,0

φs, t. 4.1

Then, we haveH K M 1. The quotient spaceB 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. LetCγ be the set of all continuous functions φ : −∞,0×−∞,0 → Rⁿ for which a limit lim_{s,t → ∞}e^{γs t}φs, texists, with the norm

φ

Cγ sup

s,t∈−∞,0×−∞,0e^{γs t}φs, t. 4.2 Then we haveHKM1.

Example 4.3. Letα, β, γ ≥0 and let φ

CLγ sup

s,t∈−α,0×−β,0

φs, t ₀

−∞e^{γs t}φs, tdt ds 4.3

be the seminorm for the space CLγ of all functionsφ : −∞,0×−∞,0 → Rⁿ which are continuous on−α,0×−β,0measurable on−∞,−α×−∞,0∪−∞,0×−∞,−β, and such thatφCLγ <∞. Then,

H1, K

₀

−α

₀

−βe^{γs t}dt ds, M2. 4.4

5. Global Result for Infinite Delay

In this section we present a global existence and uniqueness result for the problems1.8–

1.10and1.12–1.14. Let us define the space Ω:

u:R²−→Rⁿ:u_x,y∈Bfor x, y

∈E0, u|_J∈CJ,Rⁿ

, 5.1

whereE0: 0,∞× {0} ∪ {0} ×0,∞.

Definition 5.1. A functionu∈Ωsuch that its mixed derivativeD²_xyexists and is integrable on J is said to be a global solutionis of1.8–1.10ifusatisfies equations1.8and1.10onJ and the condition1.9onJ.

For eachp∈N, we consider following set, C_p u:

−∞, p

×

−∞, p

−→Rⁿ:u∈B∩CJ0,Rⁿ, ux,y0 for x, y

∈E

, 5.2

and we define in C₀:

u:R²−→Rⁿ:u∈B∩C0,∞×0,∞,Rⁿ, u_x,y0 for x, y

∈E₀

5.3

the seminorms by

u_p sup

x,y∈E

u_x,y

B sup

x,y∈J0

u x, y sup

x,y∈J0

u

x, y, u∈C_p.

5.4

Then,C₀is a Fr´echet space with the family of seminorms{up}.

Theorem 5.2. Assume that

H1the functionf:J×B → Rⁿis continuous and

H2for eachp∈N, there existsl_p ∈CJ0,Rⁿsuch that for andx, y∈J₀ f

x, y, u

−f

x, y, v≤l_p x, y

u−v_B, for eachu, v∈B. 5.5

If

Kl^∗_pp^{r1 r2}

Γr1 1Γr2 1 <1, 5.6 where

l^∗_p sup

x,y∈J0

l_p x, y

, 5.7

then, there exists a unique solution for IVP1.8–1.10onR².

Proof. Transform the problem1.8–1.10into a fixed point problem. Consider the operator N:Ω → Ωdefined by

Nu x, y

⎧⎪

⎨

⎪⎩ φ

x, y

,

x, y

∈J, μ

x, y 1

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, u_s,t

dt ds;

x, y

∈J.

5.8

Letv·,·:R² → Rⁿbe a function defined by

v x, y

⎧⎨

⎩ φ

x, y ,

x, y

∈J, μ

x, y ,

x, y

∈J. 5.9

Then,v_x,yφfor allx, y∈E0. For eachw∈CJ,Rⁿwithwx, y 0; for allx, y∈E0, we denote bywthe function defined by

w x, y

⎧⎨

⎩

0,

x, y

∈J, w

x, y ,

x, y

∈J. 5.10

Ifu·,·satisfies the integral equation,

u x, y

μ

x, y 1

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−tr2−1f

s, t, u_s,t

dt ds, 5.11

we can decomposeu·,·asux, y wx, y vx, y;x, y ≥0, which implies thatu_x,y w_x,y v_x,y, for everyx, y≥0, and the functionw·,·satisfies

w x, y

1 Γr1Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, w_s,t v_s,t

dt ds. 5.12

Let the operatorP:C₀ → C₀be defined by Pw

x, y

1 Γr1Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

×f

s, t, w_s,t v_s,t

dt ds;

x, y

∈J.

5.13

Obviously, the operatorNhas a fixed point is equivalent toPhaving a fixed point, and so we turn to prove thatPhas a fixed point. We will use the alternative to prove thatPhas a fixed point. Letwbe a possible solution of the problemw Pwfor some 0 < λ <1. This implies that for eachx, y∈J0, we have

w x, y

λ Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1 f

s, t, w_s,t v_s,t

dt ds. 5.14

This implies byH1that w

x, y≤ f_p^∗p^{r1 r2} Γr₁ 1Γr2 1

1 Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tw_s,t v_s,t

Bdt ds,

5.15

where

f_p^∗sup f

x, y,0: x, y

∈J₀

. 5.16

But

w_s,t v_s,t

B≤w_s,t

B v_s,t

B

≤K sup u

s,t

: s,t

∈0, s×0, t Mφ

B Kφ0,0.

5.17

If we namezs, tthe right-hand side of5.17, then we have w_s,t v_s,t

B≤zs, t. 5.18

Therefore, from5.15and5.18we get w

x, y≤ f_p^∗p^{r1 r2} Γr₁ 1Γr2 1

1 Γr1Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tzs, tdt ds.

5.19

Replacing5.19in the definition ofw, we have that z

x, y≤ Kf_p^∗p^{r1 r2}

Γr1 1Γr2 1 Mφ

B

Kl^∗_p Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

zs, tdt ds.

5.20

ByLemma 2.4, there exists a constantδδr1, r2such that z_p≤

Kf_p^∗p^{r1 r2}

Γr1 1Γr2 1 Mφ

B

×

1 δKl_p^∗

Γr1 1Γr2 1

:M.

5.21

Then, from5.19, we have

w_p ≤M l^∗_pp^{r1 r2} Γr1 1Γr2 1

f_p^∗p^{r1 r2}

Γr1 1Γr2 1 :M^∗. 5.22

Since for everyx, y∈J0, w_x,yB≤zx, y, we have w_p ≤maxφ

B,M^∗

:R^∗. 5.23

Set

U

w∈C₀:w_p≤R^∗ 1 ∀p∈N

. 5.24

We will show thatP:U → C_pis a contraction operator. Indeed, considerw, w^∗∈U. Then for eachx, y∈J₀, we have

Pw x, y

−Pw^∗ x, y

≤ 1 Γr1Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

×f

s, t, w_s,t v_s,t

−f

s, t, w^∗_s,t v_s,tdt ds

≤ 1 Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

l_ps, tws,t−w^∗_s,t

Bdt ds

≤K l^∗_pp^{r1 r2}

Γr₁ 1Γr2 1w−w^∗_p.

5.25

Thus,

Pw−Pw^∗

p ≤ Kl^∗_pp^{r1 r2}

Γr1 1Γr2 1w−w^∗_p. 5.26

Hence by5.6,P : U → C_pis a contraction. By our choice ofU, there is now ∈ ∂_nUⁿ such thatwλPw, forλ∈0,1. As a consequence ofTheorem 2.3, we deduce thatNhas a unique fixed point which is a solution to problem1.8–1.10.

Now, we present an existence result for the problem1.12–1.14.

Definition 5.3. A functionu ∈Ωsuch that the mixed derivativeD_xy² ux, y−gx, y, u_x,y exists and is integrable on J is said to be a global solutionis of 1.12–1.14 if u satisfies equations1.12and1.14onJand the condition1.13onJ.

Theorem 5.4. Letf, g:J×B → Rⁿbe continuous functions. Assume thatH1, H2, and the following condition hold.

H3For eachp 1,2, . . ., there exists a constantc_pwith 0 < Kc_p < 1/4 such that for any x, y∈J0, one has

g x, y, u

−g

x, y, v≤c_pu−v_B, for anyu, v∈B. 5.27 If

4c_p Kl^∗_pp^{r1 r2}

Γr1 1Γr2 1 <1, for eachp∈N, 5.28

then, there exists a unique solution for IVP1.12–1.14onR². Proof. Consider the operatorN₁:Ω → Ωdefined by

N₁u x, y

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ φ

x, y

,

x, y

∈J, μ

x, y g

x, y, u_x,y

−g

x,0, u_x,0

−g

0, y, u_0,y g

0,0, u_0,0 1

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

×f

s, t, u_s,t

dt ds,

x, y

∈J.

5.29

In analogy toTheorem 5.2, we consider the operatorP₁ :C₀ → C₀defined by

P₁w x, y

g

x, y, w_x,y v_x,y

−g

x,0, w_x,0 v_x,0

−g

0, y, w_0,y v_0,y g

0,0, w_0,0 v_0,0 1

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−t_r2−1

×f

s, t, w_s,t v_s,t

dt ds, x, y

∈J.

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, w_x,y v_x,y

−g

x,0, w_x,0 v_x,0

−g

0, y, w_0,y v_0,y g

0,0, w_0,0 v_0,0 λ

Γr1Γr2 _x

0

_y

0

x−s^r1−1

y−tr2−1f

s, t, w_s,t v_s,t dt ds,

5.31

for someλ∈0,1. Then fromH₁–H₃,5.15, and5.18we get for eachx, y∈J0, w

x, y≤ f_p^∗p^{r1 r2}

Γr1 1Γr2 1 4c_pz x, y g

x, y,0 gx,0,0 g

0, y,0 g0,0,0 1

Γr₁Γr2 _x

0

_y

0

x−s^r1−1

y−tr2−1l_ps, tzs, tdt ds.

5.32

Replacing5.32in the definition ofzx, y, we get

z x, y

≤ 1

1−4Kc_p Mφ

B 4Kφ0,0 4Kg

0,0, φ0,0 4Kg_p^∗ Kf_p^∗p^{r1 r2}

Γr₁ 1Γr2 1

!

l^∗_p Γr₁Γr2

_x

0

_y

0

x−s^r1−1

y−t_r2−1

zs, tdt ds,

5.33

wherel^∗_px, y l^∗_p/1−4Kc_pandg_p^∗sup{gx, y,0:x, y∈J0}.