EXISTENCE RESULTS FOR FUNCTIONAL PERTURBED DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH STATE-DEPENDENT DELAY IN BANACH SPACES Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 1, P. 112-130

УДК 517.957

DOI 10.46698/10065-2825-9087-1

EXISTENCE RESULTS FOR FUNCTIONAL PERTURBED DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH STATE-DEPENDENT DELAY IN BANACH SPACES

M. Helal12

1 Science and Technology Faculty, Mustapha Stambouli University, B. P. 763, Mascara 29000, Algeria; 2 Laboratory of Mathematics, Djillali Liabes University, B.P. 89, Sidi Bel-Abbes 22000, Algeria E-mail: [email protected], [email protected]

Abstract. In this paper, we provide sufficient conditions for the existence of solutions of initial value problem, for perturbed partial functional hyperbolic differential equations of fractional order involving Caputo fractional derivative with state-dependent delay by reducing the research to the search of the existence and the uniqueness of fixed points of appropriate operators. Our main result for this problem is based on a nonlinear alternative fixed point theorem for the sum of a completely continuous operator and a contraction one in Banach spaces due to Burton and Kirk and a fractional version of Gronwall's inequality. We should observe the structure of the space and the properties of the operators to obtain existence results. To our knowledge, there are very few papers devoted to fractional differential equations with finite and/or infinite constant delay on bounded domains. Many other questions and issues can be investigated 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 satisfied by the operators are optimal. This paper can be considered as a contribution in this setting case. Examples are given to illustrate this work.

Key words: partial differential equation, fractional order, solution, left-sided mixed Riemann-Liouville integral, Caputo fractional-order derivative, state-dependent delay, fixed point. AMS Subject Classification: 26A33, 34K30, 34K37, 35R11.

For citation: Helal, M. Existence Results for Functional Perturbed Differential Equations of Fractional Order with State-Dependent Delay in Banach Spaces, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 112130. DOI: 10.46698/l0065-2825-9087-l.

1. Introduction

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].

© 2023 Helal, M.

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 state-dependent 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

№U) (t,x) = f{t,X,U(p1(t,x,u(tt:c)),P2(t,x,u(tx)))) + (t>X>U(t,x)),P2(t>X>U(t,x)}^ , if (t'x) € J,

U(t, x) = 0(t, x), if (t, x) € J, (2)

Ui, ») = „(t), ^ J, (3)

i u(0, x) = ^(x),

where <p(0) = ^(0), J := [0,a] x [0,6], a,6,a,£ > 0, J := [-a,a] x [-£,6]\[0,a] x [0,6], is the standard Caputo's fractional derivative of order r = (ri,r2) € (0,1] x (0,1], f,g : J x C ^ Rn, p1 : J x C ^ [-a, a], p2 : J x C ^ [—,0,6] are given functions, 0 € C := C(J, Rn) is a given continuous function with 0(t, 0) = ^>(t), 0(0, x) = ^(x) for each (t, x) € J, ^ : [0, a] ^ Rn, ^ : [0,6] ^ R" are given absolutely continuous functions and C is the space of continuous functions on j. We denote by U(t,X) the element of C defined by U(t,x)(s,r) = u(t + s, x + t), (s, t) € «7, 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

№U) (t,x) = f x, U(pi (t,X,U(t ;x) ),P2 (t,X,U(t,x) )) ) + ^t,x,U(pi (t>X>U(t,x)),P2(t,X>U(t,x))^ , if (t,x) € J,

u(t,x) = 0(t,x), if (t,x) € J7', (5)

iVt, 0) = w(t), , x

S (t,x) € J, (6)

i u(0, x) = ^(x),

where ^ are as in problem (1)-(3), J' := (-to, a] x (-to, 6]\[0, a] x [0,6], f,g : J x B ^ Rn, p1 : J x B ^ (-to, a], p2 : J x B ^ (-to, 6] are given functions, 0 : J' ^ Rn is a given continuous function with 0(t, 0) = <^(t), 0(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.

(4)

2. Preliminaries

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

By C(J, Rn) we denote the Banach space of all continuous functions from J into Rn with the norm ||u||^ = sup(tx)eJ ||u(t, x)||, where || ■ || denotes a suitable complete norm on Rn. As usual, by AC (J, Rn) we denote the space of absolutely continuous functions from J into Rn and L:(J, Rn) we denote the space of Lebesgue-integrable functions u : J ^ Rn with the norm ||u||Li = f0af0b ||u(t,x)||dxdt.

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

Definition 2.1 [19]. Let r = (ri,r2) € (0, to) x (0, to),0 = (0,0) and for u € LJ Rn), the expression

t x

№)(*, x) = r(ri)1r(r2) 11(t- syi-1 (x - rp-1 u(s, r) drds,

0 0

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

In particular, (i|u)(t, x) = u(t, x), (/^u)(t, x) = /0 J0x u(s, t)drds for almost all (t, x) € J, where ct = (1,1), For instance, /^u exists for all r1,r2 € (0, to) x (0, to), when u€L1(J, Rn). Note also that when u € C(J, Rn), then (/£u) € C(J, Rn), moreover (/£u)(t, 0) = (Feu)(0, x) = 0, (t, x) € J. By 1 — r we mean (1 — n, 1 — r2) € (0,1] x (0,1]. Denote by Dfx := ¿Jgj, the mixed second order partial derivative.

Definition 2.2 [19]. Let r € (0,1] x (0,1] and u € L1 (J,Rn). The Caputo fractional-order derivative of order r of u is defined by the expression

(CDr0u) (t, X) = ( ll~V 7"—— u ) (t, x).

e dtdx

The case a = (1,1) is included and we

have (Du)(t,x) = (cD£u)(t,x) = (Dt2xu)(i,x) for

almost all (t, x) € 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 u : 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 < ri, r2 < 1 such that

t x

, . , . if u(s,T) , ,

V(t, X) ^ Lü(t, x) + c / / --r-7-r- drds,

v ; " v ; J J (t s)ri(x t)7"2 0 0

then there exists a constant 5 = 5(r1, r2) such that

t x

, . , . _ f f w(s,r) , ,

v{t,x) ^ Uj{t,x) + OC / / 7-r—7-r—drds,

v ; v ; J J (t - s)ri(x - r)r2 00

for every (t, x) € 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

(а) the operator equation u = A(u) + B(u) iia.s a solution, or

(б) the set S = {« € X : u = \A(u) + )} is unbounded for A € (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 [-^,6], Rn), a, b > 0.

Definition 3.1. A function u € 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 € Lq(J, Rn) and consider the following problem

i (cD0u)(t,x) = f(t,x) + g(t,x), (t,x) € J, (1)

|u(t, 0) = <p(t), u(0,x) = ^(x), <p(0) = ^(0). ()

For the existence of solutions for the problem (1)-(3), we need the following lemma. Lemma 3.1. A function u € C (J, Rn) is a solution of problem (1) if and only if u(t,x) sati sfies

u(t,x)= z(t,x) + (I0f)(t,x) + (I0g)(t,x), (t,x) € J, (2)

where z(t,x) = <^(t) + ^(x) — ^>(0).

< Let u(t,x) be a solution of problem (1). Then, taking into account the definition of the fractional Caputo derivative (cD0u)(t,x), we have /¿-r(D2Xu)(s,T) = f(t,x) + g(t,x). Hence, we obtain IQ(IQ-rD2Xu)(s,T) = IQf(t, x) + IQg(t,x), then Iq(D2xu)(s,t) = IQf(t, x) + IQg(t, x). Since Iq(D2xu)(s, t) = u(t, x) — u(t, 0) — u(0, x) + u(0,0), we have u(t, x) = z(t, x) + (Iqf)(t, x) + (Iqg)(t,x). By the definition of the left-sided mixed Riemann-Liouville integral of order r of f and g, we have

t X

•»(¿,x) = ¿(¿,x) + 1 [ ¡it — s)ri_1 (x — r)7"2-1 f(s, t) drds 1 (ri(r2) J J

0 0 t x

+ rcrorcra) / /(t " S)ri_1 " T)r2_1T) dTdS' 0 0

so

t x

u(i, x) = x) + r(ri)1r(r2) j j\t- s)^-1 (x - r)^"1 [f(s, t) + «?(*, x)] drds,

0 0

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 € C(a,b) is a solution of problem (1)-(3) if and only if u satisfies the equation

t X

u(t, x) = z(t, x) + p^p^) J Jti- s)ri~l (x - TY2~l /(s'r' w(a,r)) drds

0 0 t x

+ r(ri)r(r2) s)ri_1 ^ " T)r2_1 r' drds'

0 0

for all (t, x) € J and the condition (2) on

Set R :=R(p-p-) = {(P^s, t, u), p2(s,t, u)) : (s,t, u) € J x C, Pj(s,t, u) ^ 0; i = 1,2}. We always assume that pQ : J x C — [—a,a], p2 : J x C — [—5,6] are continuous and the function (s,t) ——► 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: (H1) The functions f,g : J x C ^ Rn 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

KaTlbT2

r(ri + l)r(r2 + 1) < ' 1 j

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

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

( ~

<(t,x), (t,x) € J,

(Fu)(t, x) = z(t, x) + J f(t - sy^x ~ rr-1

0 0

x fis,T,u(pi(s,r,u(S!T)),p2(s,t,u(StT)))) drds, (t,x) € J,

and _

'0, (t,x) € .J,

(Gu)(t, x) = r^M f fa ~ " T)r2_1

' 0 0

x g(s, T, u(pi (s,T,U(s,T )),P2(s,T,U(s,T)))) dTdS, (t, x) € 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 {un} be a sequence such that un ^ u in C(ah). Let n > 0 be such that \\un\\ ^ n Then t x

\UFun)(t,x)-(Fu)(t,x)\\^—-^—— j j\{t - s)ri~l {x - T)r2~l\

T(ri)T(r2)

00

x ||f (s, T, un(pi (s,T,unia>T)),P2(s,T,un{a>T)))] — f (s, T, u(pi(s,T,uia>T)),P2(s,T,uia>T)))) II dTds

t x

1

< r(n)r(r2) J 1{-x~t)T2 1 ll/i^^Mpii™^^)^^^^)))

0 0 T)

t x

f( Ml^ ^ ll/(T^n(y))-/(v^)lloo f in V1-1 -/(s,r,u(/0l(s>r>U(iiT))>/02(s>r>U(iiT))))||drds< -r(n)r(r2)- I /(t"s)

1 j " rir2r(n)r(r2) " r(n + l)r(r2 + l)

Since f is a continuous function, we have

ari br2\\f (■, ■,un(..)) — f (■, ■,u(..))\co

ik^O-<F«)L<—rt,;+i)r(r2+vi -

Thus F is continuous.

oo

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 € = {u € C(a,b) : ||u||^ ^ n}, we have ||F(u)||^ ^ I*. By (H3) we have for each (t, x) € J,

t X

||(Fu)(t>a0|| < \\z(t,x)\\ + r(r01r(r2) J J(t - s^"1 (x - rP"1

0 0

t x

1

X ||/(s, T, W(pi(s>^(a>T))>p2(s>1->U(a>T)))) || dxds < \\z(t,x)W+ r(ri)p(r2) J J(t-s)ri l(x-T)r2 1

0 0

t x

1

xp(s,r) drds + r(ri)r(r2) J J (t-s?1 Hx-tY2

00

t x

" /IU n

< IkfoaOII+ TV AW \ / (t - sY1-1 (x - tY2-1 drds +

r(ri)r(r2) J J v ' r(ri)r(r2)

00

t x

x I j{t- sr~l [X - rr~l drds < ||z(t, x)|| + ariy2-

00

Thus

IK^Iloo < ll-lloo + r(n + ijrir!'Tl) ^ :=r"

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

Let (tQ,xQ), (t2,x2) € (0, a] x (0, b], tQ < t2, xQ < x2, be a bounded set of C(a,b) as in Step 2, and let u € . Then

||(Fu)(t2,x2) — (Fu)(ti,xi)|| < ||z(ti,xi) — z(t2,X2)\\ + ——r

ti xi

^[(t2 — s)ri-1(x2 —T)r2-1 — (ti — s)Qi-1(xl —T)Q2-1]|f ^ u(pi(s)T)«(s,T)),P2(s)T,«(s,T))))|dTds 00

t2 x2

+ r(ri;r(r2) //fe - -r-1 (x2 ~ tY2'1 ||/(s,r,u(pi(StTtU(sT))tP2(StTtU(sT))))H drds

ti xi ti x2

(s,t,u(SiT))^\\drds

+ r(n)r(r2) J y(i2"s)ri l^~TY2

0 X1 t2 X1

1

to)

t1 0

+ r(n)r(r2) I /(i2_s)ri 1^2"T)r2

tl X1

<||z(il>a:0-z(i2>®2)|| + M^Jfe 11 [(u-sr-^-rY2-1

00

t2 X2

- (¿2 - s)^-1 (x2 - r)^"1] drds + / /(t2 - S)ri_1 ^ - T)r2_1 dTdS

tl X1

ti X2

+ [ f(h -sr-Hx2-rr-idrds+Moo + Mo°r]

r(ri)T(r2) J J v 2 y v 2 y r(ri)r(r2)

0 xi

t2 Xi

>\U + \\q\U n

x / (t2 — s)ri-1 (x2 — T)r2-1 dTds < IIz(ti,xi) — z(t2,x2)II +

r(ri + 1)r(r2 + 1)

ti 0

x22 (t2 — ti)ri + 12i (x2 — xi)r2 — (t2 — ti)ri (x2 — xi)r2 + ti xi2 — #x?

i IIpIU+IMM n + vii^ rr. \r2 i Iblloo + Iklloo v /, , ™

+ r(r1 + l)r(r2 + l) (i2_il) +r(r1 + l)r(r2 + l)

As ti — t2, xi — x2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases ti <t2 < 0, xi < x2 < 0 and ti ^ 0 ^ t2, xi ^ 0 ^ x2 is obvious. As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we can conclude that F : C(a h) - C(a h) is continuous and completely continuous. Step 4. We show that G is a contraction.

Let v,w € C([—a, a] x [—p,b], Rn). Then, for (t,x) € [—a, a] x [—p,b],

t X

\\(Gv)(t,x)-(Gw)(t,x)\\^ j J J Kt-sr-'Wix-rY2-1]

00

k

x \\f{s,T,V(fil(StTtU(sT)^P2(StTtU(sT)^)-f(s,T, W(pi(StTtU(sT)^P2(StTtU(sT)^)\\drds ^ r(n)r(r2)

t x

xj ¡(t — s)ri-i (x — T)r2-i ^(„^)) ,„(„^))) — W(pi(.,„„,)) „(.,„„,^c dTds

00

t x

k k tri xr2

* ft^WW)ll" - '"llc J J(i -(x -** * r(ri + 1№ + 1)»« -

00

Consequently,

k ari br2

l|(g") - (g")|lc,.,M < r(r, + l)r(r2 + 1) ""

Since by (3), G is a contraction.

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

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

t x

u(t,x) = Xz(t,x) + T J J(t-s)ri-l(x-T)r2-lf(s,T, u^8>TiU(atT))M8iTiU(atT)))) drds

00

t x

+ A J J (f _ S)n-1 {x _ rr-lg ^ T> dTdS.

0 0

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

t X

|u(t,x)|| ^ A||z(t,x)|| +

r(ri)r(r2)

(t - s)r1-1 (x - T)

r2-i

00

p(s, T) + q(s, T) |u(p1(s,r,«(s,T)),P2(s)r,«(s,T)}} ||C

dTds +

A

r(ri)r(r2)

t X

J J(t - s)r1-1 (x - T)r2-1 00

0 S» 1"»- ^-- T, 0)

dTds

+

+

A

t X

r(ri)r(r2)

(t - s)r1-1 (x - T)r2-1 |g(s,T, 0)| dTds < ||z(t,x)|| +

ar16r2l

00

r(r1 + 1)r(r2 + 1)

ar1 br2 g*

t X

+

r(r1 + 1)r(r2 + 1) r(n)r(r2)

(t s) 1 (x T) 2 yu(p1(s,r,M(s,T )},p2(s,r,M(s,T )}}||c dTds

00

+

k

t X

r(r1)r(r2)

(t - s)r1 1 (x - T)r2 1 |Kp1(s,t)«(s,T)),p2(s)t)«(s,T))}||c dTds < ||z(t,x)|

r2-1

00

+

ar1 br2 (||p|U +g*)

+

,+k

t X

r(n + l)r(r2 + l) ' r(n)r(r2) I S^ril(X TY2 1\\Hpi{s,T,u(S)T)),p2{s,r,u(S)T)))\\cdTds,

00

where g* = sup^)j |g(s,T, 0)|.

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

Let (t*, x*) € [—a, t] x [—^,x] be such that y(t, x) the previous inequality, we have for (t, x) € J,

= ||u(t*,x*)||. If (t*,x*) € J, then by

ar1 br2 (hpn~ +g*K IMU+k

t X

») < »)»++J J(i - s)"~I(i"T)"" "(s'T)

00

If (t*,x*) € J, then y(t,x) = ||0||c and the previous inequality holds.

If (t,x) € J, Lemma 2.1 implies that there exists 5 = 5(rQ,r2) such that we have

y(t,x) <

|z(t,x)N +

aribrH\\p\\oo+g*)

r(n + l)r(r2 + l)

t X

l + i ¡{t-sr-\x-rr-Urds

r(r1 )r(r2)

00

<

|z(t,x)N +

aribrK\\p\\oo + 9*) r(n + l)r(r2 + l)_

1+

¿ar1 br2

+ k)

r(r1 + 1)r(r2 + 1)_

:= M.

Since for every (t,x) € J, ||u(t,x)||c ^ y(t,x), we have ||u|^ ^ max(||0||c, 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). >

1

x

oo

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) € 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, ||(-, -)||b) is a seminormed linear space of functions mapping (—to, 0] x (—to, 0] into Rn, and satisfying the following fundamental axioms which were adapted from those introduced by Hale and Kato for ordinary differential functional equations:

(Aq) If y : (—to, a] x (—to, b] ^ Rn continuous on J and y(t,x) € B, for all (t, x) € E, then there are constants H, K, M > 0 such that for any (t, x) € J the following conditions hold:

(i) y(t,x) is in B;

(ii) ||y(t,x)|| < H||y(t

,x) ||B;

(iii) ||y(t>x)|B < K suP(s)T)e[0)t]x[0,x] ||y(s,T)| + M suP(s,T)€E(M) ^(s.T^B.

(A2) For the function y(-, ■) in (Aq), y(t,x) is a B-valued continuous function on J.

(A3) 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 0 : (—to, 0] x (—to, 0] ^ Rn which are continuous on [—a, 0] x [—5, 0], a, 5 ^ 0, with the seminorm ||0||b = sup(s,T)g[_a,0]x[-(g,0] ||0(s,t)||. Then we have H = K = M = 1. The quotient space B = B/|| ■ ||b is isometric to the space C([—a, 0] x [—5,0],Rn) of all continuous functions from [—a, 0] x [—5,0] 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 7 € R and let CY be the set of all continuous functions 0 : (—to, 0] x (—to, 0] ^ Rn for which a limit e7(s+T) 0(s,t) exists, with the norm ||0||c7 =

sup(S;T)e(_TO)0]x(_TO;0] eY(s+T) ||0(s, t)||. Then we have H = 1 and K = M = max{e_Y(a+b), 1}.

Example 4.3. Let a, 5,7 ^ 0 and let

0 0

||0||cl7 = sup ||0(s,t)|| + / / eY(s+T) ||0(s,t)| dTds

(s,T )e[_a,0]x[_^,0] J J

— ^o —^o

be the seminorm for the space CLY of all functions 0 : (—to, 0] x (—to, 0] ^ Rn which are continuous on [—a,0] x [—5,0] measurable on (—00, —a] x (—to,0] U (—to,0] x (—to, —5], and such that ||0||cl7 < to. Then H = 1, K = /_a eY(s+TWs, 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 (—to, b] ^ Rn : u(t,x) € B for (t,x) € E and u|j € C(J,Rn)}.

Definition 4.1. A function u € 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' := p-) = {(pQ(s, t, u), p2(s,T,u)) : (s,T,u) € J x B, p,(s,T,u) ^ 0; i = 1, 2}. We always assume that pQ : J x B ^ (—to,a], p2 : J x B^ (—to,b] are continuous and the function (s, t) ^ 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:

(H) There exists a continuous bounded function L : R'(p- p-) ^ (0, to) such that

||0(s,t) IIb ^ L(s, t) ||0||b, for any (s, t) € 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 € Q, then

IKs,T)I|b = (M + L') ||0||b + K sup l|u(0,n)ll,

(0,n)e[O,max{O,s}| X [0,max{0,T }]

where L' = sup(s,T)€r/ L(s,t).

Theorem 4.1. Assume (H^) and that the following hypothesis holds: (H1) The functions /, g : J x B ^ Rn are continuous.

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

(H3) There exist € C(J,R+) such that ||f(t,x,u)|| ^ p(t,x) + q(t,x)||u||B, for (t, x) € J and each u € B. If

lKari br2

—;-s—;-r < 1, (1)

r(ri + 1)r(r2 + 1) then the IVP (4)-(6) has at least one solution on (—to, a] x (—to, b].

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

' 0(t, x), (t,x) € J,

*(*> + wim^j Jo Lx(t - sr-1 (x - Tp-1 (Nu)(t, x) = ^ x т, u(pi(s)T,«(s,T)),P2(s)T,«(s,T))^ dTds (2)

+ r(n)r(r2) fo Jo (t ~ S)ri 1 (x ~ TY2 1

x g(S T, u(pi(s,T,«(s,T)),P2(s,T,«(s,T)))) ^^ (t, x) € J-

Let v(-, ■) : (—to, a] x (—to, b] ^ Rn be a function defined by,

| z(t, x), (t, x) € J, v(t,x) = v ' v 7 ~ [0(t,x), (t,x) € J.

Then v(t,x) = 0 for all (t, x) € E.

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

, . | w(t, x) (t, x) € J, w(t,x) = < ~

[0, (t,x) € J.

If u(-, ■) satisfies the integral equation

t x

u(t, x) = z(t, x) + r(ri)1r(r2) J J(t- sr-1 (X - r)-"1

00

X f{s,T,w{pi{s>t>Uis t))>P2{s>t>Uis t))) +v{pi{s>t>u(s t))>p2{s>t>uis t)))) drds + r(ri)p(r2)

t x

X//_ ■S)ri~1(® - + «(piis.t^.oJ.pais.t,«^,,)))) dTds,

00

we can decompose «(•,•) as u(t,x) = w(t,x) + v(t,x)\ (t,x) € J, which implies u{t,x) +v(t,x)i (t,x) € J, and the function w(-,-) satisfies

t x 0 0

X f{s,T,W(pi(s>t>U{s^>p2(s>t>U{sA^ +V(pi(s,t,u{S!t)),p2(s,t,u{S!t)))) drds + p^Jp^y

t x 00

Set C0 = {w € C(J,Rn) : w(t, x) = 0 for (t,x) € E}, and let || ■ ||(0,b) be the seminorm in C0

defined by ||w||(o,6) = suP(t,x)eE ||w(t,x)Mb + suP(t,x)eJ |w(t,x)| = suP(t,x)eJ |w(t,x)|, w € C0-C0 is a Banach space with norm || ■ ||(0,b). Let the operators A, B : C0 ^ C0 defined by

t x

(■Aw)(t,x) = r(ri)r(ra) / j(t-sp-i(x-ry

0 0

X f{s>T>W(pi(s,t,u(s,t)),P2(s,t,u(s,t))) +V(pi(s,t,v-(s,t)),p2(s,t,vis,t)))) drds

and

t x 0 0

x g{s,T,W(pi(SitiU(sA)iP2(SitiU(sA)) +V(pi(s>t,u(sA),p2(s,t,u(sA))) drds.

-■,u(s,t))'P2(s,t,u(s,t))) ' v(Pl(s,t,u(s,t

Then the operator N has a fixed point is equivalent to finding the fixed point of the operator equation (Aw)(t,x) + (Bw)(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 C0. Then

t x

II(Awn)(t,x) - (Aw)(t,x)|| < T{rjT{r2) j j{t-s)^-l{x-ry2~l

0 0

x ||/(s> Ti wn(p1(s,T,unia>T)),p2(s,T,un{a>T))) + vn(p1(s,T,unia>T)),p2(s,T,un{a>T))))

— f (s, T, W(pi(StTtU(^g^)>P2(S,T,U(SiT))) + v(p1(s,T,uia>T)),p2(s,T,uia>T)))] II drds t x

< r(ri)r(r2) //(i_s)ri~1(a;~r)i'2~1 ||/(s'r'«Jn(i,r)+^n(s,r))-/(s,T,«;(S)r)+v(S)r))|| drds. 0 0

Since f is a continuous function, we have

(Awn) - (Aw) 11 <

r(ri + 1)r(r2 + 1)

anb 2 /(•, •, wn(v) + vn(-,-)) ~ /(T,w(v) +v(v))||co

^-TT7-H^t-7T——-—--> o as n ->• oo.

r(ri + 1)r(r2 + 1)

Thus A is continuous.

oo

Step 2. A maps bounded sets into bounded sets in C0.

Indeed, it is enough show that, for any n > 0. there exists a positive constant I such that, for each w € Bn = {w € C0 : ||w||(a,b) ^ n}, we have ||A(w)||to ^ I. Lemma 4.1 implies that

eg ^ 11^(5,r) || og + ||V(s,r) || og < Kn + K¡0(0,0)! + (M + L')||0||b. Set n* := Kn + K||0(0, 0)|| + (M + L')||0||b• Let w € B^. By (H3) we have for each (t,x) € J,

t x

iioMfo x)ii < r(ri;r(r2) j j a- sp-1 (x - rp-1

0 0

x ||/(s> Ti w(p1(s,T,u(SiT)),p2(s,T,u(SiT))) + v(p1(s,r,u(SiT)),p2(s,r,u(SiT)))) II drds t x t x

* fe) / /(i"^" p(s-T) dTds + fe) / /(i"^

0 0 0 0

X (x - r)7-2 1 <?(s, r) ||^(/)i(s,r,u(SiT)),/)2(s,r,u(SiT))) + '"(/3i(s,r,M(s>T)),/32(s,r,M(s>T))) \ag drds t x t x

< fife / /(i - S)"~1(I - T)"~1IiTIb+fife / /(i - S)"~1(I - T)"1~'IiTds

0 0 0 0

< IIpIIoo + IMloo??* ,r,r2 < Iblloo + IIgM* r1ftr, ,= p* ^ r(n + l)r(r2 + 1) " r(n + l)r(r2 + 1) '

Hence ||A(w)|U < I*.

Step 3. A maps bounded sets into equicontinuous sets in C0.

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

I\(Aw)(t2,x2)-(Aw)(tl,xl)\\ < r(ri)1r(r2) j j [(t2-a)ri-1(®2-T)P2-1(ii-a)ri-1(®i-r)ra-1]

0 0

x ||/(s

t2 x2 ti xi

II/(s, r, W(pi(S)T)it(siT))>P2{s,T,U(s>t))) + v(p1(s,T,u(SiT)),p2(s,T,u(SiT)))] II drds

ti x2

0 xi

x ||/(s

t2 xi

+f(fe//^-^1^-^1

ti 0

x ||/(s

ti xi

< 11 [(fl - a)'1"1 («1 - rr- - (h - s)^-1 (x2 - tp-1] drds

00

t2 x2

+ + [ [{t2 - sp-1 (x2 - rp-1 drds + +

r(ri)r(r2) J J v ' r(ri)r(r2)

ti xi

ti x2 t2 xi

" " " " „*

x J j(t2 - syi-1 (x2 - rP~l drds + J j(t2-s)ri-1(x2-T)r2-1dTds

0 xi ti 0

< r(r!+^)r(!r+i) -iOri r - (t2-tly^x2-xiy2 + px\2-qxr2]

i* IUII i ||„|| „*

+f^wtk(i2 -- +wfrwtk K1 -(i2 --

+ J^ztlf00?^ ^-riK2 -(-2-*ir] < lbllo°+ MooV*

r(ri + 1)r(r2 + 1)v ' L 2 v ' J r(ri + 1)r(r2 +1)

x [2x22 (t2 - ti)ri + 2t2i (x2 - xi)r2 + tiixi2 - t2ix22 - 2(t2 - ti)ri (x2 - xi)r2] .

As ti — t2, xi — x2 the right-hand side of the above inequality tends to zero. The equicon-tinuity for the cases ti < t2 < 0, xi < x2 < 0 and ti ^ 0 ^ t2, xi ^ 0 ^ x2 is obvious. As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we can conclude that A : C0 — C0 is continuous and completely continuous. Step 4. B is a contraction. Let w,w* € C0. Then we have for each (t, x) € J

t x

II (Bw)(t, x) - (Bw*yt, x)\\< r(ri)1r(r2) j j (t- syi-1 (x - Tp"1

0 0

x ||<?(s, T, W(p1(s,T,U(SiT-)),p2(s,T,U(S!T-))) + v(p1(s,T,'u(s>T)),p2(s,T,'u(s>T))))

„ , ) Ti 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)))) || drds t x

^ r(ri)r(r2) J j{t~S)1 (X~T)2 ^\\W(pi(s,t,u(SiT)),p2(s,t,u(SiT))) 00

t X

~'u;*(pi(s,r,M(s>T)),p2(s,r,M(s>T)))|U ^ r(n)r(r2) J 1(X~TY2 l\\W{s,r)-w*{s,r)\\^

00

t x

f f _

^ w—vFT—V / / ~~ s)ri_1 (x — t)t2~1 sup \\w(s,t) — w*(s,t)\\drds

i(ri)i(r2) J J (s,r)g[0,T]x[0,Xl 00

t x

f f . \T1_1 / \ To_1 7 7 II_ _II x II_ _||

^ TT7—TTT7-7 / (t — S)1 (x — t) 2 drds «; —10* , ,, ^ —-----7- \\w— w* ,

r(n)r(r2) JJ 11 N(a'6) r(n + l)r(r2 + l) 11

00

Therefore \\(Bw) - (Bw*)\\^tb) ^ v(r{+i)Tb(rl+i) H^~w*\\(a,b)• Since by (1), B is a contraction.

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

Let w € then and w = AA{w) + A-B(^) for some 0 < A < 1. Thus for each (t,x) € J, we have

t x 00

t x

x

00

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

t x

IKi' ^ ^ r^to) J J{t"sr_1 " T)r2_1

0 0

t x

A

x [p(s, r) + r) ||w(,)T) + t;(8>t) |U drds + / / (* ~ s)ri 1 - 1

0 0

dTds

g r W{pi{s,t,U(SjT)),p2{s,t,U(BjT))) + v(p1(s,T,U(BjT)),p2{s,T,U(BjT))) ^ ^ ^ ^

t x

+ rife) / /(«-»)-' <* - I*»'- **» * rfeffe

00

t x

aribr2g* t Halloo f fu V2-i\\

+ r(n + l)r(r2 + 1) + J J{t~S)ri~ {X~Tr~ \\W^)+V(s,r)hdTds

00

t x

I f fu ,ri-i, y2-iii- ^ II ^ ^ ^ aribrHMoo + 9*)

+ ™ J J (t ~ S) (X~T) + dTdS ^ r(n + l)r(r2 + 1) 00

t x

+ ^il J J«- ^ - T)r2_111^) + H* 00

where g* = sup(s,T)eJ |g(s,T, 0)| and

„ si IliTJ^^II „ +

(3)

\w(s,t) + V(s,t) IU ^ \\W(s,r) IU + \y

< K sup {w(s,f) : (s,f) € [0, s] x [0,t]} + (M + L') ||0||b + K ||0(0, 0)||.

If we name y(s,r) the right hand side of (3), then we have ||l5J(S)r) ^ y(t,x), and therefore, for each (t, x) € J we obtain

t x

II u Ml ^ aribr2(||p||^ + g*) IMU +1 [[/, \n-1 / w A m

IK*'^ r(n + l)T(r2 + 1) + J J (t~ S) (X~T) y^dTds- &

00

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

Kari br2 (IIpIU + g*)

y(t,x) < (M + L') ||0|b + K I|0(0, 0)|| +

r(n + 1)r(r2 + 1)

t x

+ ffl^f / /(' - ^ <* - "" ')

00

Then by Lemma 2.1, there exists 5 = ¿(r1, r2) such that we have

t x

+ + J y (i — s)ri_1 (x — r)7"2-1 Rdrds,

00

where

R = (M + L'nM, + K 11.(0,0)11 +

Hence

+ r(n + l)r(r2 + 1) -

Then, (4) implies that

ari br2 ~ Iklloo < r(ri + 1)r(r2 + 1} [Moo+9* + R(h\\oo+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 r . _ \u(t - ai(u(t,x)),x - cr2(ti(i, a;)))| + 2 ( ou){t,x) - 1Qet+x+4(1 + _ ai(u(t}XyjX_ (J2(u(t,x)))\),

if (t, x) € J := [0,1] x [0,1], (1)

u(t, 0) = t, u(0,x) = x2, (t, x) € J, (2)

u(t,x)= t + x2, (t,x) € J := [—1,1] x [—2,1]\[0,1] x [0,1], (3)

where ^ € C(R, [0,1]), ^ € C(R, [0, 2]).

p!(t,x,<p) = t — ^1(^(0, 0)), (t,x,<p) € J x C([—1, 0] x [—2, 0], R),

02(t,x,<p) = x — ^2(^(0, 0)), (t,x,<p) € J x C([—1, 0] x [—2, 0], R),

and

= (10e^+4)(l + |y|)' ¥>^([-1,0] x [-2,0],

For each ip,ip £ C([-1,0] x [-2,0], R) and (t, x) € J we have

(¿0 -g{t,x,Tp)I < - ^llc-

Hence condition (#2) is satisfied with k = We shall show that condition (3) holds with a = b = 1. Indeed

kar1 br2 1

< 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,<p)| < M, for each (t,x,<p) € 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

(cr>r \u \ — 3 + \u(t — ai(u(t,x)),x — a2(u(t,x)))\ \ M)u){t,x) —

9et+x+5(1 + |u(t - CTi(u(t, x)), x - ^(u(t, x)))|)'

if (t, x) € J := [0,1] x [0,1], (4)

u(t, 0) = t, u(0,x) = x2, (t, x) € J, (5)

u(t,x) = t + x2, (t, x) € J, (6)

where J := (-to, 1] x (-to, 1]\[0,1] x (0,1], cti € C(R, [0,1]), ^ € C(R, [0, 2]).

B7 = {u € C((-to, 0] x (-to, 0], R) : lim e7(0+n)u(Q, n) exists € R}.

The norm of B7 is given by ||u||7 = sup^)^-«,^-^] e7(0+n)|u(Q, n)|.

Let E := [0,1] x {0} U {0} x [0,1], and u : (-to, 1] x (-to, 1] — R such that u(t,x) € for (t, x) € E, then

lim e7(0+n)u(tx)(Q,n) = lim e7(0-t+n-x)u(Q, n) = e-Y(t+x) lim e7(0+n)u(Q,n) < to.

Hence u(t x) € . Finally we prove that

l|u(t,x)IIy = Ksup{|u(s,T)| : (s,t) € [0,t] x [0,x]} + Msup {Hu^Hy : (s,t) € E^x)},

where K = M = 1 and H = 1.

If t + Q ^ 0, x + n ^ 0 we get ||u(t,x) ||Y = sup{|u(s, t)| : (s, t) € (-to, 0] x (-to, 0]}, and if t + Q ^ 0, x + n ^ 0, then we have ||u(t,x)||Y = sup{|u(s,t)| : (s,t) € [0, t] x [0,x]}. Thus, for all (t + Q, x + n) € [0,1] x [0,1], we get

l|u(t,x) ||Y = sup{ |u(s, t) | : (s, t) € (-to, 0] x (-to, 0]} + sup {|u(s, t) | : (s, t) € [0, t] x [0, x]}.

Then ||u(t>x)||7 = sup{|u(s,T) ||Y : (s,t) € E}+sup{|u(s, t) : (s,t) € [0, t] x [0, x]|}. (BY, ||-||7) is a Banach space. We conclude that is a phase space.

pi(t,x,^) = t - ai(^(0, 0)), (t,x,^) € J x ,

p2(t, x, = x — ct2(^>(0, 0)), (t, x, € J x B

Jd

(9ei+:c+5)(l + \ip\)

and

f(t,x,ip) = , (t,x) e J, (p e B7,

g(t,x,(f ) = i i—TV € J, € B7.

_3

(9ei+®+5)"(l + For each ip,Tp G B7 and (t, x) € J we have

- x, <£•)| < ¿5 -

Hence condition (iï2) is satisfied with ^ = -¿f- We shall show that condition (1) holds with a = b = K = 1 we get

lari br2 K 1

< 1,

r(n + 1)r(r2 + 1) 3e5r(n + 1)r(r2 + 1)

which is satisfied for each (r1,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,<p)| < 3 + M, for each (t,x,<p) € [0,1] x [0,1] x B7. 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.

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Received October 16, 2021 Mghamed Helal

Science and Technology Faculty, Mustapha Stambouli University, B. P. 763, Mascara 29000, Algeria, Associate Professor;

Laboratory of Mathematics, Djillali Liabes University, B. P. 89, Sidi Bel-Abbes 22000, Algeria, Associate Professor

E-mail: [email protected], [email protected]

https://orcid.org/0000-0002-7845-1018

Владикавказский математический журнал 2023, Том 25, Выпуск 1, С. 112-130

РЕЗУЛЬТАТЫ СУЩЕСТВОВАНИЯ ДЛЯ ФУНКЦИОНАЛЬНО ВОЗМУЩЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ДРОБНОГО ПОРЯДКА С ЗАПАЗДЫВАНИЕМ В БАНАХОВЫХ ПРОСТРАНСТВАХ

Хелал М.1'2

1 Факультет науки и технологий, Университет Мюстафа Стамбули, Алжир, 29000, Маскара, B. P. 763;

2 Лаборатория математики, Университет Джиллали Лиабесе, Алжир, 22000, Сиди-Бель-Аббэс, B. P. 89 E-mail: [email protected], [email protected]

Аннотация. В данной работе мы приводим достаточные условия существования решений начальной задачи для функционально возмущенных гиперболических дифференциальных уравнений в частных производных дробного порядок с участием дробной производной Капуто с запаздыванием, зависящим от состояния, сводя исследование к поиску существования и единственности неподвижных точек соответствующих операторов. Наш основной результат для этой задачи основан на нелинейной альтернативной теореме о неподвижной точке Бертона и Кирка для суммы вполне непрерывного оператора и сжатия в банаховых пространствах и дробной версии неравенства Гронуолла. Чтобы получить результаты существования необходимо принимать во внимание как структуру пространства, так и свойства возникающих операторов. Насколько нам известно, очень мало работ, посвященных уравнениям дробных производных с конечным и/или бесконечным постоянным запаздыванием на ограниченных областях. В этом направлении возникает множество проблемных вопросов относительно существования решений в весовых пространствах непрерывных функций, единственности решения, строения множества решений, а также того, являются ли оптимальными условия, которым подчинены рассматриваемые операторы. Данную статью можно рассматривать как вклад в указанную проблематику. Приведены также иллюстрирующие примеры.

Ключевые слова: уравнение в частных производных, дробный порядок, решение, левосторонний смешанный интеграл Римана — Лиувилля, дробная производная Капуто, зависящая от состояния запаздывание, неподвижная точка.

AMS Subject Classification: 26A33, 34K30, 34K37, 35R11.

Образец цитирования: Helal M. Existence Results for Functional Perturbed Differential Equations of Fractional Order with State-Dependent Delay in Banach Spaces // Владикавк. мат. журн.—2023.—Т. 25, № 1.—C. 112-130 (in English). DOI: 10.46698/l0065-2825-9087-l.