Existence results for functional perturbed differential equations of fractional order with state-dependent delay in Banach spaces

It is well known that differential equations of fractional order play a very important role in describing some real world problems. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [1]). There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Miller and Ross [2], Podlubny [3], the papers of Abbas and Benchohra [4, 5], Benchohra et al. [6] and the references therein.

The theory of functional differential equations has emerged as an important branch of nonlinear analysis. Differential delay equations, or functional differential equations, have been used in modeling scientific phenomena for many years. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case it is called a distributed delay, see for instance the books [7, 8] and the paper [9].

However, complicated situations in which the delay depends on the unknown functions have been proposed in modeling in recent years (see for instance [10] and the references therein). These equations are frequently called equations with state-dependent delay. Existence results, among other things, were derived recently for various classes of functional differential equations when the delay is depending on the solution. We refer the reader to the papers by Anguraj et al. [11], Hartung [12], and Hernandez et al. [13]. In [14], the authors considered a class of semilinear functional fractional order differential equations with statedependent delay.

The first result of this paper deals with the existence of solutions to fractional order initial value problems (IVP for short), for the system

( Dou)(t,x) = f (t,x, U(p1(t,x,u(tx)),p2(t,x,u(tx))))                             (1) + g(t,x,u(pi (t,X,U(t,x)),p2(t,X,U(t,x)))Y    if (t,x) € J, u(t, x) = ф(t, x),   if (t,x) € J,                                  (2) И' 0)=        (t,x) € J,                           (3) [ u(0, x) = ^(x), where y(0) = ^(0), J := [0, a] x [0,b], a, b, а, в > 0, J := [—a, a] x [—в, b]\[0, a] x [0,b], cD0 is the standard Caputo’s fractional derivative of order r = (ri,r2) € (0,1] x (0,1], f, g : J x C ^ Rn, pi : J x C ^ [—a, a], p2 : J x C ^ [—в,Ь] are given functions, ф € C : = C(J, Rn) is a given continuous function with ф(t, 0) = ^(t), ф(0, x) = ^(x) for each (t, x) € J, y : [0, a] ^ Rn, ^ : [0, b] ^ Rn are given absolutely continuous functions and C is the space of continuous functions on J. We denote by U(tx) the element of C defined by U(t,x)(s, т) = u(t + s,x + т), (s, т) € J, here U(t,x) (•, •) represents the history of the state u.

The second result deals with the existence of solutions to fractional order partial differential

equations ( Dou)(t,x) = f (t, x, U(p1(t,x,u(ttX)),P2(t,x,u(t,X))))                               (4) + g(t,x,u(pi (t,X,U(t, x)),P2(t,X,U(t, x}))),    if (t,x) € J, u(t,x) = ф(t,x),   if (t,x) € J*,                                  (5) ^u(t, 0) = y(t),          A c 7 Iu(0,x)= ^(x),   (t,x) € J,                               (6) where y, ^ are as in problem (1)-(3), J* : = (—to, a] x (—to, b]\[0, a] x [0, b], f, g : J x B ^ Rn, pi : J x B ^ (—to, a], p2 : J x B ^ (—то,Ь] are given functions, ф : J * ^ Rn is a given continuous function with ф(t, 0) = y(t), ф(0, x) = ^(x) for each (t, x) € J and B is called a phase space that will be specified in Section 4.

Motivated by the previous papers, we consider the existence result for each of our problems (1)–(3) and (4)–(6). Our analysis is based upon on a fixed point theorem due to Burton and Kirk for the sum of contraction and completely continuous operators and a fractional version of Gronwall’s inequality. We look for sufficient conditions ensuring existence of solutions for each of our problems. The present results extend those considered with integer order derivative and those with finite and/or infinite constant delay on bounded domains in [15–18].

As far as we know, no papers exist in the literature related to fractional order hyperbolic functional differential equations with state-dependent delay. The aim of this paper is to initiate this study.

• 2.    Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

By C(J, R n ) we denote the Banach space of all continuous functions from J into R n with the norm ||u ^ ^ = sup ( _t,x ) _G j ||u(t,x) | , where || • || denotes a suitable complete norm on R n . As usual, by AC (J, R n ) we denote the space of absolutely continuous functions from J into R n and L ¹ ( J, R n ) we denote the space of Lebesgue-integrable functions u : J ^ R n with the norm ^{| u |} L i = Jo Jo ^{lu ( t,x ) ldxdt} .

Now, we give some definitions and properties of fractional calculus.

Definition 2.1 [19]. Let r = (r1,r2) G (0, to) x (0, to), У = (0, 0) and for u 6 L1(J, Rn), the expression tx

(I 0 ^{u )( t} , x) = _Г( _{Г 1} 1_Г( _{Г 2} ) j j ⁽ — s^{) r i - 1 (}x ^{— T) r} 2 ^{- 1 u (}s, ^T ) (Ms, 00

where Г0 is the gamma function, is called the left-sided mixed Riemann-Liouville integral of order r of u.

In particular, (I g u)(t,x) = u(t, x), (i g u)(t, x) = J t J X u(s,T )(t(s for almost all (t, x) G J, where a = (1,1), For instance, Igu exists for all r 1 ,r ₂ G (0, to ) x (0, to ) , when u G L ¹ (J, R n ). Note also that when u G C(J, R n ), then (i g u) G C(J, R n ), moreover (I g u)(t, 0) = (i g u)(0, x) = 0, (t, x) G J. By 1 — r we mean (1 — Г 1 ,1 — r ₂ ) G (0,1] x (0,1]. Denote by D 2 _x := -dd- , the mixed second order partial derivative.

Definition 2.2 [19]. Let r G (0,1] x (0,1] and u G L ¹ (J, R n ). The Caputo fractional-order derivative of order r of u is defined by the expression

Du)(t,x) = (l, ^{- r} A    : g,x).

The case a = (1,1) is included and we have (Dg u)(t, x) = (^cD g u)(t,x) = (D 2 x u)(t,x) for almost all (t, x) G J.

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

Lemma 2.1 [20]. Let и : J ^ [0, to) be a real function and w(^, •) be a nonnegative, locally integrable function on J. If there are constants c > 0 and 0 < Г1, Г2 < 1 such that tx

I f       ^{u ( s,T} )        , ,

u(t,x) < w(t, x) + c                          dTdS, v v ’ 7     0 0 (t — s)g1 (x — T)r2

then there exists a constant 6 = 6(r1,r2) such that tx

Я ш(s,T )

dτds,

0 0 (t — s)r1 (x — T)r2       ’ for every (t, x) G J.

Theorem 2.1 ( Burton-Kirk [21]) . Let X be a Banach space, and A, B : X ^ X two operators satisfying: (i) B is a contraction, and (ii) A is completely continuous. Then either

• (a)    the operator equation u = A(u) + B (u) has a solution, or

• (b)    the set E = { u G X : u = AA(u) + AB ( у )} is unbounded for A G (0,1) .

• 3.    Existence Results for the Finite Delay Case

In this section, we give our main existence result for problem (1)–(3).

Before starting and proving this result, we give what we mean by a solution of this problem.

Let the space C ( _a,b ) := C ([ - a, a] x [ — в, b], R n ), a, b > 0.

Definition 3.1. A function u E C ( _a,b ) is said to be a solution of (1)-(3) if u satisfies equations (1) and (3) on J and the condition (2) on J.

Let f, g E L 1 (J, R n ) and consider the following problem

( ( ^c D 0 u)(t, x) = f (t, x) + g(t, x), (t, x) E J, (^u( t, ⁰⁾ = y^(t), u^(0, x) = ^^(x), y⁽⁰⁾ = ^⁽⁰⁾.

For the existence of solutions for the problem (1)–(3), we need the following lemma.

Lemma 3.1. A function u E C (J, R n ) is a solution of problem (1) if and only if u(t,x)

satisfies

u(t,x) = z(t,x) + (I Q f)(t,x) + (I Q g)(t,x),   (t,x) E J,

where z(t,x) = ^(t) + ^(x) — ^(0).

• <1 Let u(t,x) be a solution of problem (1). Then, taking into account the definition of the fractional Caputo derivative ( ^c D Q u)(t, x), we have I 1 ^{- r} (D 2 _x u)(s,т) = f (t,x) + g(t, x). Hence, we obtain I Q (I 1 ^{- r} D 2 _x u)(s,T) = I q f (t, x) + I q g(t,x), then I 1 (D ² _x u)(s,T) = I q f (t, x) + I 0 g(t, x). Since I 1 (D 2 _x u)(s, т) = u(t, x) — u(t, 0) — u(0, x) + u(0, 0), we have u(t, x) = z(t, x) + (I 0 f )(t, x) + (I q g)(t,x). By the definition of the left-sided mixed Riemann-Liouville integral

  of order r of f and g, we have

  u⁽t,^x) = zM + r^

  
  

tx j1(t — s)r1-1 (x — т)r2-1 f (s, т) dтds tx

⁺ r( _r 1 )1r( _{r 2} ) / /^{(t — s)} r ^{1 - 1 (x — т ) r 2 - 1} g^(s,^т)^dтds, о 0

so

^u(t,^x)= ^z(t,^{x) +} _r(r _i )¹_r(r ₂ )

tx j /(t — s^ (x — т)r2-1 [f (s, т) + g(t, x)] dTds, where z(t, x) = ^(t) + ^(x) — ^(0). Now let u(t, x) satisfy (2). It is clear that u(t,x) satisfies (1). >

As a consequence of Lemma 3.1 we have the following auxiliary result.

Corollary 3.1. The function u E C(a,b) is a solution of problem (1)-(3) if and only if u satisfies the equation tx

^u(t,x) = ^{z(t,x) +} _Г( _г 1 )Г( _{Г 2} ) j j ^{(t —} sr ^{1 - 1 (x — т) r} 2 ^{- 1 f} ( ^s,T,u ( s,т ) ) ^dтds

1   2              _tx

+ Г(Ч)Г(Г27 //(t — s)r1-1 (x — т )r2-1 g(s,т,u(s,т))dтds, for all (t, x) E J and the condition (2) on J.

Set R :=R ( _p - ,_p -) = { (P 1 (s, т, u), P 2 (s,т, u)) : (s,т, u) E J x C, P i (s,т, u) <  0; i = 1, 2 } . We always assume that p 1 : J x C ^ [ — a, a], p 2 : J x C ^ [ — в, b] are continuous and the function (s, т) ।—> u ( _s,_T ) is continuous from R into C.

Our main existence result in this section is based upon the fixed point theorem due to Burton–Kirk. We will need to introduce the following hypothesis:

(H 1) The functions f, g : J x C ^ R ⁿ are continuous.

• (H2) There exists k >  0 such that ||g(t,x,u) — g(t, x,v) | <  k^u — v | c for any u, v € C and (t, x) € J.

• (H 3) There exist p,q € C(J, R + ) such that | f (t, x, u) || <  p(t, x)+q(t, x) | u | c for (t, x) € J and each u € C .

Theorem 3.1. Assume that hypotheses (H 1) - (H3) hold. If

< 1, then the IVP (1)-(3) has at least one solution on [—a, a] x [—в, b].

• <1 Transform the problem (1)-(3) into a fixed point problem. Consider the operators F,G : C^ ^ C ( ab ) defined by,

  and

  
  

  Mx),

  
  

  tx

  
  

  ^(Fu)(t, ^x) — < ^z(t,^x) + r( r _i )r( r ₂ ) // ^(t

  
  

  0 0

  
  

  — s) ^{r i - 1} (x —

  
  

  т ) r ^{2 - 1}

  
  

  ^{X f} ( s, ^T , u ( p i ( s,T,U^ sT ) ) ,P 2 ( s,T,U (s__T ) )) ) ^dTds ,

  
  

  0,

  
  

  tx

  
  

  ⁽G^u)(t, ^{x) —}< r( r _i )r( r ₂ ) / /^(t

  
  
  
  

  — s) ^{r i - 1} (x —

  
  

  т ) r ^{2 - 1}

  
  

  ^X g ( s, ^T , u ( p i ( s,T,U (s__T ) ) ,p ₂( s,T,U (s__T ) )) ) dTds,

  
  

  (t, x)

  
  

  (t, x)

  
  

  (t, x)

  
  

  (t, x)

  
  

  €

  
  

  €

  
  

  €

  
  

  €

  
  

  J,

  
  

  J,

  
  

  J,

  
  

  J.

  
  

The problem of finding the solutions of the IVP (1)–(3) is reduced to finding the solutions of the operator equation (Fu)(t,x) + (Gu)(t,x) — u(t,x), (t,x) € J. We shall show that the operators F and G satisfies all the conditions of Theorem 2.1. The proof will be given in several steps.

Step 1. First, we show that F is continuous.

Let { u _n } be a sequence such that u n ^ u in C ( a,b ) • Let П >  0 be such that | u _n || <  n

Then

tx

|| (Fu „ )(t.x) — (Fu)(t,x) || <  _{r( r 1 )} ¹ _{r( r 2 )} //lit — s) r ^{i - 1} (x — т) r ^{2 - 1} 1

X ||f (s,т, un(pi (s,T,un(s, T )),p2(s,T,un(s, T))) ) — f (s, т, u(pi(s,T,u(s , T )),p2(s,T,u(s, T) )))^ dтds tx

< гО>2у/> ^{— s)} r ^{i - 1 (x} " ^т' r ^{2 -}

^sup li fts, ^T, ^u n ⁽ p i ^{( s},^T,u n ( s_iT ) ⁾ ,p ₂ ⁽ s,^T,u n ( s_-T )

( S,T ) £J                                 ' '               ' '

tx

—

⁽                                ,_N^      I f ( • , ^u nw ) — f ( • , • ,u) I -             ,ri - 1

^f \s ^{,T,u (}p i ^{( s,T,u} (s , T ) ⁾ ,P 2 ^{( s,T},u ( s, Т } ⁾⁾) II d ^T ds <              Г( _{Г 1} )Г( _{Г 2} )             J J ^{(t    s)}

x (x — т ) r ^{2 1} dтds <

t r ¹ x r ² IlfG • ," ,-, • ) ^f ( • , • ^,u ( ^ , ^ ) ^{) |} ^ _<  ^a r 1 ^b r ^{2 | f} ( • , • ^,u n ( ^ , ^ ) ^{) — f} ( • , • ^,u ( ^ , ^ ) ^{) |} ^

Г1Г 2 Г(Г 1 )Г(Г 2 )

r(r i + 1)r(r 2 + 1)

.

Since f is a continuous function, we have

^Fu n ) ^{— (Fu) |} ^ <

Q ^{r i} b r ^{2 | f}( • , • ^,u n ( ^ , ^ ) ^{) — f} ( • , • ^,u ( ^ , ^ ) ^{) |} ^ Г(Г 1 + 1)Г(Г 2 + 1)

^ 0

as n ^ to .

Thus F is continuous.

Step 2. F maps bounded sets into bounded sets in C ( a,b ) .

Indeed, it is enough show that, for any n > 0, there exists a positive constant I* such that, for each u E Bn = {u € C(a,b) : ||uH^ ^ n}, we have IF(u)|^ ^ I*. By (Нз) we have for each (t, x) E J, tx

B (Fu)(t,x) B < | z(t,x) | + _r(r _{i )}¹_r(r _{2 )} j jb - s) r ^{1 - 1} (x — т ) r ^{2 - 1} 0 0

tx

X f (s,T,u(pi (s,T,u(SiT )),P2(s,T,u(SiT ))))|| dTds ^ |z(t,x)|+ Г(Г1)Г(Г2) / /(t - s)ri-1(x - т tx

X p(s,T) dTds +      1      j j(t-s)r1-1 (x-TY2-1q(S,T)||u(pi(s,T,u(s,T)),p2(s,T,u(s-T))) He dTds tx

^ ^( t,^{x) 1 +} p^Ffa) / /^(t - s^{) r i 1 (}x - ^{T ) r} 2 ^{1 dTd} s ⁺ Г(Г 1 )Г (Г2 )

tx

X / At - s) r ^{1 - 1} (x - T Y^{2 -1} dTds < | z(t,x) | + _Г( " ^Р ” ” ₁+Г(’.”°₊''₁₎ a r ¹ b r ² ■

Thus

^{| p |} ^ ⁺ Hq^ ⁿ _an rp    *

^Fu) ¹¹ » ^ ^{|| z}U ⁺ Г(Г 1 + 1)Г(Г 2 + 1) a ^{b :}= ^l "

Step 3. F maps bounded sets into equicontinuous sets in C ( _a,b ) .

Let (t ₁ ,x ₁ ), (t ₂ ,x ₂ ) E (0, a] X (0, b], t ₁ < t ₂ , x ₁ < x ₂ , B _n be a bounded set of C ( _a,b ) as in

Step 2, and let u E B n . Then

|^{| (Fu)(}t 2 ^,x 2 ) ^{- (Fu)(t} 1 ,x 1 ⁾ || ^ |^{| z(t} 1 ,x 1 ^{) - z(}t 2 ,^X 2 ^{) H +} _Г( _{Г 1} 1_Г( _{Г 2} )

X y_jX' i [ (t 2 - s) r ^{1 - 1} (X 2 - T ) ^{r 2 - 1} - (t 1 - s) ^{r i - 1} (X 1 - T ) ^{r 2 - 1} ]|| f (A T.U^ tu^ t , ) ,„ 2 ( .,_T,u („ ) )) ) || dTds

⁺ Г(Г 1 )Г(Г 2 )

t 2 X 2

j j(t2 - s)ri-1 (x2 - T)r2-1 У(s,T,u(pi(s,T,U(s,T)),P2(s,T,U(s,T))))| dTds ti Xi t x2

⁺ Г( _{Г 1} )Г( _{Г 2} )     ./"(t ² - ^{s) i     (}x ² - ^{T ) 2        ( S,T,U (} P i ⁽ s'^T'^U (s T ) ^{) ,}P 2 ^{( s,T},u (s T ) ^{)) ) |} d ^T ds

0 Xi t2 x1

+ Г(Г1)Г(Г2) У ./(t2 - s) i   (x2 - T) 2    l|f (S,T,U(Pi(s’T’U(s T )),P2(s,T,u(s ,T ^Я dTds tl Xi

< H z(t 1 .X 1 ) - z(t 2 .X 2 ) H + И^У^ / J [ (t 1 - s) r ^{i - 1} (X 1 - T ) r ^{2 - 1}

t 2 X 2

• - (t 2 - s) r ^{i - 1} (X 2 - T ) r ^{2 - 1} ] dTds + >^±^^ [ /"(t 2 - s) ^{r i - 1} (X 2 - T ) r ^{2 - 1} dTds

Г(Г1)Г(Г2)   J J tl Xi t1 X2

, H p H ~ + Iklk n f Л.    r- - 1^    _■—-i л_л II p H ~ ⁺ Iklk n

⁺ Г(Г 1 )Г(Г 2 ) J J ⁽t ² - ^{s)    (}x ² - ^{T1          +} Г(Г 1 )Г(Г 2 )

0 X 1

t 2 X 1

X / A^ - s) r ^{1 - 1 (x} 2 ^{- T}Г- dTds <  || z^(t 1 ,x 1 ) ^- z^(t 2 ^,x 2 ^{) | +} pl^ ⁺p^^A

Г(Г 1 + 1)Г(Г 2 + 1)

t 1 0

X       (t 2 - t 1 ) ^{- 1} + t - ¹ (X 2 - X 1 ) r ² - (t 2 - t 1 ) ^{- 1} (X 2 - X 1 ) r ² + t - 1 x - ² — t ^ x 2 ² ]

H p H ~ ^{+ |} q h ^ n       -        -     ¹ p ^ ⁺ ° q ° - n

⁺ Г(Г 1 +1)Г(Г 2 +1)^{(t 2 - t 1 ) (}x ^{2 - x 1 )                         [}t ^{2 - (t 2 - t 1 ) ](}x ^{2 - x 1 )}

+ Г(г1 + 1)^ (t2 - t^r1 Ix^ - (x2 - x1)r2-1] C l|z(t1 ,x1) - z(t2’x2)|| llpllra + llq°^n ro rr^.    . \-1 , r+m \r2 । rn. -2 rn. -2          r,Vm,„ t-A-2 1

⁺ Г( _{Г 1} + 1)Г( _{Г 2} + 1) ^x ^{2 (}t 2 - t 1 ) ⁺2t 2 ⁽x 2 - x 1 ) ⁺ t 1 x 1 ^- t 2 ^X 2 ^{- 2(t} 2 ^- t 1 ) ^(x 2 ^{- x} 1 ) J

As t 1 ^ t 2 , X 1 ^ X 2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t 1 < t 2 < 0, X 1 < X 2 < 0 and t 1 C 0 C t 2 , X 1 C 0 С X 2 is obvious. As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we can conclude that F : C ( a,b ) ^ C ( a,b ) is continuous and completely continuous.

Step 4. We show that G is a contraction.

Let v,w G C([-a, a] X [-в, b], Rn). Then, for (t, x) G [-a, a] X [-в, b], tx

|^{| (Gv)(}t,^{x) - (Gw)(}t,^x) || ^c Г(Л)Т(Г27 / /l^{(t - s) r 1 - 1} l l ^{(x - T) - 2 - 1} I 00

k

Г(Г 1 )Г(Г 2 )

X |f (s) T, V(p1(s,T,u(s,T)),P2(s,T,u(s,T))))   f (s’ T, W(P1(s,T,u(s,T)),P2(s,T,u(s,T) ))) II dTds C tx

^Xj I (^{- - s) 1     (x - T) 2     || v (} p 1 ⁽ s,^T,u ( s,T ) ⁾ ,P 2 ^{( s,T},u ( s,T ) ))

^{W (} P 1 ^{( s,T,u} (s,T ) ) ,P 2 ^{( s,T,u} (s,T ) )) ^ C d ^Tds

tx

k

Г(Г 1 )Г(Г 2 )

¹T r ²

(t - s) r ^{1 - 1} (x - T) r ^{2 - 1} dTds C _{r( r 1 + 1 )r},₍r _{2 + 1)} | v - w | c .

Consequently, kar1 b-2

^Gv) - G b ^C Г(Г 1 +_Ur_{( r 2 + 1}) O v ^{- w} « ^C •

Since by (3), G is a contraction.

Step 5 (a priori bounds). Now it remains to show that the set E = { u G C(J, R ) : u = AF(u) + AG( U^ ) } for some A G (0,1) is bounded.

Let u G E, then and u = AF(u) + AG(^ for some 0 < A < 1. Thus for each (t,x) G J, we have tx

u(t,x)= Az(t’x) + r(r1 )Г(г2 ) j j (t - s)r1-1(x - T)r2-f (s,T,u(p1 (s,t,U(s,t)),P2(s,T,U(s,t >))) dTds tx

⁺ r(G)W/^(t " ^{s) r} ‘ ^{- 1 (x} "  ^T ' r ^{1 - 1} g(^s’^T’ 00

u ⁽ P 1 ^{( s,T,U} (s,T) ^{) ,}P 2 ^{( s,T,U} (s,T ) ))

λ

dτds.

This implies by (H2) and (H3) that, for each (t, x) € J, we have

Mt,^x) |l C A | z(t,^x) B + _{r( r i )}¹_r(r _{2 )} J j (t - s) r ^{1 - 1} (x - T) r ^{2 - 1}

0 0

x [ ^p(s,T) ^{+ q(s,T}) || ^u ( p i ( s,^T,u ( _s,_T ) ) ,p 2 ( s,^T,u ( _s,_T ) )) |l c ] ^dT d ^{s +} Г( _{Г 1} )Г(г ₂ )

tx x / /(t — s)r1-1 (x — t)r2-1

/       ^{U (} p i ^{( s,T},u (_s,_T) ⁾ ,p 2 ^{( s},^T,u (_s,_T ))) \       ,         .

dτds

g I ^s,T , --------------д--------------) ^— g^{( s} , ^T , ⁰⁾

tx

⁺ tv [ At ^{— s)} r ^{1 - 1 (x — T) r} 2 ^{- 1} | g^(s,T’^{0) 1} d^{Tds C} H z^x) ^{1 +} tv °4.ii гА^+п

Г(Г 1 )Г(Г 2 ) J J                                                     Г(Г 1 + 1)Г(Г 2 + 1)

tx

+ -- a— +         —T [ At — s) r ^{1 - 1} (x — T ) r ^{2 - 1} ||u(,,₁                      аТтНг-» dTds

Г(г1 + 1)Г(г2 + 1)    Г(г1)Г(г2)7 P ' v 7    11 (P1(s,T,uVT)),p2(s,T,U(s,T»)lIC tx

⁺ Г( _{Г 1} ) Г( _{Г 2} ) У ^^ - ^{S) 1    (}x - ^T) ^{2     | U (} P 1 ^{( s,T,}u ( s,T ) ^{) ,}P 2 ^{( s,T,u} (s,T ) )) H C ^dT d ^{S C !} z^(t, x^{) !}

a r ¹ b r ² (|| p|k +g * )      | q | _ + k    tf x i           i „                              II .

+ Г(П + 1)Г(Г2 + 1) + Г(Г1)Г(Г2) J J(t s)   (x T)    ||u(p1(s,T,u(s,T Mw(..t )))|c dTds, where g* = sup(s,T)Gj |g(s,T, 0)|.

Consider the function y defined by y(t,x) = sup {| u(s,T) | : — a C s C t, - в C t ^ x } , 0 C t C a, 0 C x C b.

Let (t*,x*) € [—a,t] x [—e,x] be such that y(t,x) = "u(t*,x*)!- If (t*,x*) € J, then by the previous inequality, we have for (t, x) € J, ar1 br2

^y(t,^{x) C !} z(t,^{x) ! +} Г_{( п} + 1)Г_{( Г 2} + 1) ⁺ Г(г ₁ )Г(г ₂ ) J J ⁽t — ^{s) 1    (}x — ^T) ^{2   y(s}’^T) ^dTds-

If (t * ,x * ) € J, then y(t,x) = ! Ф ! с and the previous inequality holds.

If (t,x) € J, Lemma 2.1 implies that there exists 5 = 5(r i ,r 2 ) such that we have

tx

^( h q h ^ +k)        _₄1-l-l(_T __T\^r2-1

^{1 +} Г(Г 1 )Г(Г 2 ) J J ^{(t s)    (x T}) ^dTds

< Ги„ la" b r ² ( « p » ~ + g * ) Th »" ' b r ² ( « q « ~ + к) 1

^C [“^z(t'^x) “ ⁺ r2 + 1)J I^{1 +}Г(Г1 + 1)Г(Г2 + 1)J ^:= M^-

Since for every (t, x) € J, ! u ( _t,_x ) ! C C y(t,x), we have ||u ! ^ C тах( ! Ф ! с , M) := M * . This shows that the set E is bounded. As a consequence of Theorem 2.1 we deduce that F + G has a fixed point u which is a solution of problem (1)-(3). >

• 4.    Existence Results for the Infinite Delay Case

  • 4.1.    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 semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato (see [22]). For further applications see for instance [23, 24] and their references.

For any (t, x) G J denote E ( t,x ) := [0, t] x { 0 }U{ 0 } x [0, x], furthermore in case t = a, x = b we write simply E. Consider the space (B, ||(- , ^HB ) is a seminormed linear space of functions mapping ( -to , 0] x ( -to , 0] into R n , and satisfying the following fundamental axioms which were adapted from those introduced by Hale and Kato for ordinary differential functional equations:

(A i ) If y : ( -to , a] x ( -to , b] ^ R n continuous on J and y ( _tx ) G B, for all (t, x) G E, then there are constants H, K, M > 0 such that for any (t, x) G J the following conditions hold:

• ⁽ⁱ⁾    y ( t,x ) is in B;

• (ii)    | y(t,x) | ^ H | y ( t,x ) | B ;

• ⁽ⁱⁱⁱ⁾    l|y( t,x ) ^| B <  ^{K su} P ( s,t ) e [0 ,t ] x [0 ,x ] l|y ^(s,T ) h ^{+ M su} P ( s,t ) e E _{( t},_{x )} h( s,T )I| b .

(A 2 ) For the function y( ^ , • ) in (A i ), y ( _tx ) is a B-valued continuous function on J.

(A 3 ) The space B is complete.

Now, we present some examples of phase spaces [25, 26].

Example 4.1. Let B be the set of all functions ф : ( -то , 0] x ( -то , 0] ^ R n which are continuous on [ - a, 0] x [ - в, 0], a, в ^ф 0, with the seminorm ^|b = sup ( s,_T ) _G [ - _a, o] x [ — e, o] ^( s,T) | . Then we have H = K = M = 1. The quotient space B = B/ | • | b is isometric to the space C([ - a, 0] x [ - в, 0], R n ) of all continuous functions from [ - a, 0] x [ - в, 0] into R n with the supremum norm, this means that partial differential functional equations with finite delay are included in our axiomatic model.

Example 4.2. Let 7 G R and let C y be the set of all continuous functions ф : ( -то , 0] x ( -то , 0] ^ R n for which a limit lim ^ ( _s,_T ) ^^^ e ^{Y ( s} + ^T ) ф(s,т) exists, with the norm ^| c _Y = ^suP ( s,T ) e ( -~ , 0] x ( -^ , 0] e ^{Y ( s + T} ) Н ^ф(^^т) h - ^{Then we have H} = ^{1 and K} = ^M = ^{max {} e ^{— Y ( a + b )} ,^{1 }} .

Example 4.3. Let a,в,Y ^ф 0 and let

^^ CL y

sup

( s,t ) G [ - a, 0] x [ - e, 0]

^(s,T)l + / / — ^ —^

e ^{Y ( s + T} ) ^(s,t ) | dTds

be the seminorm for the space CL y of all functions ф : ( -то , 0] x ( -то , 0] ^ R n which are continuous on [ - a, 0] x [ - в, 0] measurable on ( -to , - a] x ( -to , 0] U ( -to , 0] x ( -то , - в], and such that ^| cl _y <  to . Then H = 1, K = J ⁰ _a J — в e ^{Y ( s + T} ) dTds, M = 2.

• 4.2.    Main Results. Let us start in this section by defining what we mean by a solution of the problem (4)-(6). Let the space Q := { u : ( -to , a] x ( -то ,Ь] ^ R n : U ( _t,_x ) G B for (t, x) G E and u | j G C(J, R n ) } .

Definition 4.1. A function u G Q is said to be a solution of (4)-(6) if u satisfies equations (4) and (6) on J and the condition (5) on J ‘ .

Set R := R ' p-^ = { (p i (s,r,u), p 2 (s,T,u)) : (s,T,u) G J x B, p i (s,T,u) <  0; i = 1, 2 } . We always assume that p i : J x B ^ ( -to , a], P 2 : J x B ^ ( -to , b] are continuous and the function (s, т) ^ U ( _s, _T ) is continuous from R into B.

Our main result in this section is based upon the fixed point theorem due to Burton and Kirk.

We will need to introduce the following hypothesis:

(Н ф ) There exists a continuous bounded function L : R ( p - p — ^ (0, to ) such that ^(s,t ) ^ B C L(s,T ) | ф | в , for any (s,T ) G R ' .

In the sequel we will make use of the following generalization of a consequence of the phase space axioms [27, Lemma 2.1].

Lemma 4.1. If u G Q, then

II^u(s,t ) ^ B = ^(M + L ' ) Н ^ф Н в ⁺ K           ^suP           IMM)!!,

( 0,n ) G [0 , max { 0 ,s } j X [0 , max { 0 ,T } ]

where L ' = SUP ( s,T ) _G R ^L(s,T) •

Theorem 4.1. Assume (Н ф ) and that the following hypothesis holds:

• (H1) The functions f, g : J x B ^ R n are continuous.

• (H2) There exists I > 0 such that | g(t, x, u) — g(t, x, v) || C l | u — v | b , for any u,v G B and (t, x) G J .

• (H 3) There exist p, q G C(J, R + ) such that ||f (t,x,u) | C p(t,x) + q(t,x) | u | B , for (t, x) G J and each u G B .

If

< 1, then the IVP (4)-(6) has at least one solution on (—to, a] x (—то,Ь].

• <1 Transform the problem (4)-(6) into a fixed point problem. Consider the operator N : Q ^ Q defined by

  (Nu)(t, x) = <

  
  

  ^ф(t,x),

  z⁽t, ^{x) +} _Г ( г ₁ ) _Г ( г ₂ ) / 0 jo ^{(t — s) r 1 1 (x}

  
  
  
  

  T ) r ^{2 - 1}

  
  

  ^{x f} ( s,^T, ^u ( p i( s,t,u ( s,t )) ,p 2( s,t,u ( s,t ) )) ) d^Td^s

  ⁺ Г( г 1 ) ¹ Г( г 2 ) / 0 / 0 ^{(t —} s^{) r i 1 (x — T) Г 2 1}

  ^X g ( s,^T, u ( p i ( s,T,U ( s,T )) ,P 2( s,T,U ( s,t ) )) ) ^dTds,

  
  

  (t, x) G J,

  
  

  (t, x) G J.

  
  
  
  

  Let v (^ • ) : ( —to , a] x ( —to , b] ^ R n be a function defined by,

  z(t, x), (t, x) G J, v(t, x) = <                     ~

  [ф(t, x), (t, x) G J.

  Then V ( _t,_x ) = ф for all (t, x) G E.

  For each w G C (J, R n ) with w(t,x) = 0 for each (t,x) G E we denote by w the function defined by

  
  

  w(t, x) = ^

  
  

  w(t, x) 0,

  
  

  (t, x) G J, (t, x) G J.

  
  

  If u(> •) satisfies the integral equation

  
  

  u(t, x) = z(t, x) +

  
  
  
  

  t

  
  

  x

  
  

  r(r i ) Г(Г 2 )

  
  

  j j (t — s) r ^{1 - 1} (x — T)r²

  
  

  - 1

  
  

^{X f} (s, ^T, ^{W (} p i ^{( s},t,^u ( s,t ) ⁾ ,p 2 ^{( s},^t,^u ( s,t ) ^{)) + V (} p i ⁽ s,t,^u ( s,t ) ⁾ ,P 2 ⁽ s,t,u ( s,t ) ⁾⁾) ^dTds ⁺ Г( _{Г 1} ) r(r ₂ )

tx

X j У^ — S) 1   (x — T) 2    g(S, T, W(P1(s,t,U(s,t)),P2 (s,t,U(s,t))) + v(pi (s,t,U(s,t) ),P2(s,t,U(s.t)))) dTds, we can decompose u(^, •) as u(t,x)  =  w(t, x) + v(t, x);  (t, x)  € J, which implies

U(t,x) = w(t,x) + V(tx), (t,x) € J, and the function w(^, •) satisfies tx

^w(i'^x) = r(r i )1r(r 2 ) / /^{( t - s) r i - 1 (x - T) r 2 - 1}

0 0

^{X f} ( s, ^T, ^{W (} p i ^{( s,t,u} ( s,t ) ^{) ,}p 2 ^{( s,t,u} ( s,t ) ^{)) +} v ⁽ p i ^{( s,t,u} ( s,t ) ) ,P 2 ^{( s,t,u} ( s,t ) ⁾⁾ ) dTds ⁺ r( _r 1 )r(r ₂ )

tx

^X j У^ ^{- S) i    (}x ^{- T}) ²     g ( s, ^T, w ( P i ( s,t,U ( s,t ) ) ,P 2 ( s,t,U ( s,t ) )) ^{+ V} ( p i ( s,t,U ( s,t ) ) ,P 2 ( s,t,U ( s,t ) )) ) dTds"

Set Cq = {w € C(J, Rn) : w(t, x) = 0 for (t,x) € E}, and let || • h(a,b) be the seminorm in Co defined by ||wH(a,b) = SUp(t,x)GE |w(t,x) ||b + SUp(t,x)Gj |w(t,x)| = SUp(t,x)Gj ||w(t,x)||, w € CQ. Cq is a Banach space with norm || • |(a,b). Let the operators A,B : Cq ^ Cq defined by tx

^(Aw)(t, x) = _Г( _{Г 1} )1_Г( _{Г 2} ) 11 ^{(t - s)} r ^{1 - 1 (x - T}F— 1

0 0

X f (s, T, W(pi(s,t,u(s,t)),p2(s,t,u(s,t))) + V(pi (s,t,u(s,t)),p2(s,t,u(s,t))) ) dTds and

tx

^{( Bw )(}t, ^x) = _Г( _{Г 1} )1_Г( _{Г 2} ) I l ^{(t - S)} r ^{i - 1 (}x ^{- T ) Г 2 - 1}

^X g (^{s,T,W (} p i ⁽ s;t^,u (s,_t) ^{) ,}p 2 ^{( s};t^,u (s,t) ^{)) + V (} p i ^{( s,}t^,u ( s,t ) ^{) ,}P 2 ^{( s,t,u} ( s,t ) ⁾⁾ ) d ^T d ^s -

Then the operator N has a fixed point is equivalent to finding the fixed point of the operator equation (Aw)(t, x) + (B w)(t, x) = w(t, x), (t, x) € J. We shall show that the operators A and B satisfies all the conditions of Theorem 2.1.

For better readability, we break the proof into a sequence of steps.

Step 1. F is continuous.

Let {wn} be a sequence such that wn ^ w in Cq. Then tx

|^{| (A}w n ^{)(t,x) - (Aw)(t,x)} || <  _Г( _{Г 1} )1_Г( _{Г 2} ) У j ^{(t - s) r i - 1 (x - T) r 2 - 1}

^X f^, ^T, w ^{n (} p i ^{( s,T,u} n ( s,T ) ^{) ,}P 2 ^{( s,T,u} n ( s,T ) )) ^{+ V} n ⁽ P i ^{( s,T,u} n ( s,T ) ^{) ,}P 2 ^{( s,T,u} n ( s,T ) )) )

^ Г(Г 1 )Г(Г 2 )

f (s, T, W(pi(s,t,u(st)),P2 (s,T,u(s,T ))) + V(pi(s,T,u(s,T)),P2 (s,t,u(s,t ))) ) 1 dTds tx j 1(t-s)ri-1(X-T)Г2-1 f(S’T,wn(s,T) +vn(s,T))-f(s,T,ws,T) +v(s,T))|| dTdS.

Since f is a continuous function, we have

|| ^(Aw n ) ^- Uwi _x <

t r ¹ x r ² II / G, - .«Чу) ⁺ vnW) ^- f I, ^(y) ⁺ v ₍ ^ , ^ ) > H X

r(r i + 1) Г(Г 2 + 1)

< a r ¹ b r ² II/ G • >w n (y) ⁺ v n ( - , - ) ) ^{- f} G ^x w (v) ⁺ v^X , ' X

r(r i + 1) Г(Г 2 + 1)

^ 0 as n ^ to .

Thus A is continuous.

Step 2. A maps bounded sets into bounded sets in C o .

Indeed, it is enough show that, for any n >  0. there exists a positive constant I such that, for each w € B _n = { w € C o : | w | ( a,b ) C n } , we have | A(w) | ^ <  I.

Lemma 4.1 implies that

|| w ( s,T ) ⁺ v ( s,T ) \\ b ^c || w ( s,T ) \\ b ⁺ || v ( s,T ) W b ^c K ^{n +} K 11 ^ф(0, ^{0) W + (M +} L ^HH B •

Set n* := Kn + KIIФ(0,0) II + (M + L’) 11Ф11в• Let w € Bn• By (H3) we have for each (t, x) € J, tx

^C wksibt ^{— s) r i - 1 (x — T) r 1 -}

x (x — T ) r ² t

0 0

^q(s, ^T) ^W w ( p ₁ ( s,T,u _(j

x

tx

^{1 p(s}’^T)^{dTds +} Гк¹гы/ /^{(i - s) r i - 1} 00

a,T)),p2(s,T,u(a,T )))     X(pi (s,T,u(a,T ) ) ,p2 (s,T,u(a,T tx

C T^T-^¹^ [ /(t — s) ^{r i - 1} (x — T) ^{r 2 -1} dTds + Jqx^nL [ /"( t — s) ^{r i - 1} (x — T) ^{r 2 -1} dTds

Г(Г 1 )Г(Г 2 ) J                ^v ’              Г(Г 1 )Г(Г 2 ) J                ^v ’

/ ^{| p |} ^ ⁺ ll^{q |} ^ ⁿ ,r _i r ₂ / ^{| p |} ^ ^{+ | q |} ^ ⁿ r _i ir ₂ _ *

^C Г(Г 1 + 1)Г(Г 2 + 1) t x ^C Г(Г 1 + 1)Г(Г 2 + 1) a b ■= ¹ •

Hence ||A(w) | ^ C I * .

Step 3. A maps bounded sets into equicontinuous sets in C o .

Let (t1,x1), (t2,x2) € (0, a] x (0,b], t1 < t2, x1 < x2, Bn be a bounded set as in Step 2, and let w € Bn. Then ti xi

|^{| (Aw)(t} 2 ^,x 2 ^{) - (Aw)(t} 1 ^,x 1 ^{) W C} _Г( _{Г 1} )Г( _{Г 2} ) j j [ ^(t 2 ^{- s) r i - 1 (}x 2 ^{- T) r 2 - i (}t i ^{- s) r i - 1 (}x 1 ^{- T) r 2 - i} ] 00

x Wf (S,T, w(pi (s,T,u(aiT)),P2(s,T,u(a,T))) + v(Pi (s,T,u(a,T)) ,P2 (s,T,u(a,T))) )W dTds t2 x2

⁺йг;ку/> ^{- s) r} ‘ ^{- 1 (}x 2 ^{- T ) r 2 - 1}

t i ^x i

))} || dTds

||f (s,T,w(pi(s,T,u(s,T ti X2

■■    ¹ .->• - ^{s) r i - 1 (} x 2 ^{- T} г² "¹

0 x 1

^{x || f} ( ^S,T1 ^{W (} p i ^{( s,T,u} (_s,T) ^{) ,}p 2 ^{( s,T,u} (_a,T ) )) ⁺ v ⁽ P i ^{( s,T,u} ( a,T ) ^{) ,}P 2 ^{( s,T,u} (s,T ) )) )ll d^Tds t 2 X i

■■   ¹ J>- ^{- s) r i - i (}x 2 ^{- т} 2

t i 0

t 1 Х 1

C ^Г^Г^⁰) ""  / J V (t 1 - s) r ^{1 - 1} x - тГ- - (t 2 - s     (X 2 - t) r ^{2 - 1} ] dTds

0 0

t 2 X 2

IIpIIq ⁺ Н ^ Н о П *

Г(Г 1 )Г(Г 2 )

+                -- -1^

+   Г(Г1)Р(Г2>   J J t " s)      ■ " т)+ t1 x1

t 1 X 2

V/(t 2 ^- s)- *

0 x 1

(x ₂ - т ) r ^{2 1} dTds +

t 2 x 1

IIpIIo + IkUn* [

Г( _{Г 1} )ГЫ J _У ^{(t 2} - ^{s) 1 X} - ^{T) 2 dTdS}

C ГР 1 ^Г^ Г П1)  •' (t 2 - РГ + « (X 2 - X 1 ) r ² - (t 2 - t 1 ) ^{r 1} (X 2 - X 1 ) r ² + t -    - t ? X ? ]

, IIpIU ⁺ llq h o n *            -         _ АГ2 , HpHo ⁺ H q b o n *

[^ - (t 2 - tO r¹] (X 2 - X 1 ) r ²

llplk ⁺ llqlkn *

Г(Г 1 + 1)Г(Г 2 + 1)

' I   + 1)Г(Г 2 + 1) ⁽t ² "  ^{( 1 )}     ■ "  ^{X 1 ) +} Г(Г 1 + 1_№ + 1)

I ^{I p |} ° ⁺ H ^{q |} ° ⁿ           . \Г1 ^V r2    /         №^]

+ Г(П + 1)Г(Г2 + 1) (t2 - 81) ^2 - (X2 - X1) J< х [2x22 (t2 - t1)r1 +2tr1 (X2 - X1)r2 + tr1 xr2 - t21 xr2 - 2(t2 - t1)r1 (X2 - X1)r2] .

As t 1 ^ t 2 , X 1 ^ X 2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t 1 < t 2 <  0, X 1 < X 2 < 0 and t 1 C 0 C t 2 , X 1 C 0 C X 2 is obvious. As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we can conclude that A : C o ^ C o is continuous and completely continuous.

Step 4. B is a contraction.

Let w,w* G Cy. Then we have for each (t, x) 6 J tx

| (Bw)(t,x) - (Bw * )(t.x) | C _{Г( Г 1 )} ¹ _{Г( Г 2 )} j J(t - s) ^{r 1 - 1} (x - т) r ^{2 - 1} oo

^{X | g}(s. ^т . ^{W (} P 1 ^{( s},T,u^ sT ) ⁾ ,P 2 ⁽ s,^T,u ( s.T ) )) ⁺ v ⁽ P 1 ^{( s,T,u} (s,T ) ^{) ,}P 2 ^{( s},^t,^u ( s,t ) )))

- g(s. т. w* (p1(s,T,u(s,T)),P2(s,T,u(s,T))) + V(P1(s,T,u(s,T)),P2(s,T,u(s,T)))) II dтds tx

^C Г( _{Г 1} )Г( _Г 2 ) I ]' ^{(t - s) 1     (x - т ) 2    1 || W (} P 1 ^{( s,T,u} ( s,T ) ^{) ,}P 2 ^{( s,T,u} ( s,T > ))

tx

-

w* (P1(s,T,u(s,T)),P2(s,T,u(s,T)))!Ib C Г(Г1)Г(Г2) J I(t - s) 1    (x - T) 2   llw(s,T) - w*(s,T )11в tx

^C pp //^{( t - s) r 1 - 1 (x - T) r 2 -} oo

¹ sup      | w(s,T) - w * (s,T) | dTds

( s,t ) e [o ,T ] x [o ,x ]

tx

C            г    [(t - s)r ^{1 - 1} (x - т )r ^{2 - 1} dTds llw - w *H, _MC            ^X 7 l|w - w * ^r

Г(г 1 )Г(г ₂ ) J0P ’ ^{v                            ( a}’b )     Г(г 1 + 1)Г(г ₂ + 1)           ll( a,b )

Therefore | (Bw) - (Bw * ) | ( a,b ) C _Г ( _г P^r^p) | w - w * l ( a,b ) • Since by (1), B is a contraction.

Step 5 (A priori bounds). Now it remains to show that the set E ={ w G C(J, R ): w = AA(w) + AB( W ) } for some A G (0,1) is bounded.

Let w E E, then and w = AA(w) + AB (y) for some 0 < A < 1. Thus for each (t, x) E J, we have tx

^w(t-^x) = P(n №) //^(t - ^s) r ^{1 —} 1 ^(x - ^{T) r} 2 ^{- 1} 0 0

X ^f (s, ^T, w ⁽ p i ^{( s,T,u} ( _s,T ) ⁾ ,p 2 ⁽ s,^T,u ( _s,T ) )) ⁺ v ⁽ P 1 ^{( s,T,u} (s,T ) ^{) ,}P 2 ^{( s,t,u} ( _s,t )

tx

λ

Г(Г 1 )Г(Г 2 )

X // (t - s)r ^{1 - 1} (x - T (s,T, w ^{(P 1 (S,T,} u ^C

s,t ) ⁾ ,^p 2 ^{( s},^T,^u ( s,T ) )) ~"*~^( ^p 1 ( s,^T,^u ( s,r ) ⁾ ,^p 2 ^{( s},^T,^u ( s,T ) ))

λ

dτds.

This implies by (H2) and (H3) that, for each (t, x) E J, we have tx

^{| w(t},^{x) | C} _Г( _{Г 1} )Г( _{Г 2} ) / /⁽t — ^S) r ^{1 - 1 (}x — ^{T Г}— 1

tx

^X [ p^(s,^T) + q⁽s,^T) \\ w ( s,T ) + v ( s,T ) \\ b ] d ^Tds + _Г( _{Г 1} )_Г( _{Г 2} ) / j ^{(t - s)} r ^{1 - 1 (x - T) r 2 - 1}

X

g

(

^^(p 1 ^{( s},^T,^u ( s,T ) ) ,^p 2 ^{( s},^T,^u ( s,T ) )) "^ ^^{( p} 1 ^{( s},^T,^u ( s,T ) ) ,^p 2 ^{( s},^T,^u ( s,T ) ))

A

- g (s, T, 0)

dτds

tx ar1 br2 |p|^ Г(Г1 + 1)Г(Г2 + 1)

⁺ _r(r i )r( _{r 2} ) / /^{(t - s) r 1 — 1 (x - T) r 2 — 1} ^^(s,^T,⁰⁾ । ^{dTds C} 00

a r ¹ b r ² g *

⁺ Г(Г 1 + 1)Г(Г 2 + 1)

tx

⁺ гсГ: ! ^!^ / /⁽t ^{- s)} r ^{1 - 1 (}x ^{- T) r 2 - 1} \ w ( s,T ) ^{+ v} ( s,T ) \ b ^dTds 00

, I fL W1 - 1.       АГ 2 - 1 H-     , н . . / a r ¹ b r ² (hPh ~ + g * )

⁺ Г(Г 1 )Г(Г 2 ) J J ^{(t s)     (x T})      \\ ^{W (} s,^T ) ⁺ v ^{(sT ) H b} d ^T d ^S ^ Г(Г 1 + 1)Г(Г 2 + 1)

tx

. ⁽ H^qH^ ^{+ l )} I I          \Г1 — 1 /         \Г2 — 1 II  I          11 J J

+ Г(Г1)Г(Г2) J У (t — s)     (x — T)     \\W(s,T) + v(s,T)HB dTdS, where g* = sup(s,T)Gj |g(s,T, 0)| and

\\ ^{W ( s,T} ) ⁺ v ^{( s,T} ) \\ b ^ \\ ^{W ( s,T ) H} B ⁺ ll^{V ( s},^T ) IIB

< K sup { w(s,T) : (s, T) E [0, s] X [0,t ] } + (M + L ‘ ) | ф | в + K || ф(0,0) | .

If we name y(s,T) the right hand side of (3), then we have ||w(s,T) + V(s,T) ||B C y(t,x), and therefore, for each (t, x) E J we obtain llw(t,x)| <

a ^{r 1} b r ² ( H p H ^ + g * )

Г(Г 1 + 1)Г(Г 2 + 1)

tx

⁺ Г’(г1)_Г(Г2) //^{(t - s) r 1 — 1 (}x ^{- t ) r 2 — 1} y^(s,T) d^Tds- (4) 00

Using the above inequality and the definition of y for each (t, x) G J we have

y(t,x) * (M + L ‘ ) ^ Ф ^ в + K || Ф(0,0) | +

Ka r ¹ b r ² ( | p | ^ + g * ) r(r i + 1)Г(Г 2 + 1)

tx

+ K rq , _r + 2 ' / /(t - s) r ^{1 - 1} (x - T ) r ^{2 - 1} y(s, t) dTds.

Then by Lemma 2.1, there exists 5 = 5(ri,r 2 ) such that we have

| y(t,x) | * R + 5

k (IMU +1) Г(Г 1 )Г(Г 2 )

tx j j(t — s)r1-1 (x — t)r2-1 RdTds,

where

R = (M + L ‘ ) ЦфЦ в + K | ф(0,0) | +

Ka r ¹ b r ² ( H p H ^ + g * )

Г(Г 1 + 1)Г(Г 2 + 1) "

Hence

R5Ka^ 1 b r ² (||q|k + I)

y * ^{R +} Г(Г 1 _{+ 1№} + 1)

:= R.

Then, (4) implies that

r 1 r 2

m~ * _Г( _{Г 1 +1)Г} (r _{2 +}1) йр “ “ +g * +R⁽ " ⁹ " ” +i)i := R * -

This shows that the set E is bounded. As a consequence of Theorem 2.1 we deduce that A + B has a fixed point w which is a solution of problem (4)-(6). >

5. Examples

Example 5.1. As an application of our results we consider the following fractional order perturbed hyperbolic partial functional differential equations with finite delay of the form

( ^c D o r u)(t,x) =

| u(t — y ₁ (u(t, x)), x — a ₂ (u(t, x))) | + 2 10e ^{t + x +4} (1 + | u(t — a ₁ (u(t, x)), x — a ₂ (u(t, x))) | ) ’

if (t,x) G J :=[0,1] x [0,1], (1)

u(t, 0) = t,   u(0, x) = x ² ,   (t, x) G J,

u(t,x) = t + x ² ,   (t,x) G J := [ — 1,1] x [ — 2,1] \ [0,1] x [0,1],

where a ₁ G C ( R , [0,1]), a ₂ G C ( R , [0, 2]).

and

P 1 (t,x,y) = t — ^ 1 (^(0,0)), (t,x,y) G J x C([ — 1, 0] x [ — 2, 0], R ),

P 2 (t,x,y) = x — Ст 2 (у(0’ 0)), (t,x,y) G J x C([ — 1, 0] x [ — 2, 0], R ), ^f (t’x’^) = (10e t + x +4 )(1 + | ^ | ) ’ ^{(t,x) G J}’ ^ ^{G C([ —} 1’ ^{0] x [ — 2}’ ^0]’ ^{R )}’ g(t’x’^)=( _I 5e t + X +4 ) _T+^ ’ (t,x) G J’ ^ G C([ — 1’ 0] x [ — 2’ 0]’ ^R ).

For each y,y E C([ - 1, 0] x [ - 2, 0], R ) and (t,x) E J we have

^(t’x.y) - g^.x.y) ^C 5^ ^ y - y ^ C .

Hence condition (H2) is satisfied with k = 514. We shall show that condition (3) holds with a = b = 1. Indeed kar1 br2       =            1

r(ri + 1)r(r2 + 1)   5e4r(ri + 1)r(r2 + 1) < ’ which is satisfied for each (ri.r2) € (0,1] x (0,1]. Also, the function f is continuous on [0,1] x [0,1] x [0, to) and |f (t, x, y)| C |y|, for each (t, x, y) € J x C([-1, 0] x [-2, 0], R). Thus conditions (H1) and (H3) hold. Consequently Theorem 3.1 implies that problem (1)-(3) has at least one solution defined on [-1, 1] x [-2, 1].

Example 5.2. We consider now the following fractional order perturbed hyperbolic partial functional differential equations with infinite delay of the form

( ^c D o ^r u)(t,x) =

3 + |u(t - ^1(u(t’ x)), x - a2(u(t, x)))| 9et+x+5(1 + |u(t - ^i(u(t, x)), x - a2(u(t, x)))|) ’ if (t, x) E J := [0,1] x [0,1],

u(t, 0) = t,   u(0,x)= x ² , (t,x) E J,                           (5)

u(t,x)= t + x ² , (t,x) E J,                              (6)

where J := ( -to , 1] x ( -to , 1] \ [0,1] x (0,1], a i E C( R , [0,1]), a ₂ E C( R , [0, 2]).

B _Y = { u E C (( -to , 0] x ( -to , 0], R ) : lim   e ^{Y ( 6 + n )} u(9, n) exists E R } .

W^yH^

^The norm ^of B Y ^is given by llu ^ Y = ^sup ( e,n ) e ( -^ , o] x ( -^ , o] e^ +^H ⁹.^ ¹ -

Let E := [0,1] x {0} U {0} x [0,1], and u : (-to, 1] x (-to, 1] ^ R such that U(t,x) E by for (t, x) E E, then lim eY(6+n)uy x)(9’n) = lim eY(6-t+n-x)u(0’n) = e-Y(t+x) lim eY(6+n) u(9,n) < to. W,n)\H^        (,)       W,n)\H^                             W,n)\\^

Hence U ( _t,_x ) E b y . Finally we prove that

|u(t,x)hY = K SUP {|u(s,T )| : (s,T) E [0,t] x [0,x]} + M suP {|u(s,t)^Y : (s,T) E E(t,x)}’ where K = M = 1 and H = 1.

If t + 9 C 0, x + n C 0 we get | u ( _t,_x ) | _Y = sup {| u(s, т) | : (s, т) E ( -to , 0] x ( -to , 0] } , and if t + 9 ^ 0, x + n ^ 0, then we have | u ( _t,_x ) | _Y = sup {| u(s,T) | : (s,T) E [0,t] x [0,x] } . Thus, for all (t + 9, x + n) E [0,1] x [0,1], we get

^{| u} ( t,x ) 11 Y = sup { ^{| u(s}’ ^T) ^{1 : (s,T}) ^{E ( -TO} ’ ^{0] x ( -TO} ’ ^0] } ⁺ sup { ^{| u(s}’ ^T) ^{1 : (s,T}) ^{E [0}’^{t] x [0}’ ^x] } • ^{Then | u} ( t,x ) h 7 = ^su p ^{{| u} ( s,t ) H y ^{: (s,T}) ^{E E } +}S^up ^{{| u(}s’^T) ^{: (s,T}) ^{E [0,t] x [0,x] |}} . ⁽ b y ’ ||-Hy ) is a Banach space. We conclude that B γ is a phase space.

p 1 (t, x, y) = t - a 1 (y(0, 0))’   (t, x, y) E J x B _Y ,

P2(t, x, у) = x - ^(у(0, 0)), (t, x, у) E J x By, f (t,x,v) = (get+x+5 )(1 + |у|) ’ (t,x) E J’ V E BY’ and                                 3

g^(t’ x’ ■ = (_{9 e} t + x +5 )(l ₊ | _V | ) ’ ^{(t X) E} J’ V ^E B Y •

For each у, У E B y and (t, x) E J we have

\ g^{( t} ’x’V^{) -} g^{( t} ’ ^x ’ ^{V )} \ < ~ || v - ^v ^ B y •

Hence condition (H 2) is satisfied with I = з^. We shall show that condition (1) holds with a = b = K = 1 we get

ℓa ^{r 1} b ^{r 2} K

r(r i + 1)Г(г 2 + 1)   3e ⁵ r(r i + 1)Г(г 2 + 1) < ¹

which is satisfied for each (r i ,r 2 ) E (0,1] x (0,1]. Also, the function f is continuous on [0,1] x [0,1] x [0, to ) and | f (t,x,v) | C 3+ | v | , for each (t,x,v) E [0,1] x [0,1] x B _Y . Thus conditions (H1) and (H3) hold. Consequently Theorem 4.1 implies that problem (4)-(6) has at least one solution defined on ( -to , 1] x ( -to , 1].

Acknowledgment. The author is grateful to the referees for the careful reading of the paper and for their helpful remarks.