Perturbed Partial Functional Di¤erential Equations On Unbounded Domains With Finite Delay
Mohamed Helal
yReceived 1 March 2019
Abstract
In this paper we investigate the existence of solutions of perturbed partial hyperbolic di¤erential equations of fractional order with …nite delay and Caputo’s fractional derivative by using a nonlinear alternative of Avramescu on Fréchet spaces.
1 Introduction
In this paper we are concerned with the existence of solutions to fractional order initial value problem (IV P for short), for the system
(cDr0u)(t; x) =f(t; x; u(t;x)) +g(t; x; u(t;x)); if(t; x)2J; (1)
u(t; x) = (t; x); if(t; x)2J ;~ (2)
u(t;0) ='(t); u(0; x) = (x); (t; x)2J; (3)
where'(0) = (0); J := [0;1) [0;1); J~:= [ ;+1) [ ;+1)n[0;1) [0;1); cD0r is the standard Caputo’s fractional derivative of order r = (r1; r2)2(0;1] (0;1]; f; g :J C ! Rn are given functions, 2C:=C([ ;0] [ ;0];Rn)is a given continuous function with (t;0) ='(t); (0; x) = (x)for each (t; x)2J; ': [0;1)!Rn; : [0;1)!Rn are given absolutely continuous functions and Cis the space of continuous functions on[ ;0] [ ;0]:
We denote byu(t;x) the element ofC de…ned by
u(t;x)(s; ) =u(t+s; x+ ); (s; )2[ ;0] [ ;0];
hereu(t;x)(:; :)represents the history of the stateu.
In recent years, fractional di¤erential and partial di¤erential equations have become more important in some mathematical models of real phenomena, especially in control, biological and medical domains.
In these models, the investigated simulating processes and phenomena usually are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. We can …nd numerous applications of di¤erential equations of fractional order in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [7, 17, 23, 24, 26]). There has been a signi…cant development in ordinary and partial fractional di¤erential equations in recent years; see the monographs of Abbaset al. [2], Aissani et al. [5], Kilbas et al. [19], Lakshmikantham et al. [21], and the papers by Agarwal et al [3, 4], Belarbiet al. [8], Benchohraet al. [11], and the references therein.
The theory of functional di¤erential equations has emerged as an important branch of nonlinear analy- sis. Di¤erential delay equations, or functional di¤erential equations, have been used in modeling scienti…c
Mathematics Sub ject Classi…cations: 26A33, 34K30, 34K37, 35R11.
ySciences and Technology Faculty, Mustapha Stambouli University of Mascara B.P. 763, 29000, Mascara, Algeria. Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbés, B.P. 89, 22000, Sidi Bel-Abbés, Algeria
82
phenomena for many years. Often, it has been assumed that the delay is either a …xed constant or is given as an integral in which case it is called a distributed delay; see for instance the books by Hale and Verduyn Lunel [15], Hinoet al. [18], Kolmanovskii and Myshkis [20], Lakshmikanthamet al. [22], and Wu [28], and the papers [12,13].
Motivated by the previous papers, in this paper, we consider the existence of solutions for the problems (1)–(3). Our main result for this problem is based a nonlinear alternative for the sum of a completely continuous operator and a contraction in Fréchet spaces due to Avramescu [6] and a fractional version of Gronwall’s inequality. This is the main motivation to look for su¢ cient conditions ensuring existence of solutions for each of our problems. The present results extend those considered with …nite and/or in…nite constant delay on bounded domains in [1, 9]. To our Knowledge, there are very few papers devoted to fractional di¤erential equations with delay on Fréchet spaces. The aim of this paper is to continue this study.
2 Preliminaries
In this section, we introduce notations, de…nitions, and preliminary facts which are used throughout this paper. Letp2NandJ0= [0; p] [0; p]:ByC(J0;R)we denote the Banach space of all continuous functions fromJ0into Rn with the norm
kwk1= sup
(t;x)2J0
kw(t; x)k;
wherek:kdenotes a suitable complete norm onRn:As usual, byAC(J0;R)we denote the space of absolutely continuous functions fromJ0intoRnandL1(J0;R)is the space of Lebesgue-integrable functionsw:J0!Rn with the norm
kwkL1= Z p
0
Z p
0 kw(t; x)kdtdx:
De…nition 1 ([27]) Let r= (r1; r2)2(0;1) (0;1); = (0;0) andu2L1(J0;Rn):The left-sided mixed Riemann-Liouville integral of order rof uis de…ned by
(Iru)(t; x) = 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1u(s; )d ds:
In particular,
(I u)(t; x) =u(t; x); (I u)(t; x) = Z t
0
Z x 0
u(s; )d ds; for almost all(t; x)2J0;
where = (1;1):
For instance, Iru exists for all r1; r2 2 (0;1) (0;1); when u 2 L1(J0;Rn): Note also that when u2C(J0;Rn);then(Iru)2C(J0;Rn);moreover
(Iru)(t;0) = (Iru)(0; x) = 0; (t; x)2J0: Example 1 Let ; !2( 1;1)andr= (r1; r2)2(0;1) (0;1): Then
Irt x!= (1 + ) (1 +!)
(1 + +r1) (1 +!+r2)t +r1x!+r2; for almost all(t; x)2J0:
By 1 r we mean (1 r1;1 r2) 2(0;1] (0;1]: Denote by D2tx := @t@x@2 ; the mixed second order partial derivative.
De…nition 2 ([27]) Let r 2 (0;1] (0;1] and u 2 L1(J0;Rn): The mixed fractional Riemann-Liouville derivative of orderr ofuis de…ned by the expression
Dru(t; x) = (D2txI1 ru)(t; x)
and the Caputo fractional-order derivative of order rof uis de…ned by the expression (cDr0u)(t; x) = (I1 r @2
@t@xu)(t; x):
The case = (1;1)is included and we have
(D u)(t; x) = (cD u)(t; x) = (D2txu)(t; x); for almost all(t; x)2J0: Example 2 Let ; !2( 1;1)andr= (r1; r2)2(0;1] (0;1]:Then
Drt x!= (1 + ) (1 +!)
(1 + r1) (1 +! r2)t r1x! r2; for almost all(t; x)2J0:
In the sequel we will make use of the following generalization of Gronwall’s lemma for two independent variables and singular kernel.
Lemma 1 ([16]) Let : J ! [0;1) be a real function and !(:; :) be a nonnegative, locally integrable function onJ: If there are constantsc >0 and0< r1; r2<1 such that
(t; x) !(t; x) +c Z t
0
Z x 0
(s; )
(t s)r1(x )r2d ds;
then there exists a constant = (r1; r2)such that
(t; x) !(t; x) + c Z t
0
Z x 0
!(s; )
(t s)r1(x )r2d ds;
for every(t; x)2J:
3 Some Properties in Fréchet Spaces
LetX be a Fréchet space with a family of semi-normsfk kngn2N. We assume that the family of semi-norms fk kng veri…es :
kuk1 kuk2 kuk3 ::: for everyu2X:
LetY X, we say that Y is bounded if for everyn2N, there existsMn >0 such that kvkn Mn for allv2Y:
To X we associate a sequence of Banach spaces f(Xn;k kn)g as follows : For every n 2 N, we consider the equivalence relation n de…ned by : u n v if and only if ku vkn = 0 for u; v 2 X. We denote Xn = (Xj n;k kn)the quotient space, the completion of Xn with respect to k kn. To everyY X, we associate a sequencefYngof subsetsYn Xn as follows : For everyu2X, we denote[u]n the equivalence class ofuof subsetXn and we de…neYn=f[u]n :u2Yg:We denoteYn,intn(Yn)and@nYn, respectively, the closure, the interior and the boundary ofYn with respect to k kn in Xn:For more information about this subject see [14].
De…nition 3 Let X be a Fréchet space. A function N : X ! X is said to be a contraction if for each n2Nthere exists kn 2(0;1)such that
kN(u) N(v)kn knku vkn for allu; v2X:
Theorem 2 (Nonlinear Alternative of Avramescu [6]) Let (X;j:jn)be a Fréchet space and let A; B : X !X two operators. Suppose that the following hypothesis are ful…lled:
(i) Ais a compact operator.
(ii) B is a contraction operator with respect to a family of seminorms jj:jjn equivalent with the familyj:jn. (iii) the setE =fu2X :u= A(u) + B(u)for some 2(0;1)g is bounded.
Then there isu2X such thatu=Au+Bu.
4 Existence of Solutions
In this section, we give our main existence result for problem (1)–(3).
For each p2Nwe consider the following set
Cp=C([ ; p] [ ; p];Rn) and we de…ne inC0:=C([ ;1) [ ;1);Rn)the semi-norms by:
kukp=fsupku(t; x)k: t p; x pg: ThenC0is a Fréchet space with the family of semi-normsfk kpg:
Before starting and proving this result, we give what we mean by a solution of the problem (1)–(3).
De…nition 4 A functionu2C0 is said to be a solution of (1)–(3) ifusatis…es equations(1)and(3)onJ and the condition(2) onJ~.
For the existence of solutions for the problem (1)–(3), we need the following lemma:
Lemma 3 A functionu2C0 is a solution of problem (1)–(3) if and only ifusatis…es the equation
u(t; x) = z(t; x) + 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1f(s; ; u(s; ))d ds
+ 1
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1g(s; ; u(s; ))d ds;
for all(t; x)2J and the condition(2)on J ;~ where
z(t; x) ='(t) + (x) '(0):
Our main existence result in this section is based on the nonlinear alternative of Avramescu. We will need to introduce the following hypothesis:
Theorem 4 Assume the following conditions hold:
(H1) The functionsf; g:J C([ ;0] [ ;0];Rn)!Rn are continuous.
(H2) For eachp2N; there exists`p2C(J0;Rn)such that for each(t; x)2J0
kg(t; x; u) g(t; x; v)k `p(t; x)ku vkC; for each u; v2C:
(H3) There exist p; q2C(J;R+)such that
kf(t; x; u)k p(t; x) +q(t; x)kukC; for(t; x)2J0 and eachu2C:
If
`ppr1+r2
(r1+ 1) (r2+ 1) <1; (4)
where
`p= sup
(t;x)2J0
`p(t; x);
then there exists a unique solution for IVP (1)–(3) on [ ;1) [ ;1):
Proof. Transform the problem (1)–(3) into a …xed point problem. Consider the operatorsF; G:C0!C0 de…ned by
F(u)(t; x) = 8>
><
>>
:
(t; x); (t; x)2J ;~
z(t; x) + 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1
f(s; ; u(s; ))d ds; (t; x)2J:
and
G(u)(t; x) = 8>
><
>>
:
0; (t; x)2J ;~
1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1
g(s; ; u(s; ))d ds; (t; x)2J:
The problem of …nding the solutions of theIV P (1)–(3) is reduced to …nding the solutions of the operator equation(F u)(t; x) + (Gu)(t; x) =u(t; x);(t; x)2J. We shall show that the operatorsF andGsatis…es all the conditions of Theorem2.
For better readability, we break the proof into a sequence of steps.
Step 1. F is continuous. Let fungbe a sequence such thatun!uinC0:Then
k(F un)(t; x) (F u)(t; x)k 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1 kf(s; ; un(s; )) f(s; ; u(s; ))kd ds:
Sincef is a continuous function, we have
k(F un) (F u)kp pr1+r2kf(:; :; un(:;:)) f(:; :; u(:;:))kp
(r1+ 1) (r2+ 1) !0 asn! 1: ThusF is continuous.
Step 2: F maps bounded sets into bounded sets in C0: Indeed, it is enough show that, for any >0;
there exists a positive constant ` such that, for eachu2B =fu2C0:kukp g; we havekF(u)kp `. Letu2B :By (H3);we have for each(t; x)2J0;
k(F u)(t; x)k kz(t; x)k+ 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1 kf(s; ; u(s; ))kd ds
kz(t; x)k+ 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1p(s; )d ds
+ 1
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1q(s; ) ku(s; )kCd ds
kz(t; x)k+ kpkp
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1d ds
+ kqkp
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1d ds
kz(t; x)k+ kpkp+kqkp
(r1+ 1) (r2+ 1)pr1+r2: Thus
k(F u)kp kzkp+ kpkp+kqkp
(r1+ 1) (r2+ 1)pr1+r2 := ~`:
Step 3: F maps bounded sets into equicontinuous sets inC0: Let(t1; x1);(t2; x2)2J; t1< t2; x1 < x2; B be a bounded set as in step2, and letu2B :Then
k(F u)(t2; x2) (F u)(t1; x1)k kz(t1; x1) z(t2; x2)k+ 1
(r1) (r2) Z t1
0
Z x1
0
[(t2 s)r1 1(x2 )r2 1 (t1 s)r1 1(x1 )r2 1]jjf(s; ; u(s; ))jjd ds
+ 1
(r1) (r2) Z t2
t1
Z x2
x1
(t2 s)r1 1(x2 )r2 1jjf(s; ; u(s; ))jjd ds
+ 1
(r1) (r2) Z t1
0
Z x2
x1
(t2 s)r1 1(x2 )r2 1jjf(s; ; u(s; ))jjd ds
+ 1
(r1) (r2) Z t2
t1
Z x1
0
(t2 s)r1 1(x2 )r2 1jjf(s; ; u(s; ))jjd ds
kz(t1; x1) z(t2; x2)k+kpkp+kqkp (r1) (r2) Z t1
0
Z x1
0
[(t2 s)r1 1(x2 )r2 1 (t1 s)r1 1(x1 )r2 1]d ds
+kpkp+kqkp (r1) (r2)
Z t2
t1
Z x2
x1
(t2 s)r1 1(x2 )r2 1d ds
+kpkp+kqkp (r1) (r2)
Z t1
0
Z x2
x1
(t2 s)r1 1(x2 )r2 1d ds
+kpkp+kqkp (r1) (r2)
Z t2
t1
Z x1
0
(t2 s)r1 1(x2 )r2 1d ds
kz(t1; x1) z(t2; x2)k+ kpkp+kqkp
(r1+ 1) (r2+ 1)[xr22(t2 t1)r1 +tr21(x2 x1)r2 (t2 t1)r1(x2 x1)r2+tr11xr12 tr21xr22] + kpkp+kqkp
(r1+ 1) (r2+ 1)(t2 t1)r1(x2 x1)r2 + kpkp+kqkp
(r1+ 1) (r2+ 1)[tr21 (t2 t1)r1](x2 x1)r2 + kpkp+kqkp
(r1+ 1) (r2+ 1)(t2 t1)r1[xr22 (x2 x1)r2 1] kz(t1; x1) z(t2; x2)k+ kpkp+kqkp
(r1+ 1) (r2+ 1)[2xr22(t2 t1)r1 +2tr21(x2 x1)r2+tr11xr12 tr21xr22 2(t2 t1)r1(x2 x1)r2]:
The right-hand side of the above inequality tends to zero ast1 !t2; x1 !x2. The equicontinuity for the casest1< t2<0; x1< x2<0and t1 0 t2; x1 0 x2is obvious.
As a consequence of steps 1 to 3 together with Arzela-Ascoli theorem, we can conclude thatF :C0!C0
is a compact operator.
Step 4: Gis a contraction. Letu; v2C0:Then we have for each(t; x)2J0
k(Gu)(t; x) (Gv)(t; x)k 1
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1 kg(s; ; u(s; )) g(s; ; v(s; ))kd ds
1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1`p(s; )ku(s; ) v(s; )kC 1
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1`p(s; ) sup
(s; )2[0;T] [0;X]ku(s; ) v(s; )kd ds
`p(s; ) (r1) (r2)
Z p 0
Z p 0
(t s)r1 1(x )r2 1d dsku vkp: Therefore
k(Gu) (Gv)kp
`ppr1+r2
(r1+ 1) (r2+ 1)ku vkp: By (4),Gis a contraction.
Step 5:(A priori bounds). Now it remains to show that the set E =
n
u2C(J;R) :u= F(u) + G(u
) for some 2(0;1) o
is bounded. Letu2 E. Then andu= F(u) + G(u)for some0< <1:Thus for each(t; x)2J0;we have
u(t; x) =
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1f(s; ; u(s; ))d ds
+ (r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1g(s; ;u(s; ) )d ds:
This implies by(H2)and(H3)that, for each(t; x)2J0;we have
ku(t; x)k jjz(t; x)jj+ 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1[p(s; ) +q(s; )ku(s; )kC]d ds
+ (r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1 g(s; ;u(s; )
) g(s; ;0) d ds + (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1jg(s; ;0)jd ds
jjz(t; x)jj+ pr1+r2kpkp
(r1+ 1) (r2+ 1)+ pr1+r2g (r1+ 1) (r2+ 1) + jjqjjp
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1ku(s; )kCd ds
+ 1 (r1) (r2)
Z t 0
Z x 0
(t s)r1 1(x )r2 1`p(s; )jju(s; )jjCd ds
jjz(t; x)jj+ pr1+r2(kpkp+g ) (r1+ 1) (r2+ 1) +(jjqjjp+`p)
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1ku(s; )kCd ds;
whereg = sup(s; )2Jjg(s; ;0)j. Consider the functiony de…ned by
y(t; x) = supfku(s; )k: s t; xg; 0 t p; 0 x p:
Let (t ; x ) 2 [ ; t] [ ; x] be such that y(t; x) = ku(t ; x )k: If (t ; x ) 2 J0; then by the previous inequality, we have for(t; x)2J0;
y(t; x) jjz(t; x)jj+ pr1+r2(kpkp+g ) (r1+ 1) (r2+ 1) +jjqjjp+`p
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1y(s; )d ds:
If(t ; x )2J ;~ theny(t; x) =k kC and the previous inequality holds. If(t; x)2J0, Lemma 1 implies that there exists = (r1; r2)such that we have
y(t; x) jjz(t; x)jj+ pr1+r2(kpkp+g ) (r1+ 1) (r2+ 1) 1 + (jjqjjp+`p)
(r1) (r2) Z t
0
Z x 0
(t s)r1 1(x )r2 1d ds
jjz(t; x)jj+ pr1+r2(kpkp+g )
(r1+ 1) (r2+ 1) 1 + pr1+r2(jjqjjp+`p)
(r1+ 1) (r2+ 1) :=M:
Since for every(t; x)2J0;ku(t;x)kC y(t; x); we have
kukp max(k kC; M) :=M :
This shows that the set E is bounded. As a consequence of Theorem2 we deduce that F +Ghas a …xed pointuwhich is a solution of problem (1)–(3).
5 Application
As an application of our results we consider the following partial perturbed hyperbolic functional di¤erential equations of the form
(cDr0u)(t; x) = ju(t 1; x 2)j+ 2
10cpet+x(1 +ju(t 1; x 2)j); if(t; x)2J := [0;1) [0;1); (5)
u(t;0) =t; u(0; x) =x2; (t; x)2J; (6)
u(t; x) =t+x2; (t; x)2J~:= [ 1;1) [ 2;1)n[0;1) [0;1); (7) where
f(t; x; u(t;x)) = ju(t 1; x 2)j
(10cpet+x)(1 +ju(t 1; x 2)j); (t; x)2J;
g(t; x; u(t;x)) = 2
(10cpet+x)(1 +ju(t 1; x 2)j); (t; x)2J;
and
cp= 3pr1+r2 (r1+ 1) (r2+ 1):
For eachu; v 2C([ 1;0] [ 2;0];R)and(t; x)2J0= [0; p] [0; p];we have jg(t; x; u) g(t; x; v)j 1
5cpet+xku vkC: Hence condition(H2) is satis…ed with`pet+x=5c 1
pet+x. Since
`p= sup 1
5cpet+x; (t; x)2J0 1 5c:
We shall show that condition (4) holds for each(r1; r2)2(0;1] (0;1]and allp2N . Indeed
`ppr1+r2
(r1+ 1) (r2+ 1) = 1 15 <1:
Also, the functionf is continuous onJ0 [0;1)and
jf(t; x; ')j j'jfor each(t; x; ')2J0 C([ 1;0] [ 2;0];R):
Thus conditions (H1) and(H3) hold. Consequently Theorem 4 implies that problem (5)–(7) has at least one solution de…ned on[ 1;1) [ 2;1).
6 Conclusion
Fractional calculus is a wide subject that requires extensive tools and various methods. In our consideration in this paper, we have presented a contribution to the study of di¤erent classes of Darboux problem for partial hyperbolic functional perturbed of fractional order involving the Caputo fractional derivative with
…nite delay in Fréchet spaces.
In most of this paper su¢ cient conditions were considered to get the existence and uniqueness results of solutions for our problem by reducing the research to the search of the existence and the uniqueness of …xed points of appropriate operators by applying a nonlinear alternative of Avramescu on unbounded interval.
There are many directions in which we can extend the work done. We should observe the structure of the space and the properties of the operators to obtain existence results. Many other questions and issues can be investigeted regarding the existence in the space of weighted continuous functions, the uniqueness, the structure of the solutions set and also whether or not the condition satis…ed by the operators are optimal.
Acknowledgment. The author is grateful to the referees for the careful reading of the paper and for their helpful remarks.
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