Fractional integrodifferential equations with nonlocal conditions and generalized Hilfer fractional derivative Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 11. № 4 (2019). С. 150-169.

FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND GENERALIZED HILFER

FRACTIONAL DERIVATIVE

H.A. WAHASH, M.S. ABDO, S.K. PANCHAL

Abstract. We study some basic properties of the qualitative theory such as existence, uniqueness, and stability of solutions to the first-order of weighted Cauchy-type problem for nonlinear fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative. The fractional integral and derivative of different orders are involved in the given problem and the classical integral is involved in nonlinear terms.

We establish the equivalence between the weighted Cauchv-type problem and its mixed type integral equation by employing various tools and properties of fractional calculus in weighted spaces of continuous functions. The Krasnoselskii’s fixed point theorem and the Banach fixed point theorem are used to obtain the existence and uniqueness of solutions of a given problem, and also the results of nonlinear analysis such as Arzila-Ascoli theorem and some special functions like Gamma function, Beta function, and Mittag-Leffler function serves as tools in our analysis. Further, the generalized Gronwall inequality is used to obtain the Ulam-Hvers, generalized Ulam-Hvers, Ulam-Hvers-Rassias, and generalized Ulam-Hvers-Rassias stability of solutions of the weighted Cauchv-type problem. In the end, we provide two examples demonstrating our main results.

Keywords: fractional integro-differential equations, nonlocal conditions ,ф—Hilfer fractional derivative, existence and Ulam-Hvers stability, fixed point theorem.

Mathematics Subject Classification: 34K37, 26A33, 34A12, 47H10.

1. Introduction

A fractional calculus is an extension of the ordinary calculus to non-integer orders. The fractional calculus is more than three centuries old, but it attracted much attention in recent decades due to it is a solid and growing employing both in the theoretical and applied concepts, see [15], [17], [19], [25]. The fractional derivatives were developed in the past epoch by Riemann-Liouville, Grunwald-Letnikov, Riesz, Erdlvi-Kober, Caputo, Hadamard, Hilfer and others. In the past years, fractional differential equations appeared as rich and nice field to be studied due to their applications to the physical and life sciences and to engineering as is witnessed by blossoming literature. Many researchers worked with the fractional derivatives and the results can be found in [1], [4], [5], [8]—[13], [18], [23], [24], see also the references therein.

The properties of fractional integrals and fractional derivatives of a function with respect to another function have been introduced by Kilbas with co-authors in [17]. Howover, recently, in [6], Almeida have introduced a fractional differentiation operator, a so-called -©Caputo fractional operator. On the other hand, Hilfer [15] introduced a fractional derivative, which in particular gives the Riemann-Liouville and the Caputo fractional derivative operator. In

Fractional integro-differential equations with nonlocal conditions and generalized Hilfer

FRACTIONAL DERIVATIVE.

©Wahash H.A., Abdo M.S., Panchal S.K. 2019.

Submitted November 11, 2018.

[13], Furati with co-authors considered a nonlinear fractional differential equation involving Hilfer fractional derivative:

u(t) = f (t,u(t)), t > a, 0 < a < 1, 0 ф ft ф 1, (1,1)

u(a+) = ua, 7 = a + ft - aft, (1,2)

where (■) and Z^-7(■) are Hilfer fractional derivative of order a and type ft and Riemann-

Liouville fractional integral of order 1 — 7, respectively, ua Є R, The authors used some fixed point theorems to study the existence and uniqueness of global solutions in the weighted space to problem (1.1)—(1.2). The stability of the solution of a weighted Cauchv-type problem was also analyzed. In [24], Wang and Zhang proved the existence of the solutions to equation (1,1) with the nonlocal condition

m

/J-7u(a+) = ^ Afcu(rk), Tfc Є (a,b], 7 = a + ft(1 — a), (1.3)

fc=1

by using Krasnoselskii and Schauder fixed point theorems, Vivek et al, [23] established the existence, uniqueness and Ulam stability results for an implicit differential equations of Hilfer-tvpe fractional order:

D^f u(t) = f (t,u(t),D^f u(t)), t> 0, 0 <a< 1 , 0 ф ft ф 1,

m

/q1-7u(0+) = ^ Afcu(rk), Tfc Є [0,6], 7 = a + ft(1 — a),

k=1

via Schaefer fixed point theorem and Banach contraction principle.

Lately, Sousa and Oliveira [21] have recently proposed a ft-Hilfer fractional operator and extended the results of few previous works [13], [15]. In [22], Sousa and Oliveira disccused the existence, uniqueness, Ulam-Hyers and Ulam-Hvers-Rassias stabilities of the implicit fractional differential equation involving ft-Hilfer fractional derivative. Very recently, in [20], Sousa with co-authors proposed a generalized Gronwall inequality for a fractional integral with respect to another function ф. They also considered Cauchv-type problem (1.1)—(1.2) involving the ft-Hilfer fractional derivative (■) introduced by Sousa and Oliveira in [21] and they

established results on existence, uniqueness, and continuous dependence.

Motivated by the above works, we prove the existence, uniqueness, and Ulam-Hyers and Ulam-Hvers-Rassias stabilities for a nonlinear fractional integro-differntial equation with nonlocal condition and ft-Hilfer fractional derivatives of the form:

u(i) = ў (t,u(t),xu(t)) , 0 <a< 1, 0 ^ ft Ф 1, t Є (a,b], (1,4)

m

ll+T^u(t) |t=a= Ua + 5^Cfcu(Tfc), Tfc Є (a,b), а Ф 7 = a + ft — aft, (1.5)

fc=1

where ua Є R, D^f’^(■) is the generalized Hilfer fractional derivative introduced by Sousa and de Oliveira in [21], (■) is the generalized fractional integral in the sense of Riemann-

Liouville, for у : D x R ^ R, xu(t) '■= /Q h(t, s,u(s))ds. Here D : = {(t,s) : a ф s Ф t ф b}, f : (a,b] x R x R —у R is appropriate function, rfc, к = 1, 2,... ,m are given points satisfying a < t1 < r2 < ... < тт < b and cfc are real numbers.

This paper is organized as follows. In Section 2 we introduce some notations, basic definitions, and preliminary facts, which will be used in the paper. In Section 3, we list the hypotheses and we also show that problem (1,4)-(1,5) is equivalent a the mixed type integral equation. We also prove the existence and uniqueness of solution to problem (1,4)-(1,5), The Ulam-Hyers and Ulam-Hvers-Rassias stabilities in a weighted space for such equations is discussed in Section 4,

In Section 5 we provide examples demonstrating our main results. Finally, the conclusion is given in the last section.

2. Preliminaries

In this section, we gather some essential facts, definitions, and lemmata concerning fractional calculus and fractional differential equations.

Let J = [a,b], — ж < a < b < to, be a finite interval in R. We denote by C(J, R) and Cn(J, R), n Є Z+ := N U {0}) the spaces of continuous and u-times continuously differentiable functions on J with the norms

П П

II/lie = imiJx |/(t)l, II/İle» = ^11/wllc = ^ max I / W(i)|,

i= 1 i=0

respectively, where C(J, R) = C0(J, R). And LP(J, R) (p ф 1) is the space of measurable functions f : J ^ R with the norm

II/IIlp =^ja ^(t)|P) dt-

We introduce the following weighted spaces of continuous functions:

Ст,ф(J, R) = {f : ^, b] ^ R : (ф(ф — ф(а))1 f (t) Є С^, R)} , 0 ^ 7 < 1

Сфф(J, R) = {f Є Cn-1(J, R); f(n) Є С1]ф(J, R)} , 0 ^ у < 1, n Є N.

Obviously, СТ;Ф(J, R) and С'фф(J, R) are the Banach spaces with the norms

II/llc7;^ = IK^CO — Ф(a))1 f Шс = max І(Ф(і) — Ф(а))1 f(t)l,

and

II/II

Cn /

Т;ф

n— 1

EI

k=0

(fc)|

+ II f (ra)„

c +1 \J и c.

respectively, where С1;ф (J, R) = 0° ;ф(J, R).

A well-known function frequently used in the solution of fractional differential equations is the Mittag-Leffler function given by

EaJ3(z) = V f I ,, Re(a), Re(^) > 0, z Є C,

k=0 i(a + P)

where r(z) = J e—xxz—1dx, z > 0, is the Euler gamma function. Moreover, if a 0

we have

2 and /3 = 1,

where

is the error function.

E і (z) Ф e* (1 + erf (z)) 2 fz 2

erf (z) = —= e * dt

л/р J0

Definition 2.1. [17] Let f be an integrable function defined on (a, b) and ф be an increasing function having a continuous derivative ф' on (a, b) such that ф'(Ф) = 0 for all t Є J and a > 0 is a constant. The left-sided fractional integral of order a of function f with respect to ф is defined by

IZ* f (t) =

r(«)

V''(s)№W — As))" V (s)*.

t

1

(2.1)

In particular, if ф(і) = t, we obtain the known classical Riemann-Liouville fractional integral.

Definition 2.2. [7], [17] Let n — 1 < a < n, and f/ф Є Cn(J, R) be two functions such that ф is increasing and ф'(t) = 0 for all t Є J. The left-sided fractional derivatives of order a of f with respect to 'ф in the sense of Riemann-Liouville and Caputo are given by

D/ S (t)

1 d 'ф'(і) dt

■п—афф

and

С^<а‘фФ £(+\ jn-афф

I

1 d

'ф'(і) dt

f (t),

f (t),

'аФ f (t) = C

respectively, where n = [o] + 1 and [o] denotes the integer part of a real number a. The fractional derivative we deal with is a ^-Hilfer type operator defined as follows.

Definition 2.3. [21] Let n — 1 < a < n Є N, 0 ф ф ф 1 and f/ф Є Cn(J, R) two functions such that 'ф is an increasing and 'ф'(к) = 0 for all t Є J. The left-sided ф-Hilfer fractional derivative of order a and type ф of function f is determined as

7T! (t)

I'

■'(п—а);гф

1 d

j- (1~')(n-a);^ j (j.)

On the other hand, we have

D/* ! (t)

where

т'(п-а);ф П7;У 1a+

'Ф'(t) dt

W/ f (t), 7 = a + 0 (n — a),

(2.2)

1 d

f w = ^ {t) dti

j- (1-')(п-а);ф j (_£)

In particular, The ф-Hilfer fractional derivative of order 0 < a < 1 and type 0 ф ф ф 1 can be written in the following form

Г t

K?;* f «

1

Ф (з)(ф(і) — ффф1 a f (s)ds

r(7 — 7

where у = a + Ф(1 — a), p-a,2(.) are defined by equation (2.1) and

1 d

go* // f (t).

Dlf f (0

Tv* f «)•

ф'(к) dt

Lemma 2.1. [3, 14] Let a > 0 and ф > 0. The following semigroup property holds:

(i) : If f Є L”(J. R) (p > 1), then І/1™f (t) = І/ыf (t). a.e. t Є J.

(ii) : If f Є С-,я(J. R), then /7f (t) = 1$'*f (t). t Є (a,b\, 0 ф 7< 1

(iii) : If f Є C(J. R). then І/Iі/f (t) = І/'"*f (t). t Є J•

As a + Ф > 1, Statement (i) holds at each point in J.

Lemma 2.2. [21] Let 0 < a < 1. 0 ф ф ф 1 and 0 ф / < 1. If f Є L1(J. R) and д'|1-а);^ f is well-defined as an element of Ь1(Т. R), then

j а;ф a+

Moreover, if f Є C1(J. R). then

f (t) = i:

|(1 -Ы);гф -^'(1—(х);гф

a+

a+

7 (7

~\ n

“І П

+

For 0 < a < 1, 0 ^ fi Ф 1 and 7 = a + fi(1 — a) we introduce the weighted spaces

(J, R) = {/ є С^,я(J, R); D^f'*f Є (J, R)},

and

e<_r.t(J, R) = {/ є С1.г,ф(J, R),B7/ є (J, R)}

(2.3)

Since 77*f = 7+1 ^7+*f, it is obvious that С'1_гф(J, R) c 7-7(J, R).

Lemma 2.3. [2] Let 0 < a < 1, 0 ^ fi ^ 1 and 7 = a + fi (1 — a). If f Є CJ_T^ (J, R), then

and

7* 7f f «) = TT -7“ / «)

Wf ryf / M = Tf1-^ * / 7

(2.4)

(2.5)

Lemma 2.4. [17] If a > 0 and 8 > 0, then if-fractional integral and derivative of a power function are given by

rf (m—Ф(а))і_1 = —v-wr+^1,

and

Dff (f(t) — f>(a))a_l = 0, 0 < a < 1.

Lemma 2.5. [20] Let a > 0, and 0 ^ 7 < 1. Then Iff (■) is bounded from C1-1;ф(J, R) into С1_Т'ф (J, R). In particular, if 7 ^ a, then, Iff (■) is bounded from С1-Ъф (J, R) into C (J, R).

Lemma 2.6. [21] Let a > 0, 0 ^ у < 1, and f Є C1-1;ф(J, R). If a > y, then Iff f Є C(J, R) and

ТТ І M = lim+T? S (t) = 0.

Theorem 2.1. [21] Let 0 <a< 1, 0 ^ fi ф 1. If f Є C1-1 (J, R), then

7"- f M = f (<) —

I

(1_j3)(1_a)' Ф

a+

^ [ф(і) — ф(а)]а+[і{1_а)_1

Г(о + fi (1 — о))

Moreover, if 7 = a + fi (1 — a), f Є Cj_7 ; ^ (J, R) and iff-1 ' ^ f Є C(_7. ^ (J, R), then

T1-7 ' ф f(n)

ryf 7? f (t) = f (t) — 1 №) — ф(а)]-'-1.

Theorem 2.2. [25] (Banach fixed point theorem) Let (X,d) be a nonempty complete metric space with T : X ^ X is a contraction mapping. Then map T has a fixed point x* Є X such that Tx* = X*.

Theorem 2.3. [25] (Krasnoselskii’s fixed point theorem) Let X be a Banach space, let S be a bounded closed convex subset of X and let T]_, T2 be mapping from S into X such that Т1X + T2y Є S for every pair x,y Є S. If T1 is contraction and T2 is completely continuous, then the equation T-]_x + T2x = x has a solution in S.

3. Main results

In this section, we show that problem (1.4)-(1.5) is equivalent to a mixed type integral equation. We also prove the unique solvability of this problem. To this aim, we shall apply the fixed point theorems by Krasnoselskii and Banach. First we make the following assumptions.

(A1) f : (a,b] X R x R ^ R is a function such that f (-,u,xu) Є Cf-^T^J, R), for any и Є C\-1;ф (J, R) there exists M > 0 such that

|/(t,u,xu) — f (t,v,xv)l ^ M [\u - v\ + \xu — xv\] , Vi Є (a,b],u,v Є R. (3.1)

(A2) h : D X R —> R is continuous on D and there exists L* > 0 such that

\xu — xv\ Ф L* \u — v\, u,v Є G C R,

where

Xu(t) = h(t, s, u(s))ds, D = {(t, s) : a ^ s ^ t ^ b}.

J 0

(A3) The inequality

M + ML* \ і

П := —;-------— CkФ-Ф (Tk, a) < 1

1 - В

k= 1

holds, where

В = ^ Ck (Tk ,a) = 1, (Tk ,a)-.--

k= 1

Г(7)

(Ф^) — Ф(а))

i_1

and B(-, ■) is the Beta function.

Lemma 3.1. [2] Let 0 < a < 1 and 0 ^ /3 ^ 1. Then a function и solves the Cauchy problem

u(t) = f (t,u(t),xu(t)) , t Є (a,b},

lf+T’^и(Ф \t=a= ua, у = a + 0(1 — a),

if and only if и solves the following Volterra integral equation

u(t)

^a

ГЫ

(Ф(Ф — Ф(о)їі 1 +

1

Г(а)

ФІ(Ф(Ф(Ф — Ф(Ф)а f (s, u(s),xu(s)) ds,

where

Xu(s)= h(t, s,u(s))ds.

Lemma 3.2. Let 0 < a < 1, 0 ^ /3 ^ 1 and 7 = a + 0(1 — a). Assume that f (-,u(-), xu(')) Є Сі_7Г0 (J, R). If и Є Cf_rгф (J, R) then и satisfies the problem (1.f)-(1.5) if and only if и satisfies

the mixed type integral equation

where

and

(f) Фф(t,a)

u(t) —

1 — В

+

£

k= 1

1 rt

Ck

Гтк

Фф (Tk, s)f (s, u(s),xu(s)) ds + Ua

(3.2)

КФм') :=

Г(а)

№t) — Ф(о)]1т1

Фф(t,s)f (s,u(s),xu(s)) ds,

Г(7)

Фф (Tk, S) = Ф'(S) [tf(Tk) — ф(8)]

a—1

В = ^ Ckф1ф (Tk, a) = 1.

1

t

t

0

(3.3)

Proof. First, we prove the necessary condition. According to Lemma 3.1, a solution to equation (1.4) can be expressed as

«M =и<) -ffl)|7-1 ip* «(<0

1 f *

+ гТ~) Ф'ФФФФ) - ^(s))a-1f (s,u(s),xu(s)) ds

We substitute t = Tk into the above equation, then multiply it by Ck to obtain

17—1

, ^ [Ф(П) - Ф(а)}1 ті—1Уф , ,

CkU(Tk) =---------------------4+ u(a>

+

Ck

PTk

ФФФ [Ф(Тк) - Ф^Г 1 f (s, u(s),xu(s)) ds.

Hence,

(3.4)

(3.5)

7 ^ м(а) = X2 иОк) +

k= 1

Г“) ^+^u(a)Yl Ck [^(Tfc) - ^(a)|7 1

(^) k= 1

Г Tк

+ ^ ГТ^Т Ф'(з)[Ф(тк) - ^(s)|^1 / (S,U(S),XU(S)) ^ + Ua

fc=1 i(am/ a

which implies

І1а—Т;ф u(a)

1

1 - В

ST' ck

rrk

Фу(rfc, s)f (s,u(s),xu(s)) ds + ua

Substituting equation (3.6) into equation (3.4), we get

(f) ¥Ф(t,a)

u(t) —

1 - В

+

£

k=1

1 H

L-k

W)

Гтк

Фф (Tk, s)f (s, u(s),xu(s)) ds + Ua

г(а)

Фф(t,s)f (s,u(s),xu(s)) ds.

(3.6)

We proceed to proving the sufficient condition. Applying the operator C—1;^ to both sides of (3.2) and employing Lemmata 2.4, 2.1, we obtain:

11—Т;Ф <Ф

1

1 - в

+

rrk

Фф (Tk, s)f (s,u(s),xu(s)) ds + ua

’ST'' ck

Г(о;) ^ a

,k=1 K 1 a

1 Ґ

Фф(t,s)f (s,u(s),xu(s)) ds.

Г(1 - у + a)

We pass to the limit as t ^ a; since 1 - x < 1 - 7 + <a, Lemma 2.6 implies

1

І1а—Гф u(a)

1- В

Ck

rrk

Ф“{Tk, s)f (sal(s),xu(0) ds + u,.

(3.7)

Substituting t = Tk and multiplying equation (3.2) by Ck we get

УCk и(ть)=y

Ck"

k= 1

k=1

Фу Cfc >a) 1 - Б

E

.fe=1

Ck

W)

n

фуCfc, s)/ (s,u(s),xu(s)) ds + ua

111 Гтк

+уr(^) j ф^(rfc,s)^(s,m(s),^m(s))

fc=1

1

Г(о)

m

E

1 -5 t! rH

ф“Cfc, s)/ (s, Xu(s)) +

Б

1 - Б

j

which implies

m

У Cfcu(rfc) + ua

1

fc=1

1 - Б

E

,fc=1

Ck

W)

fT-fc

фуCfc,s)/ (s,u(s),xu(s)) ds + ua

(3.8)

Comparing equation (3.7) and equation (3.8), we see that

I

1_7;^

a+

ЧТ;Ф

u(o) = У CfcU(Tk) + м«.

fc=1

Now we apply the operator to both sides of equation (3.2) and by Lemmata 2.4, 2.6, and 2.3 we obtain that

yr «(*) =y+ ct / (tMt),Xu(t))

=Д/3|1-«);У^ (t,u(t),xu(t)).

(3.9)

Since и Є 61%^(J, R), by the definition of CT_r^(J, R) we have D1c++ и Є С1-Т;ф(J, R). Hence,

D'

5'1_“)У

a+

f (t,u(t),xu(t)) = DI

1_/3'1_“)У

a+

,f Є С1-,1ф(J,R)

(3.10)

For each f (■, u(-), xu(')) Є С1-1;ф(J, R), by Lemma 2.5 we have

У1;15'1-““f Є С\-Гх(J. R).

Equation (3.10) and the definition of Cj_lX(J, R) yield that

£;5'1-e);y f (■,«(■),*«(■)) є сі_і;ф (j, r).

Therefore, f and Ila_ 5(1 “);^ f satisfy the assumptions of Theorem 2.1. We apply the operator рз(+1_а);ф to the both sides of equation (3.9) and by Lemmata 2.1, 2.2, 2.6 we get Т+д-оУУ^г+ФU(t) =I5^_“)-NDm_“)-N f (t, U(t), xu(t))

=/ (t,u(t),xu(t))

/i+5'1 «);у j (a,u(a),xu(a))

[CC0 - 7(a)]

5'1_“)_1

(3.11)

T(7(1 - a))

=/ (t,u(t),xu(t)).

Comparing equation (3.11) with equation (2.2) when n = 1, we get

У-У u(t) = f (t,u(t),xu(t)).

This means that equation (1.4) holds true. The proof is complete.

We proceed to proving the solvability of problem (1.4)-(1.5) in the weighted space (J, R); we shall do this by means of Krasnoselskii fixed point theorems.

□

Theorem 3.1. Let

0 < a < 1, 0 ^ fi ^ 1, 7 = a + fi — afi

and Assumptions (A1), (A2) and (A3) are satisfied. Then problem (1.f)-(1.5) is solvable in the space С(-Г,ф (J, R) С (J, R).

Proof. We are going to reduce problem (1.4)-(1.5) into a fixed point problem. Consider the operator T : C1_7;ф (J, R) ^ C1_7(J, R) defined by

Ф1 (t, a) (Tu)(t)~ ф

1 — В

+

&k

rrk

k

1 Ґ

Xi Г(а) j ф*(jk'(s’ “(s)l ds + ““

(3.12)

We define

r :=1 +

+

Г(а)

В

" 1 m

фу(s,M(s),XM(s)) ds, t Є (а,Ь].

1 — В рВ(а/у)

Г(о)

1 - В

J2Ck (^(rk) — Ф(а)Т + [^(&) — ^(«)]

к=1

Вг ={м Є Сі_7 ; ф ((J, R) : ^ г},

h := max \[ф(8) — Ф(а)}1-1 f (s,u,xu)\ ,

( s,u,xu)€ J xBr X Br

and we introduce operators T1 and T2 on Br as follows:

and

ФІ (t, a)

TMt) =

T2u(t)

^ Гоў / (rk,S)^ (S,M(s),^M(s)) + Ма

k=1

1

Г(а)

фїs)/ (Аи(а Х“М) *.

Observe that Т1 + Т2 = T, where the operator T : C1_1 ; ^(J, R) ^ C1_1 ; ^(J, R) is defined by

equation (3.12). The rest of the proof is split into several steps.

Step 1: Let us prove that T1u + T2v Є Br for each u,v Є Br.

For и Є Br and t Є (a, b] we have

\T1u(t) [ф(і) — ф(а)}1 7\ ^

1 1

<-

Г(7)1 — В 1

rrk

^ ГГ)/ ф^ (rk,s) ^ (s,u(s),Xu(s))\ ds + ua

k=1 l(a) Jа

1 — В

_ 1 = 1 — в

1

<-

rrk

^ ГГ) I ф^(rk,5)ф^(s,a)ds + иа

k= 1 l(a) 3а

^ Ck ) (^(Tk) — Ф(о))а+1 1 + Ua

.k

A- В

= k Г(а)

рВ(а,х)

E

3Ck Г(а)

(t)(Tk) — ф(а))а + Ua

where we have employed that

1 f'Tk

j фаф (^k ,в)ф1 (s,a)ds = ^ (ф(тк) — ф(а))а+1 \

a

t

and

Hence, it follows that

ІІЗДІ

1

[ф(тк) - ф(а)]' [ф{тк) - ф(а)\

LB(^,l)

^1-7;^ ^ 1 — $

m

S Cfcl

,k= 1

г(«)

< 1.

W'(n.) - İ'(a»a + ua

(3.13)

In the same way, for the operator T2 we have

1 Ґ

\TMt) [ф(і) - -ф(а)\ 'I ^ [V-’M - Ла)]1-7 Ф2,((, a) \f (s, u(s), \-u(s))| ds

4B(o‘,i)

Г(о)

Г(а)

[ф(і) - ф(а)]а , for V Є Br.

Therefore,

НЗДіСі-7;*

By equations (3.13), (3.14) we get llT1 U + ЗДІСі-.^ ^ ІІТ1иІІСі-7.ф + ІТ2^ІСі-т^

1

« W6> - ^(“)\”.

ГН

(3.14)

<-

1 - В

ST^ hB(a, 7) ( I ( ^ , ( \\a ,

> tCk (Ф(Тк) - ф(а)) + Ua

k= 1

Г(а)

, LB(B,1) r /1 м<

+ w ч [W - ^(«)]

ГН

ма pB(a,x)

+

1 - В Г(а)

1

1 - В

^ ск(ф(тк) - ф(а»а + [ф(Ь) - ф(а)\

к= 1

< Г.

This proves that Т1и + T2v Є Br for each u,v Є Br.

Step 2: At this step, we are going to show that the operator T1 is a contracting mapping on Br.

For each u,v Є Br, and for each t Є (a,b], it follows from Assumptions (A1) and (A2) that \[ф(і) - ф(а)]1-1 T1u(t) - [ф(ф - ф(а)]1-1 T1b(t)\

<-

1 1 ^ ck ru

T(7)1 - В ^ Г(а)

E

(rk,s) \f (s,u(s),xu(s)) - f (s,n(s),xn(s))| ds

<-

1 1 ^ Cfc a

T(7)1 - В ^ Г(а)

E

фФ fa, s)M [\m(s) - v(s)\ + \xu(s) - хФ^ ds

<-

1 1 ^ Cfc a

E

Td(rk,s) (M + ML*) \u - v\ ds

T(7)1 - В ^ r(o)j (M + ML*) ^ ck Г. ..

^----------^2 тФпЛ I (тк, 8)Ф1(s, a) llu - vllCi-^ ds

1- В

Г(а)

(M + ML*) / \ İl II

---MTr CkФ+ (тк, a) llu -

Cl-7;^ ^

fc=1

which implies

llT1U - T1vllCi-TaP ^ П llU - ЬІІСі-т,Ф .

Due to Assumption (A3), we conclude that the operator T1 is contracting. Step 3: Here we prove the operator T2 is completely continuous on Br.

a

The continuity of f implies that operator Т2 is continuous. Also, T2 is uniformly bounded on Br. Indeed, by Step 1, for v Є Br we have

İlgile-,;, « Ш - ^(a)l” := l.

This shows that for each r > 0 there exists a positive constant l such that ||T2u||Ci_ ^ l for

V Є Br.

In order to prove the compactness of the operator T2, we take и Є Br and t1,t2 Є (a, b] with t1 < t2 and we have

I[^(*2) - Ф(а)} 1T2u(t2) - [^(ii) - -0(a)]1 1T2u(tl)\

[ф(І2) - ф(о)]1; ^

Г(а)

Щи) - ф(а)]1; Ґ1

Г(а) Ja

[ф(І2) - ф(а)]1- ц. Г^

ФФ (h,s)f (s,u(s),xu(s))ds

ФФ (t1,s)f (s,u(s),xu(s))ds

Г(а)

[ФФФ- Ф(д)}1-1 d Ґ1

Г(а) Ja

ФФ ^^Фф (s,a)ds

фФ (U и)фф (s,a)ds

^) \[^(t2) - ^(a)]1 7 ФФ+а(І2,a) - ШІ1-) - Ф(а)]1 7 Фф+а(і1,а)\

рВ(а,ҳ) ' Г(а)

|[^2) - Ф(а)Т - \ф(d 1) - ф(а)]аі

The right hand side of the above inequality is independent of u. The continuity of ф as t2 ^ t1 implies that

\[Ф(І2) - Ф(о)]1-1 T2U(İ2) - [ф(І1) - ф(а)}1-1 T2U(İ1)\ ^ 0 as U ^ І1-

This proves that Т2 is equicontinuous. Hence, T2 is relatively compact on Br. By the Arzela-Ascoli theorem, T2 is compact on Br. Thus, all assumptions of Theorem 2.3 are satisfied and Hilfer problem (1.4)-(1.5) is solvable in С1-Т.ф (J, R).

Finally, we show that such a solution is in С\-Тф(J, R). By applying Dff on both sides of equation (3.2), and using Lemmata 2.4, 2.3 we obtain

Da+u(t) = Da+%+ f (t,u(t),xu(t) = f (t,u(t),xu(t).

Since f (■, и(■), xu(■)) Є С'ф.-1г;ф)(Т, R), it follows from the definition of the space С1_—Тф(J, R) that Da+ u(t) Є C\-т.ф(J, R) and this u(t) Є С^—.ф(J, R). The proof is complete. □

4. Ulam-Hyers-Rassias stability

In this section, we study Ulam-Hyers and Ulam-Hyers-Rassias stabilities for nonlocal fractional integro-differntial equation (1.4)-(1.5). The stability results are based on the Banach fixed point theorem.

Theorem 4.1. Let Assumptions (A1) and (A2) be satisfied. If

Л :

EfrdS <T‘ •“> + *? (b-A

1 _ Г(7)‘

1

(M + ML*) В(а,х) < 1,

(4.1)

then problem (1.4)-(1.5) has a unique solution.

Proof. We consider the operator T : C1-r^(J, R) ^ С1-Т;ф(J, R) defined by equation (3.12). In view of Theorem 3.1, we know that the fixed points of T are solutions of problem (1.4)-(1.5).

Let us prove that T has a unique fixed point, which is a solution of problem (1.4)-(1.5). Indeed, for each u,v Є C1-T^ (J, R) and each t Є (a, b], we have

I №(i) - ф(а)\'-<Tu(t) - [VW - VO)]1-'rw)|

<-

1

T(7)(1 - В)^T(a)

E

C-k

Фу(П,s)\f (s,u(s),xu(s)) - f (s,n(s),x^(s))| ds

+ --- Г \j (s,u(s),xu(s)) - j (s,w(s),xw(s))\ds

ГН Ja

(M + ML*) 1 - В

пъ

Z ft) I Ф^(тк,5)Ф^(s,a) \и - v\\c1-1,p

к=1 ( ) Jа

ds

, Г(7)№) -ф(а)]1- (М + ML*) !

(М + ML*)

1 - в

Г(а)

В(а,х)

(М)Фу (s,o) \\и - v\\Cı_^

ds

Z °кФ^(Тк, а) \\U - VWcı_T,4

+ (м + ML*) В(а, 7)Фу (t, а) ||и -

1

С-к

Е гтЬ Ф?(тк •а'> + Фї (1°)

1 - В ti Г<7)

(M + ML*) й(а,7) ||и - «||Cı_ /:

7 : V

This gives:

\\Ти - Tv\\Cı_::ф ^ Л \\U - п\|

Cı—-у : -ф

By inequality (4.1), the operator T : С1-Т;ф(J, R) ^ С1-Т;ф(J, R) is a contracting mapping. Hence, we conclude that the operator T has a unique fixed point и Є С1-Т;ф(J, R) given by Banach fixed point theorem. □

We proceed to studying the Ulam-Hyers stability and Ulam-Hyers-Rassias stability. For e > 0 and for each <p Є С1-Т;ф(J, R) we consider the following inequalities:

и() - f (t,îı(t),xü(t)) ^ 7 t Є (a, b], (4.2)

M(f) - f (t,îı(t),xü(t)) ^ z^P(t), t Є (a, b], (4.3)

M(t) - f (t,îı(t),xü(t)) < L(t), t Є (а, Ь]. (4.4)

Definition 4.1. Problem (1.4)-(1.5) is Ulam-Hyers stable if there exists a real number Cf > 0 such that for each e > 0 and for each function її Є Cj-lX(J, R) satisfying inequality (4.2), there exists a solution и Є C\-lX(J, R) of equation (1.4) obeying

\u(t) - u(t)\ ^ Cf e, t Є (a,b\.

Definition 4.2. Problem (1.4)-(1.5) is generalized Ulam-Hyers stable if there exists фф Є C(R+, R+) with фф(0) = 0 such that for each function її Є C\-lX(J, R) satisfying inequality (4.2), there exists a solution и Є C\-lX(J, R) of equation (I.4) obeying

\u(t) - u(t)\ ^ фф(e), t Є (a,b].

Definition 4.3. Problem (1.4)-(1.5) is Ulam-Hyers-Rassias stable with respect to p Є С\-Т;ф (J, R) if there exists a real number Cf,v > 0 such that for each є > 0 and for each function и Є С'1-і;гф(J, R) satisfying inequality (4.3), there exists a solution и Є Cf ^(J, R) of equation (1.4) obeying

\u(t) - u(t)\ ф Cf,vep(t), t Є (a,b\.

Definition 4.4. Problem (1.4)-(1.5) is generalized Ulam-Hyers-Rassias stable with respect to p Є С\-Т;ф(J, R) if there exists a real number Cf,v > 0 such that for each function и Є C?-7. Ф(J, R) satisfying inequality (4.4) there exists a solution и Є C\.^(J, R) of equation

(1.4) obeying

\u(t) - u(t)\ Ф Cf^ıp(t), t Є (a,b\.

The next lemma is a generalization of Gronwall lemma.

Lemma 4.1. [20] Let a < 0 and ф Є Cl[a,b] be an increasing function such that ф'(Р) = 0 for all t Є [a,b]. Assume that h is nonnegative and non-decreasing, and у is a nonnegative function locally integrable on [a,b] and suppose also that x is nonnegative and locally integrable on [a, b] obeying

x(t) Ф y(t) + h(t) f (t,s)x(s)ds, t Є [a,b],

J a

then, for all t Є [a, b], we have

X(t) « V(t) + ^‘ (1 sMOds.

Moreover, if y(t) is a nondecreasing function on [a,b], then

x(t) Ф y(t)Ea [h(t)r(a)<$f(b,a)] , t Є [a,b].

Now, we are ready to prove Ulam-Hyers and Ulam-Hyers-Rassias stability for problem (1.4)-

(1.5) .

Theorem 4.2. Let Assumptions (A1) and (A2) be satisfied. Then problem (1.4)-(1.5) is Ulam-Hyers stable.

Proof. Let и Є Cf ф(J, R) be a function satisfying inequality (4.2). Applying operator Iff to the both sides of inequality (4.2) and using Theorem 2.1, we have

та;ф

a+

Dff , фU(t) - Iff f (t,u(t),xu(t))

U I

t.

This implies that

u(t) - Чи

1

t

Фу(t,s)f (s,u(s),xu(s)) ds

Ф -(b,a),

a v

where

and

Фу (b,a)

ЩЬ) - Ф(д)Г

Г(о)

Чи

Фу (t,g) 1 - в

Ск

к

•Tk

Ф£(тк, Of (s, d(s), xu(s)) ds + Ua

(4.5)

We denote by и Є с;-тф (J, R) the unique solution of the following problem

Dff’^ u(t) = f (t,u(t),xu(s)), 0 <a< 1, 0 ^ ft ^ 1, t Є (a,b]

u(t) \t=a = І^+Г,Фu(t) \t=a= Ua + ^ °ku(Tk), Tk Є (a, Ъ],

k=1

where у = a + ft — aft. Using Lemma 3.2, we obtain

1 Ґ

u(t) = Пи +wr— Фф(t,s)f (s,u(s),xu(s)) ds,

Г(«Ь/а *

(4.6)

where

Пи

Фу (t,a)

^Г(а)

,k=1 ' '

Ck fTk

Фу (Tk ,s)f (s,u(s),xu(s)) ds + Ма

1 — В On the other hand, if

и(тк) = м(тк) and І^-Т’фu(t) \== І^-Т’фu(t) \t=a,

it is easy to see that Bu = Bu. Hence, by Assumptions (A1) and (A2) and equations (4.5), (4.6), for each t Є (a,b] we have

\u(t) — u(t)\ ^

1 Ґ

u(t) — 4u — Фу(t,s)f (s,'u(s),X'^(s)) ds

Г(а)

1 Ґ

+ \4ü — Bu\ + ^^ Фу(M) \/(t/u(t),X'u(t)) — f (t,u(t),xu(t))\ds

Г(а)/а * (4.7)

Фу^, s)M [\m(s) — u(s)\ + \xu(s) — xu(s)\] ds

^—Фу (b, a) + ГТ-) а 7 Г(а) ja

«^Ф?.(M + (M +(д)L‘} [ф°*(M№) — “M!

We apply Lemma 4.1 to obtain

\u(t) — u(t)\ Ф^(Ь,а)

а v

^ - Фу (b,a)

a v

1+ f ‘ t Фїк (M

k=1

Г(о&)

ds

1+E

k=1

[(M + МА*)Ф“ (6, а)] Г(о& + 1)

-Фу (b,a)

а

t

E

,k=0

[(M + МА*)Ф“ (6, а)] Г(о& + 1)

=-Ф^(b, о)Еа [(M + ML*) Ф“ (b, а)]

Then for

we get:

Cf

Фу (b,a)

a

Ea [(M + МІ*)Фаф(b,a)}

\u(t) — u(t)\ ^ Cfe.

This means that problem (1.4)-(1.5) is Ulam-Hyers stable. The proof is complete.

□

Corollary 4.1. Under the assumptions Theorem 4.2, if there exists ftf Є C(R+, R+) with ff (0) = 0, the problem (1.4)-(1.5) is generalized Ulam-Hyers stabile.

k

k

Proof. Following the lines of the proof of Theorem 4.2, we choose <ff (є) = Cfє and <ff (0) = 0 and we obtain |u(t) — u(t)| 7 <ff(є). Hence, problem (1.4)-(1.5) is generalized Ulam-Hyers stable. □

Theorem 4.3. Let Assumptions (A1) and (A2) hold and the following condition be satisfied

(A4) There exists an increasing function <p Є Cı_7,y (J, R) and Xv > 0 such that for each t Є (a,b], the inequality holds:

K+ T(t) 7 XrT(t).

Then problem (1.f)-(1.5) is Ulam-Hyers-Rassias stable.

Proof. Let e > 0 and let и Є Cf ^(J, R) satisfty inequality (4.3). Solving equation (4.3) and taking into consideration Assumption (A4), we get

1 Ґ

u(t) —Пъ — —— Td(t,s)f (s,u(s),xu(s)) ds

T(o)

7 eX^p(t).

(4.8)

On the other hand, let и Є Cf . (J, R) be the unique solution of problem (1.4)-(1.5), that is

y1_7 , ^

1 Ґ

u(t) = Пи + wr— Фф(t,s)f (s,u(s),xu(s)) ds,

r(ab/a

(4.9)

Then, as in (4.7), for each t Є (a, b] we have |u(t) — u(t)l 7

1 Ґ

u(t) — Hu — wr— (t,s)f (s,u(s),xu(s)) ds

Г(ак/«

1 Ґ

+ \Пй — Hu\ + І (t,s) |/(t,u(t),xu(t)) — f (t,u(t),xu(t))l ds

T(o)

1 Ґ

7e\pp(t) + —— (t, s)M [\w(s) — u(s)\ + \x'u(s) — xu(s)\] ds

Г(«Ь'а *

7oXvp(t) + [ Фаф(t,s) \“(s) — u(s)\ds.

T(o)

We apply Lemma 4.1 and we get:

nt ~ (M + ML* f

\u(t) — u(t)\ ^eX^t) + eXp j ---------цЩ-------Ф?(1, s)T(s)ds

-eXtptp(t) + еХц

k= 1 t

+

ak)

(M + MM^

ГМ '

Ґ (M + MU)2 „ Г(2а) Ф*

ФФ (t,s)p(s)ds

ФФ(f, s)p(s)ds + ...

=eXvp(t) + бЛ^ [(M + MM) I<+p(t) + (M + MM)21%?*p(t) + ... 7oXvp(t) + eXv \_(M + MM) X<p<p(t) + (M + MM) (Xv) <p(t) +.. .~\

1 + ^ (M + ML* )k (K)k

k=1

Then, for

Cf,v = Xy

1 + ^ (м + ML*)k (X<r)k

we get that

\u(t) - u(t)\ ф Cf^ep(t).

This proves that problem (1.4)-(1.5) is Ulam-Hyers-Rassias stable. The proof is complete. □

Corollary 4.2. Under the assumptions of Theorem 4.З problem (1.4)-(1.5) is generalized Ulam-Hyers-Rassias stable.

Proof. We follow the lines of the proof of Theorem 4.3 and choosing є = 1, we get

\u(t) - u(t)\ ф Cf,vep(t).

□

5. Examples

5.1. Example 1. Consider the nonlocal fractional integro-differential equation involving the ^-Hilfer fractional derivative:

1

Oil3, u(t)

3E\(t + 2)(1 + \m\) 3E\(2) J0

1

+ wv^ I e-u(s)ds, (0,1],

Here

1 , 12

i0lu(0) = 2 m(-).

11 2 12

a =2, ^ =3, 7 =3, Uo = 0, Cl = 2, Tl = 3,

(5.1)

(5.2)

and Ф : [0,1] ^ R is such that %f(t) = t for all t Є [0,1] and

1

f (t,u(t),xu(t)) = t 6 +

3Ei(t + 2)(1 + \u\) 3T'i(2^/o

1

+ І Є^u(s)ds t Є (0,1],

Xu(t) = h(t, s,u(s))ds = e 2 u(s)ds.

Clearly,

since

f (t,u(t),xu(t)) Є C|.t([0,1], R+)

t3 f (t,u(t),xu(t)) Є C([0,1], R+). Let u,v Є R+ and t Є (0,1]; then it is easy to see that

and

\f (t,u(t),xu(t)) — f (t,v(t),xv(t))\ Ф 3^(2) (\u - v\ + \xu — xv\)

Г ^ 1

\xu — xv\ Ф e^|u(s)-u(s)|ds ^ - \u — v\.

Hence, Assumptions (A1) and (A2) hold with

m 1

L*

3^(2)’ 2'

We are goint to check that Assumption (A3) hold as well. Indeed, by simple calculations we see that

B = сіфр (W 0) = ci

[^(n) — ф(0)]

7—1

and

Г(7)

0.10 < 1.

~ 0.67= 1,

o

o

o

1

And since all assumptions of Theorem 3.1 are satisfied, problem (5.1)-(5.2) has at least one

solution in C3 ([0,1], R+). For t Є (0,1] and и Є R+ we have Л ~ 0.42 < 1, therefore, condition

3 ;

(4.1) holds. Hence, by Theorem 4.1, problem (5.1)-(5.2) has a unique solution in C3 ([0,1], R+).

3 ;i

5.2. Example 2. We consider the nonlocal fractional integro-differential equation involving the ^-Hilfer-Hadamard fractional derivative

1 , ^................ 1 ' *

where

1

a = -, 2’

and

D*и(ї) = 1п(л/£) cos(t)u(t) + -1 І є 2 u(s)ds,

20 20 J і

1+

In t

7+..u(1)=2 “ (3)

1

P = 0,

7 =

2’

Uq = 0,

C1 =2,

r 1

1

1 Ґ -1

Xu(t)

20

e -21 u(s)ds.

20 J1

t Є (1,e],

(5.3)

(5.4)

ф(і) = lnt, t Є [1, e],

f (t,u(t),xu(t)) = —ln(Vt )cos(t)u(t) + — e2 u(s)ds t Є (1,e]

Clearly,

since

f (t,u(t),xu(t)) Є C1;ln^([1,e], R)

(lnt)2 f (t,u(t),xu(t)) Є C([1,e], R). Let u,v Є R and t Є (1, e], then it is easy to see that

and

|/(t,u(t),xu(t)) - f (t,v(t),xv(t))l ^ 20 (lu — v\ + l^u — wD

Ixu — xvl ^ e 2 |m(s) v(s^ds ^ - lu — n|.

Hence, Assumptions (A1) and (A2) hold with

M = —,

20’

L*

1

2.

Let us check that condition (4.1) is satisfied.

Indeed, by simple calculations we see that for t Є (1, e],

[ln(n) — ln(1)]7_1

B = С1ф1 (W0) = c1-

Г(т) 2у/n !og(2)

~ 0.44 = 1,

and

Л ~ 0.29 < 1.

1

Then by Theorem 4.1 problem (5.3)-(5.4) has a unique solution in Cf lnt([1,e], R). As it has

1 2

been shown in Theorem 4.2, for t = 2 > 0, if и Є C 1lnt([1, e], R) satisfies

0"+^ u(t) — f (t,u(t),xu(t))

1 , n

^ 2, 1 Є (1,e],

1

3

2

t

t

1

і

there exists a unique solution и Є C|,lnt([1, e], R) such that

K^) - u(t)1 A 2Cf,

where

c, = (e, 1)6 2

3

40Ф* (",1)

-L- e( 40 )2 г( 3)e

1+ert(l)

~ 1.23 > 0.

Here

(e, 1) := 'ln(e) - ln(1)]‘

г(«)

C, L) := ln'(«) I<=1 [ln(e) - ln(1)]

a—1

Hence, problem (5.3)-(5.4) is Ulam-Hyers stable. Finally, we consider ip(t) = (ln t) 2, then

і

(ln t)2 p(t) = ln(i) Є C([1, e], R),

i.e., ip(t) Є Ci,inR).

In order to verity the condition

/"jlnVW A \<p<p(t), > 0,

by employing the Hadamard fractional integral and simple computations we get

/f;ln * ln t

1

Г( 2 )J 1

Thus, Assumption (A4) is satisfied with

A t\ 2 і ds 1 Г Л A 2 ds 2 1

(in(ln3)2IT A гшУі (ln A IT « -^ln2(().

\ = — > 0.

t

And tor e = 1 > 0, it и Є C|t([1,e], R) satisfies

- /(t,u(t),xu(t)) (lnt)2, і Є (1,eL

there exists a unique solution и Є C|t([1, e], R) such that

1 і 2

- u(t)1 A Cf,v-(ln t) 2 = -7=

2 \/^

К d ‘(-)

к ı

1+> '4Ğ1 <-=

-(ln t) 2.

Hence, problem (5.3)-(5.4) is Ulam-Hyers-Rassias stable. Finally, taking є =1, we get

к / n \ к и

!уС0 - y(^)| A —

Jf

■+s( -0+a ( t

(ln t)2.

2

2

Therefore, problem (5.3)-(5.4) is generalized Ulam-Hyers-Rassias stable.

Conclusions

The main results of this article have been successfully achieved by employing Krasnoselskii and Banach fixed point theorems and our most important results in the made nonlinear analysis is the study of the existence and uniqueness of solutions tothe Cauchy-type problem for a nonlinear fractional integro-differential equation introduced by the left ^-Hilfer fractional derivative. We discussed the Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stabilities. This paper contributes to the growth of the fractional calculus, especially in the case of fractional differential equations involving a general formulation of Hilfer fractional derivative with respect to another function.

There are some articles that carried out a brief study on existence, uniqueness, and stability of solutions of fractional differential equations, however, there are just a few of them devoted to Hilfer type operator and one of our aims was to contribute in this field. We expect that our results can be extended to some other fractional differential equations involving Hilfer derivative with respect to another function ф.

Acknowledgments

The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. The authors would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to the improvement of the presentation of the paper.

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Hanan Abdulrahman Wahash,

Department of Mathematics,

Dr. Babasaheb Ambedkar Marathwada University,

Aurangabad 431004 (M.S.), India E-mail: [email protected]

Mohammed Salem Abdo,

Department of Mathematics,

Dr. Babasaheb Ambedkar Marathwada University,

Aurangabad 431004 (M.S.), India E-mail: [email protected]

Satish Kushaba Panchal,

Department of Mathematics,

Dr. Babasaheb Ambedkar Marathwada University,

Aurangabad 431004 (M.S.), India E-mail: [email protected]