ON TWO-ORDER FRACTIONAL BOUNDARY VALUE PROBLEM WITH GENERALIZED RIEMANN-LIOUVILLE DERIVATIVE Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 15. № 2 (2023). С. 136-157.

ON TWO-ORDER FRACTIONAL BOUNDARY VALUE PROBLEM WITH GENERALIZED RIEMANN-LIOUVILLE

DERIVATIVE

H. SERRAI, B. TELLAB AND Kh. ZENNIR

Abstract. In this paper we focus our study on the existence, uniqueness and Hvers-Ulam stability for the following problem involving generalized Riemann-Liouville operators:

+ *)u(t)= f(t, u(t)).

It is well known that the existence of solutions to the fractional boundary value problem is equivalent to the existence of solutions to some integral equation. Then it is sufficient to show that the integral equation has only one fixed point. To prove the uniqueness result, we use Banach fixed point Theorem, while for the existence result, we apply two classical fixed point theorems due to Krasnoselskii and Lerav-Scauder. Then we continue by studying the Hvers-Ulam stability of solutions which is a very important aspect and attracted the attention of many authors. We adapt some sufficient conditions to obtain stability results of the Hvers-Ulam type.

Keywords: fractional derivatives, generalized Riemann-Liouville derivative, fixed point theorem, fractional Boundary value problem, Hvers-Ulam stability.

Mathematics Subject Classification: 34A08, 26A33, 34A12, 34B15

1. Introduction

Fractional calculus is a subject that has lately spread rapidly and its applications are used in several fields of applied sciences [22], [29], [36]. It plays essential roles, for example in engineering

[1], [23], [24], [32], structures [19], optimal control [6], chaotic systems [17], epidemiological models [12], [30]. The fractional structures of boundary value problems and initial value problems generally give a great diversity of mathematical models for the description of certain physical, chemical and biological processes that can be referred to in recently published papers

[2], [7], [10], [11], [21], [25], [26], [31], [33]. Parallelv to these real patterns caused by real phenomena, many researchers studied the existence theory of solutions for general constructions of fractional boundary value problems involving boundary conditions implying a multi-point nonlocal integral [3], [8], [9], [13], [14], [15].

S. Eezapour et al. [27] discussed the existence of numerical solutions via DGJIM and ADM methods for the follwing fractional boundary value problem implying the generalized ^-Riemann-Liouville operators:

u(t) = C Mt), u(t), vs0f u(t),..., Vs0^u(t)),

u( 0) = 0, «(1) = M£X£)) + k*(v,u(v)),

H. Serrai, B. Tellab and Kh. Zennir, On two-order fractional boundary value problem with

generalized rlemann-llouville derivative.

© Serrai H., Tellab B. and Zennir Kh. 2023. Submitted May 12, 2022.

where

0 ^ t ^ 1, 1 <g< 2, 0 < ¿i < 82 < ••• <8n < 1, q > 8n + 1,

and

< : [0,1] x Rn+1 ^ R, kj : [0,1] x R ^ R, j = 1, 2,

are continuous functions, Ve+, Vlf,..., V^ are the ^-Riemann-Liouville derivatives depending on an increasing function ^ of orders q, 81,..., 5n, respectively, and X+ '1S the ■0-Riemann-Liouville integral depending on the special function ^ of order 7 e {p,v} with p,u,p,q > ^d 0 < £, ^ ^ 1, Thabet et al, [34] considered and studied the existence of solutions of the following coupled system of the Caputo conformable fractional boundary value problems of the Pantograph differential equations formulated as

ccDT1*u(t) =Pi(t,TO(t),rn(lt)), z e [to,K], to ^ 0,

cc Dg"1 m(t) =V2(t,v(t),v(£t)), via three-point-EL-conformable integral conditions

^(to) = 0, + C2KC If/* v(8) = w*,

m( to) = 0, c1m(X) + c*2nc Iff* m(u) = w*2,

where

q e (0,1], a{,o*2 e (1,2), 5,u e (to,*:),

C1,C2,C*1,C*,W*1,W** e R, 1 e (0,1), Vx, f>2 e C([to,X] x R x R, R).

Based on some ideas and techniques used in the above cited works, we are interested in certain criteria of existence, uniqueness and Ulam-stabilitv of the generalized fractional boundary value problem

faf + ^)u(t) = f(t, u(t)), t eO = [0,1],

u(0) = 0, u(1) = pZ£'* u(rj)) + qZft u(a)),

where 0 < p1,p2 < 1, p1 + p2 > 1, f, : Ox R ^ R, (j = 1, 2) are three continuous functions, Vf+ is the ^-Riemann-Liouville fractional derivative depending on an increasing function ^ of order q e {p1,p2}, is the ^-Riemann-Liouville integral which depends on the function ^ of order 8 e {p3, p4}, with p, q,pri,,p4,v > 0 and 0 < rq,a < 1.

The structure of the paper is as follows. In Section 2, we present some useful definitions, lemmas and theorems used throughout our work: the Riemann-Liouville fractional derivative of a function with respect to another function, fixed point Theorems due to Banach, Krasnoselskii and Lerav-Sehauder, Section 3 contains result of the existence and uniqueness with an illustrative example. We finish our paper by Section 4 where we study two problems on existence using Krasnoselskii fixed point theorem and Lerav-Schauder nonlinear alternative,

2. Basic notions of fractional calculus

Before starting the proofs of our main results, we should remind the notion of fractional derivatives of a function with respect to another function as well as its essential properties. To this end, in the following and throughout this section, q > 0 is a real constant number, O = [a, b] is a finite or infinite interval, x an integrable function and ^ e Cn(0) is an increasing function such that ^'(t) = 0 for each t eO.

The ^-Riemann-Liouville fractional integral of order q of the function x is defined as

t

X^x(t) = / ¥'(s)(¥(t) - ^(s))e-1x(s) ds,

r(e)

and the ^-Riemann-Liouville fractional derivat ive of order q of the funct ion x is defined as

V^'(t) dt,

1 /1 d n n

t

(^m dt) 7 !'(s)(^(t) - *(s)r0-1x(s) ds

r(n - g) w'(t) dt

a

where n = + 1. In particular, if we choose !(t) = t, !(t) = lnt, !(t) = tu, we find respectively the well-known fractional operators of Riemann-Liouville, Hadamard and Erdelvi-Kober type.

The semigroup property holds for fractional integrals, in other words, for p,u > 0, we have

Z^+*x(t) = ^x(t). Definition 2.1. [22] Lei ! e Cn[a, b] such that !'(t) = 0 for all t e [a, b]. Then we define

ACn;*[a, b] = j 9 : [a, b] ^ R, 0[n-1] = ^ d, 0[n-1] e AC [a, b] J.

Lemma 2.1. [22] Let q > 0 and v > 0. If u(s) = [^(s) - !(a)]^-1, then

(v0Ms))(t) = rr^ [!(t) - *(a)]"- 0-1, (2.1)

and

(lOMs)) (t) = [!(t) - *(a)]0+^ (2-2)

r(^ + e)

.U a particular case of (2.1) and (2.2) we have respectively the following expressions

r(*)

r(^ - q) and

(z0°fsa(v-1)^(t) = v ' ,t

Lemma 2.2. [20] Let q > n with n e N. Then

o;t<T ^-1)^ (t) = r(^) fT(e+i'-1) r(^ + Q)

( _L d) V+* $(t) = j°;n' *$(t). V^'(t) dv

Lemma 2.3. [20] Let q> v, n - 1 < u < n, n e N. Then In particular,

^+%0+*$(t) = $(t).

Lemma 2.4. [20] Let q > 0, n = [q] + 1, $ E L[a, b] andX^+*$ E ACn* [a, b], Then / n n Tj—e; *$(a)

(zaQ+*^+*) $(t) = $(t) - ^ ;(-q;- *+a) (*(t) - *(a))Q—.

In the special ease 0 < q < 1 we have

q-l—Q", *$( )

(iff$(t) = $(t) - Xa+r(Q$(a) (¥(t) - ¥(a))Q—l. Lemma 2.5. [20] Let q > 0 and E ACn*[a, b] n Ll[a, b], then

T%*VS++*$(t) = $(t) + ki (tf(t) - tf(a))Q—1 + fc2(^(t) - tf(a))Q—2 + ■ ■ ■ + fcn(^(t) - *(a))Q—n,

where k]_,... ,kn E R an d n = [q] + 1.

Theorem 2.1. [16}(Banach fixed point theorem). Let (E,d) be a complete metric .space and T : E ^ E be a contraction mapping. Then T has unique fixed point in E.

Theorem 2.2. [16] (Krasnoselskii fixed point theorem). Let M be a nonempty, bounded, closed and convex subset of a Banach space E. Let A and B be two operators such, that:

• Ax + By E M whenever x,y E M.

• A is compact and continuous. •B

Then there exists z E M such that z = Az + Bz.

Theorem 2.3. [16] (Leray-Schauder nonlinear alternative). Let E be a Banach space, C a closed, convex subset of E, U an open subset of C an d 0 eU.

Suppose that T : U ^ C is a continuous, compact 'map (that is, T(U) is a relatively compact subset ofC). Then either:

• T has a fixed point in U, or

• There are ax E dU (the boundary ofUin C) and X E (0,1) wit h XT (x) = x. Lemma 2.6. Let

0 < pl, p2 < 1, pl + p2 > 1, p, q, p3, p4,v> 0, 0 <rj,a< 1

and f, $ j : Ox R ^ R, j = 1, 2, are three continuous functions. Then the fractional boundary value problem

(K+* + v) u(t) = f(t, u(t)), t E O = [0,1],

u(0) = 0, u(1) = u(r])) + ql0+ $2 (&, u(a))

is equivalent to the following integral equation

t

U(t) = - r^y J Ф'(8) ^(t) - Ф(в))Р2-1и(8) ds

0

t

1 I 4 /.4 T / ~.\P1+P2-1,

+ Ф'(s)^(t) - *(s)r+P2-1f(s, U(s)) ds

0

- & (t) ( - Ф'Ф (Ф(1) - Ф(^Г-1u(s) ds

20

1

+ r(pi + ^ J Ф'(s) (Ф(1) - Ф^)Г+Р2-1f(s, u(s)) ds 0

V

p Лт,/ ллЛт,^ шлл^з-1

(2.4)

Г(рз) J 0

a

q f

Г(Р4) J

0

Ф'^) (Ф(^) - Ф^))р3 s, u(s)) ds

Ф'(s) (Ф(а) - Ф^))р4 1Ф2 (s, u(s)) ds^ ,

where

= / Ф(t) - Ф(0) ç*(t) = Л Ф(1) - Ф(0^ •

Доказательство. From (2.3) and by applying the integral operator Z0+'* to both sides we get

Z*'*^'* + ^)u(t) = Z+'*f(t, u(t)). (2.5)

Then by using Lemma 2.5 we see that

Eft'*u(t) = -*u(t) + Z+'*f(t, u(t)) - сх(Ф(4) - Ф(0))Р1-1. (2.6)

Applying the Ф-Riemann-Liouville fractional operator Z0+'* to both sides of (2.6), we find:

Z0i'*^i'*u(t) = -<i'*u(t) + Z^|+^2'*f(t, u(t)) - - Ф(0))л-1.

In view of Lemma 2.5, the last equation implies

u(t) = - <i'*u(t) + Z^2'*f(t, u(t)) - C1 г(Г(+) ) (Ф(t) - Ф(0))Р1+Р2-1

0 0 Г(Р1 + p2)V (2.7)

- C2^(t) - Ф(0))Р2-1. From the first boundary condition, we get c2 = 0 and then (2.7) becomes

u(t) = -^2'*u(t) + Z0'1+P2'*f(t, u(t)) - C1 ) (Ф(t) - Ф(0))Р1+Р2-1. (2.8)

Employing then the second boundary condition, we have on the one hand

u(1) = - <+'*u(t)|t=i + u(t))|t=i - *^) (*(1) - ¥(0))" 1

i

— 1

tf'(s)(tf(1) - u(s) ds

r(p2) J 0

1

+ r(pi + p2)/ ^'(s)(^(1) - ^(s))Pl+P2-1f(s, u(s)) ds 12

- * FiTrb) (*(1) - *(0)»'1+'1-1'

and on the other hand

u(1) =p2£'*$1(V, u(v)) + *C'*$2 (*, u(a))

r(ps)

V

^ -, , T ' 1t ^s, u(s)

p ' ¥'(s)(*( /7) - ^(s))P3-1 $1 (s,u(s)) ds

0

a

+ r(^) J ^(s)^) - s, u(s)) ds

Using then expressions (2,9), (2,10), after some computations we get

1

r(p 1 + P2) ( v

1

T{ - ft) I ^'(s) (^(1) - *(s)r-1u(s) ds

r( p 0(^(1) - ^(0))P1+P2-1 V r(P2) 1

+ r(p/+pJ *'(s)(¥(1) - ^(s))Pl+P2-1 f(s, u(s)) ds 12

V

P f .T.I ! \

r(ps) q

r(p4)

tf'(s) (tf(rç) - ¥(s))P3 s, u(s)) ds

0

a

J ¥'(s)(¥(a) - ¥(s))P4—1$2 (s, u(s)) d^ .

0

i

u( t)

u(t) = - <+'%(t) + Xf++'2'*f(t, u(t)) - É*(t)

- u(1)

+ 2ft+"'*f(1, u(1)) - pZ£'*$1 (V, u(v)) - ql?f$2{a, u(a))

(2.9)

(2.10)

(2.11)

Applying the ^-Riemann-Liouville fractional operator T^* to both sides of (2,11) and using Lemmas 2,1 and 2,3, after some manipulations we get

far + „)u(t) = X--f(t, U(t)) - r(P1 + P2)(!(t) ^^+(0))r-1 ( - ^(1)

V 0+ ) () 0+ ()) (*(1)-*(0))^2-1 V 0+ () (212)

+ X£+^*f(1, u(1)) - (V, ufa)) - (a, u(a))) .

Applying the fractional derivative T^* to both sides of (2,12), due to the property

£>£,*(!(t) - !(0))p1-1 = 0,

we obtain

+ u) u(t) = f( t, u(t)). To check boundary conditions, it is easy to confirm by (2,11) that

u(0) = 0, and u(1) = pX^*^, ufa)) + qX^*^*, u(a)). Therefore, u(t) is a solution of the problem (1,1) and this completes the proof, □

In what follows, we introduce some new notations based on Lemma 2,6, In addition, we consider

the Banach space C = C([0,1], R) equipped with the norm ||u|| = max |u(t)| and we define an

te[o ,1]

operator M : C ^ C

t

(^u)(t) = - J ¥'(s)(*(t) - !(s))P2-1u(s) ds

2

t

+ r(pi + ^ J *'(s)(*(t) - *(s))^2-1f(s, u(s)) ds 12

1

- &(t) ( - r^y J !'(s) (¥(1) - *(s)r-1u(s) ds

2

1

+ r(pi + P2 J *'(s)(*(1) - !(s))P1+P2-1f(s, u(s)) ds 12

V

(2.13)

r(p 3)

!'(s) (!(/?) - !(s))p3 s, u(s)) ds

0

a

r(p4) J 0

q 1 !'(s) (!(a) - !(s))p4-1$2 (s, u(s)) d^ .

The fixed point equation

Nu = u, u e C,

is equivalent to integral equation (2.4), and the continuity of the functions f, $1, $2 ensures that for the operator M.

We are in position to formulate an existence and uniqueness theorem.

3. Unique solvability

Theorem 3.1. Suppose that the following assertions hold : There exists a real соnstant ^f > 0 such that

|f(t,u) - f(t, v)| ^ ^f|u - v|, t g [0,1], u, v g R.

(%2) : There exist two Ф-Riemann-Liouville integrable functions : [0,1] ^ R+ such that

^¿(t, u) - Ф^, v)| ^w¿(t)|u - v|, ¿g{1, 2}, t g [0,1], u, v g R.

(%з) : A real constant 7

7 _2, (ф(1) - Ф(0))- + 2M(*g) - ф(°)Г+" + ы+ы 7 _ г^Л + г(рi + Р2 + 1) + px°+wi(??) + qx°+Ш2(а)

satisfies 0 < 7 < 1.

Then fractional boundary value problem (1,1) possesses a unique solution.

Доказательство. In view of the above, we know that the solvability of fractional boundary value problem (1,1) is equivalent to the solvability of integral equation (2,4), Hence, it is sufficient to show that integral equation (2,4) has only one fixed point.

First, since Ф is ад increasing function and p1 + p2 > 1, we have 0 ^ ^(t) ^ 1. Therefore, for each t g [0,1] we can write

t

|(^u)(t) - (^v)(t)| ^г^/Ф'(в)(ФСе) - Ф(8))Р2-1|и(в) - v(s)|ds

2°

t

i

+ Г(р1 + p2)] - ФЮ)Pl+P2-1|f(s,u(s)) - f(s, v(s))|ds

0

1

+ J Ф/(s)(Ф(i) - Фф)"3-1^) - v(s)|ds

2o

1

+ Г(Р1 + ^ J Ф/(s)(Ф(i) - Ф^))"^-1^, u(s)) - f(s, v(s))|ds 0

v

+ ГЩ / *(s)№ - Ф^))"3-1^,u(s)) - Фl(s,v(s))|ds

0

а

+ Гк) I Ф^^ - Ф^Г^Ф (s, u(s)) - Ф2 (s, v(s))|ds

< Hlu - vll^(t) - Ф(0))Р2 + .Mf llu - v| (Ф(t) - Ф(0))

P1+P 2

Г(Р2 + i) Г(р 1 + P2 + i)

+ Hlu - v||^(1) - Ф(0))Р2 + lu - vl (Ф(!) - Ф(0))

p 1+P2

Г(Р2 + i) Г(р 1 + P2 + i)

+ plu - vllXo+'^l^) + qlu - vlX^^M

< /2 „(фЩ - ф(0)Г + 2^f (фЩ - ф(0)Г+Р2 + Ф Ч ГЛ! + Г(р 1 + Р2 + i) + pX+ "l(??)

+ qZ£+'%(a)) ||u - v||.

Thus,

iNu -^Vll ^ 7|u - v ||, 0 <>y< 1

(3.1)

and ^ is a contraction. Therefore, by applying the Banach principle, we conclude that M has only one fixed point and this implies the existence of a unique solution for the FBVP (1.1). The proof is complete. □

Example 3.1. Consider a fractional boundary value problem of the following form

t e [0, 1],

4/ 1U

u(0) = 0,

1 t2 / 2 t2 1 ^2+ (^03+ +4

u(1)= l2s(1 - s2y(l + 10u(s)es)ds+ I - s2)4(l + ies sin (u(s))) ds

1

1U

1

(3.2)

In this case we have

*(t) = t2,

and

1

"=4,

Pi

P = r(4),

1 2,

2=

3'

q = r(5),

p3 = 4, p4 = 5,

2

" =4,

f(t, u(t)) = t + — sin(2u(t)), 1U

$ (t, u(t)) = 1 + 10u(t) et, $2(t, u(t)) = 1 + 1et sin (u(t)).

2

4

2

1

1

Then

Mf = ^, ^i(t) = 10et, ^2(t) = 1et. o 2

Therefore, by simple computations we get

2u(¥(1) - tt(O))'2 + 2M(¥(1) - tt(0))»+'2 + + (a)

7= ro^TT) + r( p i + P2 + 1) + pXo+ Ui(v) + qXo+ ^2(a)

«0.9031 < 1.

3.1

uniquely solvable.

4. Existence results

In order to apply the Krasnoselskii fixed point theorem, it is useful to decompose the operator Mas N = M\_ + M2, where,

t

(Mu)(t) = + y (s)(¥(t) - ¥(s))^2-if(s, u(s)) ds i2

Г(Р 1 + P2) J

0

1

& (t) f Ф'(8)(Ф(1) - Фф)'1*"2-1^ u(s)) ds,

and

t

(^2U)(t) = - Г^у / Ф/(s)(Ф(t) - Ф^))^^) ds 20

+ J Ф/(s) (Ф(1) - Ф^)Г-1 u(s) ds

20

(г( Р2^Ф'

\ n

V

Ф/(s)(Ф(г/) - Фф)'3-1Ф1^, u(s)) ds

Г(рз)

0

a

q Лт,/^Лт,^ шл,^-1;

+ 1q3J *'(s)(*(*) - *(s)) s, u(s)) dsj .

40

To simplify the futher calculations, we use the following parameters:

_ 2(^(1) - !(0))p1+p2 _ 2 v(!(1) - !(0))p2

= -tT7-\-, ^2

r(p 1+ P2 + 1) ' 2 Г(р2 + 1)

o_ р(Ф(д) - Ф(0)Г 0_ я(Ф(а) - Ф(0))Р4

3 = Г(рз + 1) ' 04 = Г(р4 + 1) •

The main result in this section is the following theorem.

Theorem 4.1. Assume that Conditions (^1) and (%2) hold. In addition, let there exist three functions Y,^- E С ([0,1], R+), j E {1, 2} such, that (^4) : |f(t, u)| ^ Y(t) /or a lit E 0, u E R; CH5) : |$»(t, u)| ^ <^(t), i E {1, 2} /or a lit GO, u G R. Then boundary value problem (1,1) is solvable if

O2 + 0з|М| + 04M < 1. (4.1)

Доказательство. We begin with considering the following nonempty closed convex ball

Be = {u E C : ||u|| ^ £>},

where q satisfies the inequality

0i||Y|| + 0з|Ы| +O4Ы|

1 - ^2

with ||Y|| = supte0 T(t^d ||' || = supte0 '(t).

First step: We are going to show that Ar1 u + A2V e B0 for every u, v e B0. Let u e B0, then we can write

l(Mu)(t)| 1

t

^ У Ф/(s)(Ф(t) - Ф^))^2-1^, u(s)) ds

Г(Р1 +P2)

0

1

-&(t)J Ф/(s)(Ф(1) - Фф)^2-1^,u(s)) ds)

T(Pi +P2)

0

i

^^-r(J ¥'(s)(¥(t) - ¥(s))^2-i|f(s, u(s))| ds

r(Pi

Since ¥ is an increasing function, we have

+ &(t)J ¥'(s)(¥(1) - ¥(s))^2-i|f(s, u(s))|ds) 0

- ¥(0))P1+P2 + &(t)(tf(1) - ¥(0))P1+P2).

IIMu|| ^r(pi + 1) sup ((¥(t) - S^))^ + &(t)(¥(1) - ¥(0))P1+P2)

2||T||

(^(1) - tf(0))= Qi

T(p 1

In the same way, for v E Be we get

t

\P2-i

Then

IM v)(t)|

¥'(s)(tf(t) - ¥(s))p2-iv(s) ds

0

r(p2) - ^ ^ v(

+ **(tfe/ ¥'(s)(^(1) - ¥(s))'2-iv(s) ds 2

+ ¥/(s)(W(77) - ¥(s))P3-i$1 (s, v(s)) ds

q

r(p 4)

0

+ f ^(s)(¥(a) - ¥(s))P4-i$2(s, v(s)) ds^j

0 t

^¥'(s)(^(t) - ¥(s))P2-i|v(s)|ds

0

i

+ &(t)J ¥'(s)(¥(1) - ¥(s))'2-i|v(s)|ds)

0

v

+ *'(s)(*fa) - ¥(s))'3-i l$i(s, v(s))| ds

0

a

+ - *(s)r-i|$2(s, v(s))|ds) .

IIMvll ^(^(1) - ¥(0))'2 + fp^- ¥(0))'3

+ ^3^1 II TOfe!

(4.2)

+ (¥(-) - ¥(0))p4 (43)

r(P4 1

t

V

Bv combining (4,2) and (4,3) we get

IIMu + Mv|| ^||Mu|| + ||Mv||

^2 Q + Qi||T|| + QaI^iI + 0^" ^ e,

which shows that Ariu + A^v E Be for everv u, v E Be.

Second step: We are going to prove that M2 is a contractio n on Be, For eve rv u, v E Be we

have

KMuXt) - (Mv)(t)|

t

V i.T,/^ iTi / \\P2-i,

¥'(s)(tf(t) - ¥(s))'2-i(u(s) - v(s)) ds

r(p2) J 0

i

+ ¥'(s)(^(1) - ¥(s))'2-i(u(s) - v(s)) ds

2

V

+ PW / - ¥(s))P3-i($i(s, u(s)) - $i(s, v(s))) ds

0

a

+ rw / - *(s)r-i($2(s, u(s)) - $2(s, v(s))) ds

4

t

^ ftj ¥'(s)(^(t) - *(s)r1 |u(s) - v(s)| ds

2

1

+ W)I ¥'(s)(^(1) - ^(s))'2-i|u(s) - v(s)|ds

2

V

+ Pr^^ y ¥'(s)(¥(^) - ¥(s))'3-i|$i(s,u(s)) - $i(s, v(s))|ds

0

a

+ q^ i ¥'(s)(¥(^) - ¥(s))'4-i | $2 (s, u(s)) - $2 (s, v(s))| ds,

0

^ (^(t) - ^r+^"u - v" (^(1) - ^(0))'2

+ pe - v" ow - ¥(0))'3

+ qe *(t)|H|||u- v| - )r.

r( P4 '

Then

II^u -^v|| ^—^y (*(1) - *(0))M + fp^ (*(») - *(0))'3

+ fqrTi) - - v||

= (^2 T^MI +Q4 ||^2 ||)||u - v||. Hence, it follows from condition (4,1) that A2 is a contraction.

Third step: We are going to show that Ar1 is compact and continuous,

i) It follows from the definition of the operator M that the continuity of f implies that of

M.

ii) The uniform boundedness of the operator Ar1 on B0 is due to expression (4,2), where we

have shown that for any u G B0

||Mu|| ^ fi1|T|.

iii) In view of (%4), for all u e B0 and for ea ch t1, t2 e O such th at t1 < t2 we have:

|(Mu)(t2) - (Mu)(t1)|

1

r(P1 +P2)

!'(s^!(t2) - V(s)) W+P2-1f(5, u(s)) ds

0

t1

-J *'(s)(*(t1) - !(s))pi+p2-1f(s, u(s)) ds 0

1

- e*(t2^ *'(s)(*(1) - !(s))Pl+P2-1f(s, u(s)) ds 0

1

+ £*(t1) / ¥'(s)(¥(1) - !(s))pi+p2-1f(s, u(s)) ds

1

r(P1 +P2)

!'(s^!(t2) - V(s))Pl+P2-1f(s, u(s)) ds

t2

+ / ^(s)^) - tf(s))Pl+P2-1f(s, u(s)) ds

t1

-J ^(sK^(t1) - !(s))pi+p2-1f(s, u(s)) ds 0

1

- Vis) Ip1+p2-1f(s, u(s)) ds

(t2) - e*(t1^ v'(s)(V(1) - v(s))

0

V'(s)((V(i2) - V(s)r+'2-1 - (V(t1) - V(s))^2-1)f(s, u(s)) ds

1

r(P1 +P2)

t1) - V(s

t2

+ J V,(s)(V(i2) - V(s)) t1

1

11)J I V'(s

0

s) )Pl+P2-1f(s, u(s)) ds

(e*(t2) -mm) / v'(s)(V(1) - v(s))pi+№-1f(s,u(s))ds

(/W)((

\ n

*F^W /*>)(№)-+02-1

- (Ф(11) - *(s))" +и-1) |f(s, u(s))| ds

t2

ift +/02-1 I

+ J Ф'^Ф^) - Фф)0l+°2-1|f(s, u(s))| ds t

+ (ы<й) - Ы^) I Ф/(s)(Ф(1) - Фф)01 +02-1|f(s, u(s))|dsj

;r( +Y 11 T1) ((ф(*2 ) - Ф(0))"+02 - (Ф(М - Ф(0))"+02

1(P1 + Р2 + 1) v

+ (ы<*) - ЫМ)(Ф(1) - Ф(0))Л+02

u

as t2 ^ t1. Hence, Ar1 is equicontinuous. By Arzela-Ascoli theorem this yields the compactness of the operator Ar1. As a consequence of Krasnoselskii fixed point theorem, problem (1,1) has at least one solution. The proof is complete, □

Theorem 4.2. Assume that there exist three functions x,XbX2 e C ([0,1], R+) and three non-decreasing functions : R+ ^ R+ such, that

(«a) («7)

(«s)

|f(t, u)| ^x(t)- (||u||), for al 11 GO, u E R.

|^i(t,u)| ^ ||u||), г E {1, 2} /or all t eO, u E R.

There exists a real constant ш > 0 satisfying

Ш(1 - 02)

Then fractional boundary value problem (1,1) possesses a solution.

> 1.

Доказательство. First step: We are going to show that M maps bounded sets into bounded sets in C, For a positive number r we define the bounded sets Br of C as follows:

Br = {u E C : ||u|| ^ r}.

For u E Br, we have

|(Яu)(t)|

t

,т .^A^-1

Г(Р2)

1

Ф'ф(Ф^) - Фф)02 u(s) ds

0

t

+ r(pi + ^) / Ф^)(Ф№ - ФЮ)01+02-1f(s, u(s)) ds

0

1

- &(t) ( - ^ / Ф/(s) (Ф(1) - Фф)02-^) ds

77

Г(р 2)

0 1

+ Г(Р1 + Р2)/ Ф/(s)(Ф(1) - Ф^))01 +02-1f(s, u(s)) ds 120

Then

¥'(s)(¥(v) - ¥(s))'3 1 $i(s, u(s)) ds ¥'(s)(¥(a) - ¥(s))'4-i$2(s, u(s)) ds^j

r(p4) J

0

t

^ ft) I ¥'(s)(^(t) - ¥(s))"-l|u(s)| ds

2

t

+ r(pi + pJ ¥'(s)(¥(t) - ¥(s))'1+'2-i|f(s, u(s))| ds i2

1

+ &(t) ( - r^J ¥'(s) (¥(1) - ¥(s))'2-i|u(s)| ds 2

1

+ r(pi + pJ ¥'(s)(¥(1) - ¥(s))'1+'2-i|f(s, u(s))| ds i2

V

+ r(b / ¥'(s) (¥(»/) - ¥(s)r-i|$i (s, u(s))| ds

0

a

r(p4)

+ I ¥'(s)(¥(a) - ¥(s))'4-i|$2(s,u(s))| ds^)

^f^(¥(t) - ¥(0))'2 + r('pX|I:(iu]^ (¥(t) - ¥(0))'1+'2

u" ^(¥(1) - ¥(0))'2 + "u"^ (¥(1) - ¥(0))^

+ (¥(77) - ¥(0))'3 + «M.(¥(a) - ¥(0))'4 ^

^2 + "x", (r)Qi + ||Xi",l(r)Q3 + "X2",2(r)^4.

Second step: We are going to show that M maps bounded sets into equieontinuous sets of C Based on Assumptions (%6) - (%7), f°r ti, t2 E O with ti < t2 and each u E Br we can write

u)(t2) - (^u)(ti)|

¥'(s^¥(t2) - ¥(s))'2-iu(s) ds

r(p 2)

v

- /¥'(s)(¥(t1 ) - *(s))P2-1u(s) dsj

0

t2

(12

J ¥'(s)(¥(t2) - ¥(s))w +P2-1f(s, u(s)) ds 0

tl

-J ¥'(s^¥(ti ) - *(s))+P2-1 f(s, u(s)) dsj

0 '

1

- (e^(t2) - ^J ¥'(s)(¥(1) - ¥(s))P2-1u(s) ds

2

t1

P2-1 ^^^^^^ ^^2-1^

r(p 2)

0 t2

(/ *

\ n

*(s)((tf(t*) - *(s))P2-1 - (*(t1) - *(s))P2-1) u(s) ds

^*'(s)(*(t2) - *(s))P2-1u(s) dsj t1

t1

(/ *<

\ n

- (*(t1) - *(s))+P2-1)f(s, u(s)) ds

t2

+P2-1(

+ j *'(s)(*(t2) - *(s))"^-1f(s, u(s)) dsj

i

- (£^2) - &(ti)) ¥'(s)(¥(1) - ¥(s))'2-iu(s) ds

2

<((¥(t2) - ¥(0))'2 - (¥(ti) - ¥(0))'2) + r("X^^(rT1) ((¥(t2) - ¥(0))- +'2 - (¥(ti) - ¥(0)r +'2)

i(Pi + P2 + 1)

jyr

+ (t2) - & (ti)) ^ (¥(1) - ¥(0))'2.

We observe that as ti —y t2, u

goes to zero uniformly. Therefore, the operator M : C — C is equieontinuous and thus the operator M is completely continuous.

We are going to confirm that the set of all solutions to the equation \Mu = u is bounded for A E (0,1).

By computations similar to ones used in the first step we get

||u|| < "u"^2 + IIx",("u")Qi + ||xi",i("u")Q3 + "X2",2("u")Q4,

which leads to

||u||(1 - ^2) < 1.

(NR + |X1|^1(|u|)n3 + M^N)^

According to Assumption (%s), there exists a real constant u > 0 such that ||u|| = u and

_U(1 - »2)_ > 1

||X|| ~ (u|)fi1 + M^u)^ + ||X2|^2(u)Q4 .

We introduce the set

U = {u e C : ||u|| <M},

and note that M : U ^ C is continuous and completely continuous. Then the choice of u implies that there is no u e SW such that A^(u) = u for some A e (0,1). Then by Lerav-Sehauder nonlinear alternative we conclude that M has a fixed point u e U which corresponds to a solution of fractional boundary value problem (1.1). The proof is complete. □

5. Hyers-Ulam stability outcomes

Fractional differential equations play a very important role in mathematical analysis and especially in the modeling of physical phenomenons and these have been widely studied from different sides. Among these, the stability analysis in the Hyers-Ulam sense is a very important aspect which attracted the attention of many authors [18], [35], [28]. Based on the definition of Hyers-Ulam stability, then this notion was modified into more general types [4], [5]. In this section, we will adapt some sufficient conditions to obtain stability results of the Hyers-Ulam type for our main problem.

Definition 5.1. [18], [28] Let us consider a Banach space S and an operator K : S ^ S. The operator equation

Ku = u,

is said to be Hyers- Ulam stable if the inequality

|u(t) - Ku(t) | ^ e,

which holds for a 111 eO = [0,1], implies that there exists a constant > 0 such that for each, u e C(0, R) satisfying (5,1) one can find a unique solution u e C(0, R) of operator equation (5.1) provided that for each, t e O we have

|u(t) - u(t) | ^ ^Ke.

Definition 5.2. [18], [28] Consider the operator N : C ^ C. We say that the operator equation

u(t) = Nu(t), (5.2)

is Hyers-Ulam stable if for the inequality

|u(t) -^u(t) | ^ e, t eO, (5.3)

we can find a constant ^ such that for each, u satisfying (5,2) there exists a unique solution u of the operator equation (5,2) provided that

|u(t) - u(t) | ^ ^e

for each, t eO

Theorem 5.1. Assume that $ e C(0, R) is a solution of the inequality (5,3) satisfying the conditions

(«) |$(t)| for all t eO;

(««) P^l'* (£>£'* + i/)u(t) - f(t, u(t)) + $(t) = 0 for a 111 e O.

Then

| $(t) -^$(t)U le, t eO,

where

2 / T , T N pi +P2

l = г( + ■1)(ф(1) - Ф(°)) Г(р 1 + p2 + 1)

and N is the operator given by (2,13).

Доказательство. According to Condition (гг), for each t G О we have

Eft'*^'* + ")*(t) - fM(t)) + Ф^) = 0,

0(0) = 0, 0(1) = pZ0+'* Ф^ЭД) + qZ0+'* Ф2МЫ).

In view of Lemma 2,6, the solution of fractional boundary value problem (5,4) can be expressed as

t

0(t) = - r^y / Ф'(в)(Ф№ - ф(в))о2-10(в) ds 2

t

+ Г(Р1 + Р2)/ ¥'(s)(¥(t) - Ф^))01 +O2-1f(s,0(s)) ds 12

t

+ г(р1 + р2)/ ¥'(s)(¥(t) - Ф^))01 +02-1ф(5) ds 12

- &(t) ( - г^у J Ф'(в) (Ф(1) - Ф(в)Г-V(s) ds 0

1

+ г(р1 + р2)/ Ф'(s)(Ф(1) - Ф^)Г +p2-1f(s,0(s)) ds 0

1

+ г(р1 + р^/ Ф'(^(Ф(1) - Ф^)Г +p2-10(s) ds 0

V

p Лт,/ллЛт,^ .т.^рз-1

Ф'^) (Ф(^) - Фф)p3 s, 0(s)) ds

Г(Р4) J

0

Since t G О, the above relations imply

Г(Р з)

о

a

0 q

e -— I

V Г(Р2)У

x 0

t

0(t)-( — rv~—V Ф'(s)(Ф(t) - Ф(^Г V(s) ds

+ г(р1 + р^/ Ф'(s)(Ф(t) - Фф)"1 +P2-1f(s,0(s)) ds 0

- &(t) ( - г^) J ^(s) (Ф(1) - Ф(з))Р2-1ад ds

1

+ Г(р11+р2)/ ф/(8)(ф(х) - Ф^Г ds

0

V

P Лт,/! \ >тг / N\P3-1

Ф'ф (Ф(^) - Фф)03 S, tf (s)) ds

Г(рэ)

0

a \ r^J Ф'(б) (Ф(<Г) - Ф(Б))Р4-4(M(s)) ds) j

0 '

t

1 ' Ф'(б) (Фф - Ф(б))л+P2-10(s) ds

Г(Р 1 + P2)

0

1

^(t) î Ф/(s)(Ф(1) - Фф)01+P2-V(s) ds

Г(Р1 +P2)

0

t

^^^-r- i Ф/(s)(Ф(t) - Фф)01 +p2-1|0(s)|ds

Г(Р1 +P2)

0

1

^(t) î Ф/(s)(Ф(1) - Фф)01+02-1 1 0(s) 1 ds

Г(Р1 + P2)

0

^ Г( +2 +1) (Ф(1) - Ф(0)Гe,

Г(Р1 +P2 + 1) which can also be written

| $(t) l.e, t GO.

This completes the proof, □

Theorem 5.2. Let Assumptions ) - (%3) hold. Then the solution of fractional boundary value problem (1,1) is Hyers-Ulam stable.

Доказательство. Let $ G C(0, R) be an arbitrary solution of the following inequality

| + f)u(t) - f(t, u(t)) | ^ e, t GO,

and let г? gC(C, R) be the unique solution of the problem

fe* + ")%) = f(t, $(t)), t GO,

V J T T (5-5)

$(0) = 0, $(1) = рХ0^'Ф Ф^rç,^)) + qX0+ Ф2МИ).

Thanks to Lemma 2,6, the solution of fractional boundary value problem (1,1) can be expressed

as

t

£(t) = - ^ / *'(s)(¥(t) - *(s)r-10(s) ds 0

t

+ r(piX+^/ *'(s)(*(t) - *(s))Pl +P2-1f(M(s)) ds 0

- &(t) ( - ^ J ¥'(s) (^(i) - ^(s))P2-10(s) ds

^ 0 1

+ r(pi + ^ J *'(s)(*(1) - *(s))"+P2-1f(s,0(s)) ds 0

V

p /\t,Z^Aw^ .w,,^-1

(5.6)

T(P 3)

tf'(s) (tf(rç) - tf(s))P3 s, tf(s)) ds

0

a

q 1 tf'(s) - ^(s))p4-1$2 (s, 0(s)) ds j .

r(p4) J

0

Based on (5.6), we can write

|0(t) -0(t)| =|0(t) -^0(t)|

= |0(t) -A^(t) + ^0(t) - ^0(t)| ^|0(t) - A^(t)| + |A^(t) - M?(t)|. Then by (3.1) and Theorem 5.1 we arrive at

^ It +

which gives immediatelv

I

II II 1 -7

Therefore, the solution of fractional boundary value problem (1.1) is Hvers-Ulam stablen. The proof is complete. □

Acknowledgements The authors thank the referee for careful reading of the manuscript.

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Hacen Serrai,

Laboratory of Applied Mathematics, Kasdi Merbah University, B. P. 511. 30000, Ouargla, Algeria

E-mail: hacenserrai31@gmail.com

Brahim Tellab,

Laboratory of Applied Mathematics,

Kasdi Merbah University,

B. P. 511. 30000,

Ouargla, Algeria

E-mail: brahimtel@yahoo.fr

E-mail: hacenserrai31@gmail.com

Khaled Zennir, Department of Mathematics, College of Sciences and Arts, Qassim University, Al-Rass, Saudi Arabia

E-mail: k.zennir@qu.edu.sa