CC BY
6
DOI: 10.17516/1997-1397-2021-14-2-159-175 УДК 517.9
On the Theory of y-Hilfer Nonlocal Cauchy Problem
Mohammed A. Almalahi* Satish K. Panchal^
Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University Aurangabad (M.S), India
Received 10.08.2020, received in revised form 10.09.2020, accepted 20.11.2020 Abstract. In this paper, we derive the representation formula of the solution for y-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard's successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze Ea-Ulam-Hyers stability. Finally, one example to illustrate the obtained results.
Keywords: fractional differential equations, fractional derivatives, Ea-Ulam-Hyers stability, fixed point theorem.
Citation: M.A. Almalahi, S.K. Panchal, On the Theory of y-Hilfer Nonlocal Cauchy Problem, J. Sib. Fed. Univ. Math. Phys., 2021, 14(2), 159-175. DOI: 10.17516/1997-1397-2021-14-2-159-175.
Introduction
In recent years, the scientific community has been paying more attention to fractional calculus because it is an effective tool in modeling many phenomenaIn various fields of engineering and science, since its non-local properties are suitable for describing memory phenomena such as non-local elasticity, polymers, diffusion in complex medium, biological, electrochemical chemistry, porous media, viscosity, electromagnetism, etc. For more details, we refer the reader to monographs of Kilbas et al. [12], Samko et al. [21], Hilfer [10], Podlubny [19] and the papers [5,9]. In the recently years, Kilbas et al. in [12] introduced the properties of fractional integrals and fractional derivatives of a function with respect to another function. Sousa and Oliveira [22] proposed a ^-Hilfer fractional operator and extended few previous works dealing with the Hilfer [7,10]. Moreover, they discussed some important qualitative properties of solutions such as existence, uniqueness, and stability results in the following papers [18,22-24]. Over the last years, the stability results of fractional differential equations have been robustly developed. Very significant contributions about this topic were introduced by Ulam [28], Hyers [11] and this type of stability is called Ulam-Hyers stability. Thereafter, the Ulam-Hyers stability was extended by Rassias [20] in 1978 to a new type of stability which called Ulam-Hyers-Rassias stability. For some recent results of stability analysis, we refer the reader to a series of papers [2,3,14,17,18,24,26,30,31]. For the existence and uniqueness results of different classes of initial value problem for fractional differential equations involving ^-Hilfer derivative operator, one can see [1-4,15,27]. More recently, Wang and Li in [32] introduced four new types of Ea-Ulam stabilities. Gao et al., in [8] established the existence and uniqueness of solutions to the Hilfer nonlocal boundary value problem by using some properties of Hilfer fractional calculus, Mittag-Leffler functions, and fixed
*aboosama736242107<8gmail.com https://orcid.org/0000-0001-5719-086X
tdrpanchalsk@gmail.com © Siberian Federal University. All rights reserved
point methods. Kucche et al., in [13] obtained representation formula for the solution of Cauchy problem in the form of Mittag-Leffler function.
Motivated by [8,13,32], in this paper, we use Picard's successive approximation technique to obtain representation formula for the solution of linear Cauchy problem with constant coefficient
HDa/wy(t) = Xy(t) + h(t), n — 1 < a < n, P e [0,1], t e J := (a, b], (0.1) yl;-j]Ina-r'vy(a) = cj, j = 1,2,..., n, a < r = a + np — ap. (0.2)
/ i d \n-j
in the form of Mittag-Leffler function, where y^-^y(t) = f ^^) y(t), j = 1,2,..., n.
Furthermore, we introduce new results of some qualitative properties of solution such as existence, uniqueness, and Ea-stability results of a nonlinear y-Hilfer fractional differential equation
HDa/'¥y(t) = Xy(t) + f (t,y(t)), a e (1,2), p e [0,1], t e J := (a, b], (0.3)
m
y(a) = 0, y(b) = £ Sil^yT), Ti e (a, b] , (0.4)
i=1
where HDa+P'ydenotes the y-Hilfer fractional derivative of order a e (1,2), type P e [0,1], Y = a + ¡(2 — a), X < 0, m e N, and f : (a, b] x R —> R is given function satisfying some assumptions that will be specified later.
To the best of our knowledge, this is the first paper dealing with y-Hilfer fractional derivative with constant coefficient of order a e (1,2). In consequence, our findings of the present work will be a useful contribution to the existing literature on the topic.
This paper is organized as follows: In Section 2, we recall the basic definitions and prove some lemmas which are used throughout this paper, also we present the concepts of some fixed point theorems. In Section 3 , we derive representation formula for the solution of the problem (0.1)-(0.2) in the form of Mittag-Leffler function. Furthermore, we derive an equivalent fractional integral equation to the nonlocal problem (0.3)-(0.4). In Section 4, we study the existence and uniqueness results of y-Hilfer nonlocal problem (0.3)-(0.4) by using some properties of Mittag-Leffler function and fixed point theorems. In Section 5, we discuss Ea-Ulam-Hyers stability of solution to a given problem. In Section 6 we give one example to illustrate our results. Concluding remarks about our results in the last section.
1. Preliminary
Let [a, b] C R+ with 0 < a < b < to. For y = a + ¡(2 — a), 1 < a < 2, 0 < P < 1. Then 1 < Y ^ 2. Let y e C1 [a, b] be an increasing function with y' = 0, for all t e [a, b], the weighted space C2-Y;y [a, b] of continuous function f : [a, b] ^ R is defined by
C2-r,v [a, b] = {f : (a, b] ^ R; (y(t) — y(a))2-Yf (t) e C [a, b]} , 1 < Y < 2.
Obviously C2-ry [a, b] is the Banach spaces with the norm
11 f "c2-r.M= ) — V(a))2-Yf (t )|.
Next, define Lp ([a, b], R) the Banach space of all Lebesgue measurable functions n : [a, b] ^ R with M\L„[aM <
Definition 1.1 ([12]). Let a > 0, f e L1 [a, b] . Then, the y-Riemann-Liouville fractional integral of a function f with respect to y is defined by
1 r'
ayf(t) = fa Ja y'(s)(y(t) — y(s))a-1 f(s)ds. - 160 -
Definition 1.2 ([22]). Let n — 1 < a < n G N, and f, y G Cn[a, b] (—<x < a < b < <x>) be two functions such that y is increasing and y'(t) = 0, for all t G [a, b]. The left-sided y-Hilfer fractional derivative of a function f of order a and type 0 ^ P ^ 1 is defined by
HDa,ß,y fff) = jß{n-a);y ( 1 J1-ß)(n-a),y f(.)
Da+ f () = 1 \y'(t) dt) la+ f (
y'(t) dt
Lemma 1.1 ([12]). Let a, Y > 0. Then
(1) fyff (t) = cvyf (t)
(2) ay(y(t) - y(a))Y-1 = JO^Yy - y(a))a+Y-1
(3) HDYf(y(t) - y(a))Y-1 = 0.
Lemma 1.2 ([22]). If f e Cn[a, b], n - 1 < a < n, and 0 < ß < 1, then
ay HDajyf (t)=f (t) -f k %k fmn-k]e-ßKn-a) ; yf (ai
n-k
d
i 1 \n-k
where f[;-k]f(t)= [y-^¿J f(t).
y'(t)
Theorem 1.1 ([6]). (Banach fixed point theorem) Let X be a Banach space, K C X be closed, and G : K ^ K be a strict contraction, i.e., ||G(x) — G(y)|| ^ L ||x — y|| for some 0 < L < 1 and all x, y G K. Then G has a fixed point in K.
Remark 1.1. To simplify the notation and the proof of some results, we will introduce the following notation
Q-2(t, a) = (y(t) — y(a))Y-2 and Na-1(t, s) = y'(s)(y(t) — y(s))a-1.
Lemma 1.3 ([29]). Let a G (1,2] and P > 0 be arbitrary. The function Ea(-), Ea,a(-) and Eap(-) are nonnegative, and for all z < 0
Ea (z) := Ea,1 (z) < 1 Ea,a(z) < rjOj, Ea,P(Z) < Y^p) ■ Moreover, for any X < 0 and t1, t2 G [0, 1], we have
Ea,a+p(^Q.y(t2, a)) ^ Eaa+p(XQy(t1, a)) as h ^ t2, (1.1)
where Eap is the Mittag-Leffler function.
Proof. See [29], Lemma 2 and [33]. □
Lemma 1.4. Let a > 0, P > 0, Y > 0 and X G R. Then
%yQp¥-1(t, a)Erp(XQry(t, a)) = Q^+p-1(t, a)EYia+p(XQ7y(t, a). Proof. By Definition 1.1, we have
1 rt
to)
ÜyQßr\t, a)EY,ß(XQ7y(t , a)) = ^ / Nva-1(t, sQ-1(s, a)E%ß(XQYy(s, a))ds
i Naa-11 S)QP-1(S a) f (XQYy(s,a))n
r(a) Ja Ny (t,S)Qy (S,a) f r(yn + ß)
w 2 n 1 r t
= £ m^w) m I -¿r1*, ^'f.'*.
Via Lemma 1.1, we get
Iaa; ¥ Qy-1 (t, a)Er,ß(ÄQy (t, a)) = Qa+ß-'(t, a)EYia+ß(XQrv (t, a).
Lemma 1.5. Let a > 0 , ß > 0, k > 0 , X e R, z G R and f G C [a,b]. Then
Ikf N«-' ^ t )Ea,a(XQ y(z, t ))f(t )dt = i\.N¥a+k-1(z, t )Ea,a+k(XQy(z, t ))f(t )dt. Ja Ja
Proof. According to Definition 1.1 and Lemma 1.4, we obtain
Ikf f N«-'(z, t)Ea,a(XQy(z, t))f (t)dt =
a
= W) la Nk-1 (z, u){ I" N¥a-1(u, t )Eaa(XQf(u, t ))f(t)dt} du 1 rz i'z
= rk) LI Ny-'(u, t )Ea,a(XQf(u, t ))Nk-1(z, u)f(t )dudt 1 rz
= ^ Ja f (t)F(k)N¥a+k-1(z, t)Ea,a+k(XQf(z, t))dt = [Z(N¥a+k-1 (z, t)Ea,a+k(XQf(z, t))f (t)dt.
a
2. Equivalent fractional integral equations
□
□
In this section, we present explicit solutions to //-Hilfer fractional differential equations 0.1, 0.2 in the form of Mittag-Leffler function. Moreover, we interduce equivalent fractional integral equation of the problem 0.3-0.4.
Lemma 2.1. Let h G Cn-7//(J, R), X G R, n — 1 < a < n and в G [0,1]. Then, the solution of Cauchy problem 0.1, 0.2 is given by
n t
y(t) = £ CjQr¥J(t, a)Ea,r-j+i XQ/(t, a)] + N¥a-1(t, s)Ea,a [XQ/(t, a)] h(s)ds. (2.1) j=i Ja
Proof. The equivalent fractional integral of the linear Cauchy problem (0.1)-(0.2) is
n QY-j(t a) X rt 1 rt
y(t>=£ rY—тщCJ+ra a nt1(<■ s)y(s)ds+ra a Na^'.'w»*. (22>
For explicit solutions of Eq. (2.2), we use the method of successive approximations, that is, ) £ О"-j(t, a)
yo(t H £ Г7—7Л)Cj, (2.3)
X i'' 1 f
Ук (' )= y0(t ) + r— ja Ny-1(t, s)y-i(t )ds + — J N¥a-\t, s)h(s)ds. (2.4)
By Definition 1.1, Lemma 1.1 together with Eq. (2.3), we obtain
X f' r'
yi(t) = yo(t) + =7— Nva-1(t, s)yo(s)ds + N—-1(', s)h(s)ds
г (a) Ja J a
Г(а)
n 2 2i-1nai+e(n-a)-i(t a) 1 rt
E Cj vj Qy,-, ( ,a\, + N—-l(t,s)h(s)ds. (2.5)
t1 ]h r(ai + Bin- a)- i + 1) r(a) Ja y y '
=1 Jt{ r(ai + p(n - a) - i + 1) r(a) Similarly, using Eqs. (2.3)-(2.5), we get
n 3 Xi-1nai+e(n-a)-i(t a) ft 2 Xi-1
y2(t> = E с E г— n - —) -'i +1) +1 E шума—-1о, s)h(!>ds-
Continuing this process, the expression for yk(t) is given by
n k+1 Xi-1aai+e(n-a)-i(t a) t k Xi-1
yk(t> = E c E т— X - a) -i+'в +1E у'Ш—Л', srnd*.
Taking the limit k ^ то, we obtain the expression for yk(t), that is
n ж Xi-1aai+e(n-a)-i (in) rt ж Xi-1
y(t> = E Ci E r—— +%, - a) -i+'в + a E y (Mf1(tж*..
Changing the summation index in the last expression, i ^ i + 1, we have
n Ж xia"1+r-i(t a) rt Ж XX
y(t) = E Ci EJXay—+ E^—y' (s)ayi+a-1(t, s)h(s)ds. =1 =0 r(ai + Y - i + 1) Jah r(ai + a)^ ;
Using the definition of Mittag-Leffler function, we can obtain (2.1). □
Lemma 2.2. Let Y = a + - a/3 such that a e (1,2), в e [0,1] and f : (a, b] x R ^ R be a continuous function. Then y is a solution of the problem (0.3)-(0.4) if and only if y is a solution of the follwing integral equation
aY-1(t, a)EaY(Xay(t, a)) y(t) =-;;--- x
K
,-b
E Si / Nya+Z-1(ti, s^aa+zXayTi, s))f (s, y(s))ds-i=1 Ja
+
b
- Nya-1(b, s)Ea,a(Xaay(b, s))f (s, y(s))ds
a
+ f Nya-1(t, s)Eaa(Xay(t, s))f (s, y(s))ds, (2.6)
a
where
m
K := ay-1(b, a)EaY(Xay(b, a)) - E Ь^О-1 Т a)Ea,y(Xay(Ti, a)) = 0. (2.7)
1
Proof. In view of Lemma 2.1, the problem (0.3)-(0.4) is equivalent to
y(t) = Qry-1(t, a)Ea,Y(XQy(t, a))c1 + QYy2(t, a)Ea,Y-1(XQy(t, a))c2
+ f Nya-1(t, s)Ea,a(XQ'y(t, s))f(s, y(s))ds, (2.8)
a
where
ci = (lïï dt) ^¥У(а) = D— Vy(a) and c2 = I— *y(a).
By the first condition (y(a) — 0), QY (t, a) — œ, we get C2 — 0 and hence, Eq. (2.8)
reduce to
l't
y(t) — QV'it, a)Ea,y(XQy(t, a))d + N¥a-1(t, s)Ea,a(XQ%(t, s))f(s,y(s))ds. (2.9)
J a
Next, substitute t — t into Eq. (2.9) and multiplying both side of Eq. (2.9) by 8i, we derive that
Siy(Ti) — SQ^Ti, a)Ea,r(XQ^(Ti, a))a + Si f' NJ-1(t{, s)Ea,a(kQ%(Ti, s))f(s, y(s))ds.
a
Thus, we have
m m
£ S4fy(Ti) — C1 £ 84? qY-1 (t, a)Ea,r(XQa¥(Ti, a)) i—1 i—1
m r f Ti
+ £ Siév N¥a-1(Ti, s)Eaa(XQ"(Ti, s))f(s,y(s))ds. (2.10) i—1 Ja
m z i r
From Eqs. (2.9), (2.10) and second condition (y(b) — £ 8iIa+¥y(Ti)), we get
i=1
ci = K
m r f Ti
Zs/af N¥-1(Ti,s)Ea' aXQlT, s))f (s,y(s))ds i=1 Ja rb
- Nva-1(b, s)Eaa(XQl(b, s))f (s,y(s))ds
a
(2.11)
Substitute Eq. (2.11) into Eq. (2.9) and using Lemma 1.5, we obtain Eq. (2.6). Conversely, applying D0+y on both sides of Eq. (2.6) and using the fact DrjQr¥ 1(t, a) = 0, we can easily prove that
HDa/;yy(t ) = Xy(t)+ f (t, y(t)).
Next, take t ^ a in Eq. (2.6), we get y(a) = 0. On the other hand, applying IZ+y on both sides of Eq. (2.6) with taking t ^ Ti, and multiply by Si, we get
E SZy , E" 1 Sil^Ql-1 (Ti, a)Ear(XQy(Ti, a))
£ SiIa+ y(Ti) = k X
m f Ti r
£Si / Nya+Z-1(Ti, s)Eaia+z(XQ<y(ti, s))f(s, y(s))ds—
i=i
b
+
- f Nva-1(b, s)Eaia(XQl(b, s))f(s, y(s))ds
a
m С Ti Z 1
+ £Si (Ti, s)Ea,a+z(XQl(Ti, s))f(s, y(s))ds. (2.12)
i=i Ja
Thus, from Eq. (2.7), we can reduces Eq. (2.12) to
m 1 , .
) = - (Q^it, a)Ea,r(XQav(b, a)) - k) x i=i K v '
m T r
/ N¥a+Z-\Ti, s)Ea,a+zßQ"¥iTi, s))f(s, y(s))ds -i=1 Ja
rb
- NÇ-'ib, s)Ea,a(XQa¥(b, s))f (s, y(s))ds
a
+
+ L^/ s)Ea,a+dXQay(Ti, s))f(s, y(s))ds = y(b). (2.13)
i=l Ja
Thus, the nonlocal boundary conditions of the problem (0.3)-(0.4) are satisfied. □
3. Existence of solution
The existence and uniqueness theorems of solutions to problem (0.3)-( 0.4) are presented in this section. For our analysis, the following assumptions should be valid.
(H\) f : (a, b] x R ^ R is jointly continuous.
(H2) There exist 0 < q < 1 and a real function V € L1 ([a, b],R+) such that |f(t,y)| < V(t) for
q
all t € [a, b] and y € R.
(H3) There exist 0 < q' < 1 and a real function W € Lx ([a, b], R+) such that | f(t,x) - f(t,y)| <
q'
W(t) ^ - y| for all t € [a, b] and x, y € R. For brevity, we set
P in (a + Z - q' Y- q' - m Xna+Z-q'+r-2(T a)
P = 1 - q' ,-Tq-)) £Xn y a),
-=( o (ß №, W■ ))
l-q'
Theorem 3.1. Assume that f : (a, b] x R ^ R is continuous and satisfies (H1)-(H2). Then the problem (0.3)-(04) has at least one solution in C2-y,¥ [a, b].
Proof. Consider the operator U : C2-r¥ [a, b] —> C2-r¥ [a, b] defined as
Qr¥1(t, a)Ea,r(XQy(t, a)) Uy(t) =-^-
K
b
m T r
L8i / N¥a+Z-1(Xi, s)Ea,a+z(XQy(Ti, s))f(s, y(s))ds-i=1 Ja
+
fb
- Na-1(b, s)Ea,a(Xny(b, s))f (s, y(s))ds
J a
+ f N¥a-1(t, s)Ea,a(Xny(t, s))f(s, y(s))ds. (3.1)
a
It is obvious that the operator U is well defined. Define a bounded, closed, convex and nonempty set
H = { y € c2-y, ¥ a b] : ||y||c_r> |} ,
X
of Banach space C2-r,¥ [a, b] with
where
{ H+ B] + c}
\\Li [a,l q
A :=
B :=
C :=
1 - q
1-q
Z lSl Qav+Z-q(Ti, a),
Г(а + Z)\ a + Z - q, г=1 Qav-q(b, a) ( 1 - q\l-q
Г(а)
a - q
Q2¥r+a-q(b, a) ( 1 - q \1-q
Г(а)
a - q
Claim(1). The operator U is continuous in H^. Consider a sequence {yn}c^=1 such that yn —> y in C2-Yy [a, b]. In view of Lemmas 1.3 and 1.5, for t e (a, b], we have
Q2vY(t, a)[Uyn(t) -Uy(t)]
<
Qv(t, a) r(Y)K
|Si|
N
a+Z-1
(тi, s) 1 fyn - fyl ds +
1 i'b
+ ¡a Ja Nva-1(b, s) lfyn - fyl ds
+
<
Qi(t, a)
Z Г(а + Z + Y - 1)Ql (T a) +
ti Г(а + Z)
Q\0)a) ia Na-1 (t, s) l f (s, yn (s)) - f (s, y(s))l ds < B(a, Y - 1)Qa¥+Y-2(b, a) '
t(y)K
x\\ f (■, yn(-)) - f (; y(-))\\2-Y,v
Г(а)
B(a, y - 1)Ql(b, a) + T{a]1 \ f (■, Уп(■)) - f (■,уШ2-ш ,
Qr(Ti, a)
where B(a, Y — 1) is Beta function. As 1 < r < 2, then " < 1, it follows that
Q2 (T , a)
\\UУ -иyn\L
<
<
Qi(t, a) m lSil r(Y- 1) Qa+Z
T(r)K =1 Г(а + Z + Y - 1)
Qa+Z(T,a) +
Qv(t, a) \ B(a, y - 1)Q$(b, a)
r(Y)K
+1
Г(а)
x\\ f (■,ynt)) - f (■,У())\\
2-Yl .
Since f is continuous function and yn ^ y as n ^ to, we have
\\Uy — Uyn\\c2_r,v ^ 0. Thus, the operator U is continuous in H^.
Claim(2). U maps bounded sets into bounded sets in C2-Yy [a, b]. For each y e H^, t e (a, b], by Lemmas 1.3, 1.5 and Holder inequality, we have
<
Q2¥Y(t, a)Uy(t)
Q¥(t, a) r(Y)K
1
<
Nva+Z-1(Ti, s) lf (s, y(s))l ds
=1 Г(а + Z)
+
Г(а)
N"-1(b, s) l f (s, y(s))l ds
+^Toa a Na-1(t, s) l f(s, y(s))l ds
1
X
<
ny(t, a) r(r)K
|X|
+
<
1 r(a + Z)
a Na-1(t sv (s)ds
|X|
Ti 1
Nva+Z-1(Ti, s)V(s)ds +
r(a)
Na-1(b, s)V(s)ds
r(
ny(t, a) r(r)K
1
+
Ny
a+Z-1
+
T(a) nl-r(t, a)
=1 r(a + Z)
{N°-1(b, s))1-1 ds
(Ti, s
1
1-q
ds
1-q
(V (s)) qds
(V (s)) qds
<
r(a
ny(t, a) r(r)K
1
+
t \1-q /, t 1 \ q Na-1(t, s)) 1-qds) (V (s)) qds
|Xi|
Ti a+Z-1 \1-q / r Ti
1-q
<
r(a) nVr(t, a)
r(a) n (t, a)
1 r(a + Z)
a-1
Ny'-q (b, s)ds
Ny 1 (Ti, s)ds
(V (s))1 ds
(V (s)) qds
b a-1 \ 1-q / ft 1 \q'
+
t a-1 1
Nyi-q (t, s)ds
1-q
r(r)K 1
+
|X|
Ti a+Z-q 1 Ny i-q (Ti, s)ds
1-q / rb
(V (s)) qds
1-q / iT
+
r(a) nl-r(t, a)
1 r(a + Z)
b a-q 1
Nyi-q (b, s)ds) U (V(s))qds
1-q / ft
(V (s)) qds
<
r(a
ny(t, a) r(r)K
t a-q_1
NJ- 1
(t, s)ds
1
1 - q
r(a + Z) \a + Z - q
1-q
(V (s)) qds
L X na/Z-q(T, a)
i=1
+
a-q
<
nyq(b, a) ( 1 - q
r(a)
ny(b, a) r(r)K
1-q
Nl 1 [a
+
n2-r+a-q n
(b, a) 1 - q
T(a)
a-q
1-q
L1 [a
(A + B) + 4 ||V||l 1
) q
Thus, U : H^ —> H^, that is UH^ is uniformly bounded.
(3.2)
Claim(3). U maps bounded sets into equicontinuous set of C2-r,y [a, b]. For any y € H^, t1, t2 € [a, b] such that t1 < t2, using Lemmas 1.5 and 1.3, we have
n\ r(t2, a)Uy(t2) - ny 7(t1, a)Uy(t1)
2-ri
<
<
yNC2-r
ny(t2, a)Ear(Xny(t2, a)) - ny(h, a)Ea,r(Xgy(t1, a))
r(r -1)
+
r(a + Z + r - 1) =1 B(a, r- 1)
|K|
m
LSny+Z+r-1(Ti, a) -
B(a, r - 1) r(a)
nay(b, a)
+
r(a)
(nay(t2, a) - ny(t1, a))
b
q
q
q
q
q
q
x
x
By Eq. (1.1) as t1 —> t2, the right-hand side of the preceding inequality is not dependent on y and goes to zero. Hence
Q2 Y(t2, a)Uy(t2) - Q% 7(tu a)Uy(h) ^ 0, V \t2 - 0, y G Hç.
i2-r
(3.3)
From the above claims, together with Arzela-Ascoli theorem, we infer that the operator U is completely continuous. In the remaining part of the proof, we only need to prove that the set
A = {y e C2-Y [a, b] : y = WUy, for some W e (0,,
is bounded set. For each t e (a, b], let y e A, and y = WUy for some W e (0,1). Then l|y||2-7y ^ \Uy\\-y y. Hence, by virtue of claim (2), we obtain
IC2-
< ç.
Thus, the set A is bounded. According to Schaefer's fixed point theorem we deduce that U has a fixed point which is a solution of the problem (0.3)-(0.4). The proof is completed. □
Theorem 3.2. Assume that (Hi)-(H3) hold. If
Q y (b, a) p + Qy "(b, a) a
L i [a,I
T(r)K r(a + Z) r(a)
then the problem (0.3)-(0.4) has a unique solution in C\-r,y [a, b] .
< l,
(3.4)
Proof. In view of Theorem (3.1), we have known that the operator U defined by 3.1 is well defined and continuous. Now, we prove that U is a contraction map on C2-Y,y [a, b] with respect to the norm ||-||C2 y . For each y, y* e C2-Y,y [a, b] and for all t e (a, b] with the help of Lemmas 1.3, 1.5 and Holder inequality, we have
Q2y7(t, a)[Uy(t) -Uy*(t)]
<
Q y (t, a)Eaj(XQy(t, a))
K
ZSi / Nya+Z-1(Ti, s)Eata+z(Q(T, s)) [f(s, y(s)) - f(s, y*(s))] ds i=i Ja
b
A/a-\(u „Mr
a,a (
b
- Nya-1(b, s)Ea,a(XQay(b, s))[f(s, y(s)) - f(s, y* (s))] ds
a
r t
+ Ql-r(t, a) Nya-1(t, s)Ea,a(XQay(b, s))[f(s, y(s)) - f(s, y*(s))] ds
a
^ \St\
< Qy(t, a)
< r(Y)K X
H r(a + Z)
Ny
a+Z-l,
(Ti, s)Qy (s, a)) ds
i \i-q
i-q
(W (s))q ds) yy - y*||
II 2-y, y
+
+
<
l
r(a)
Ql-r(t, a) r(a)
Q y (t, a) l r(r)K r(a + Z)
Nva-1(b, s)Q7y-2(s, a)
l-q'
ds
1-q' / b
(W (s))q ds) ||y - y* 112—
7, y
[ (Nya-1(t, s)Q7y2(s, a)) - ds^ * (jf (W(s)) 7 dsj ||y - <
B
a + Z - q' 7 - q' - l l - q ' ' l - q'
l-q'
ISiQy
,a+Z-q'+y-2
(Ti, a) x
i=l
7, y
X
X
q
X
q
b
(Q¥(t, a) + 1 4 mK +1
Qg-q (b, a) Г(а)
B
a _ q' у _ q' _ 1
1 _ q', 1 _ q'
l-q'
L 1 M \\y _ y* lll-у,¥ <
<
Q¥(t, a) P , ^¥ r(ï)K Г(а + Z)
Qa- (b, a)a' Г(а)
L l [a,l ¥
Wy _ y*W
2-у,
(3.5)
By (3.4), the operator U is a contraction map. According to Banach contraction principle, we conclude that the problem (0.3)-( 0.4) has a unique solution in C2-r,y [aa, b] □
4. Ea-Ulam-Hyers stability
In this section, we discuss the Ea-Ulam-Hyers stability of the problem (0.3).
Lemma 4.1 ([23]). Let a > 0 and x, y be two nonnegative function locally integrable on [a, b]. Assume that g is nonnegative and nondecreasing, and let \¡f £ C1[a, b] an increasing function such that \¡f'(t) = 0 for all t £ [a, b]. If
f t
a—11
x(t) < y(t)+ g(t) Í 1(t, s)x(s)ds, t G [a, b]
a
then
x(t) < y(t) + f £ ^rSr"N;a-1(t, s)y(s)ds, t G [a, b].
Ja n=1 1 (na)
(4.1)
If y be a nondecreasing function on [a, b], then we have
x(t) < y(t)Ea {g(t)r(a)Qy(t, a)} , t e [a, b]. Remark 4.1. A function z e C2-r,y[a, b] satisfies the inequality
HDa/'yz(t) — Xz(t) — f (t, z(t))| < £EaQay(t, a), t e (a, b],
if and only if there exists a function n e C[a, b] such that
(i) \n(t)| < eEaQy(t, a), t e [a, b];
(ii) HDa/'yz(t) = Xz(t)+ f (t, z(t))) + n(t), t e (a, b].
Definition 4.1 ([32]). The problem (0.3) is Ea-Ulam-Hyers stable with respect to EaQy(t, a) if there exists CEa > 0 such that, for each £ > 0 and each z e C2-r,y[a, b] satisfies the inequality (4.1), there exists a solution x e C2-r,y[a, b] of the problem (0.3) with
\\z — x\\2-y,y < CEa£Ea(KQy(t, a)), t e [a, b], k > 0.
Lemma 4.2. Let 1 < a < 2, 0 ^ P ^ 1, if a function z e C2-r,y [a, b] satisfies the inequality (4.1), then z satisfies the following integral inequality
z(t ) _Az _ Í Nva-1(t, s)Ea,a&QY(t, s))f(s, z(s))ds
a
<
< e
1
I SI Q ¥ (Ti, a)Ea,Z+lQ Y(Ti, a) + + lj EaQ Y(b, a)
q
where
QY 1 (t, a)Ea,Y(XQi(t, a))
K
Z Si / 'N¥a+Z-1(Ti, s)Eao+Z(XQav(Ti, s))f (s, z(s))ds i=1 Ja
b
- Nva-1(b, s)Ea,a(XQl(b, s))f (s,z(s))ds
a
y (
Proof. Indeed by Remark 4.1, we have
HDa/'yz(t) = Xz(t) + f (t,z(t)) + n(t), t e (a, b]. By Lemma 2.2, we obtain
QY 1(t, a)Ea,Y(XQl(t, a))
z(t) = Az -
Z Si / Nva+i-1(Ti, s)Eata+Z(XQ%(Ti, s))n(s)ds i=1 Ja
- fbNva-1(b, s)Ea,a(XQav(b, s))n(s)dsj + f Nva-1(t, s)Ea,a(XQav(t, s))f (s, z(s))ds
aa
+ it NO-1 (t, s)Ea,a(XQl(t, s))n(s)ds.
a
It follows from Lemma 1.3 and the fact QY, 1 (t, a) = Qy( ' a) < 1, that
Q ( , a)
z(t) -A - i NO-1(t, s)Ea,a(XQ$(t, s))f(s, z(s))ds
a
<
<
r(Y)K
lSil
=1 Г(а + Z)
£ No+z-1(Ti, s) ln (s) l ds + tO I Nvcl-1(b, s) ln (s) l ds
1t
+ ГО Ja NO-1(t, s) ln(s)l ds <
<
T(Y)K
lSil
- Г(а + Z)
pTi 1 fb
Nva+Z-1(Ti, s)EaQl(s, a)ds + — Nva-1(b, s)EaQav(s, a)ds
a Г( а ) a
£ ft
+ ГС) Ja 'Ny-1(t, s)EaQl(s, a)ds.
By definition of Mittag-Leffler function and Theorem 1.1, we get
z(t) -Az - i NO-1(t, s)Ea,a(XQl(t, s))f(s, z(s))ds
a
<
< £
^ £
i m too
TYK ZlSlQ| (Ti,a) Z
Q 7^'(Ti, a) , ( 1 ^ Qa¥in+1)(b, a)
Г((п + 1)a + Z + 1)
W + V Z г
\(n + 1)a + 1)
<
1
TO Qan
T(Y)K =1 1
Z lSl Ql(Ti, a) Z
QCn(Ti, a) ( 1
+
n=0
r(na + Z + 1) \r(Y)K (1
to Qan
+1 Z
n=0
QO (b, a) r(na
rYKZ S Ql(Ti, a)Ea,Z+1Ql(Ti, a) + щк + ^ EaQa¥(b, a)
□
1
£
£
In the forthcoming theorem, we prove the Ea-Ulam-Hyers stability result for the problem (0.3). For that, the following assumption should be valid.
H4 There exist Lf > 0 such that \ f (t, x) - f (t, y) \ < Lf \x - y\ for all t e [a, b] and x, y e R.
Theorem 4.1. Assume that (H1) and (H4), are satisfied. Then Eq. (0.3) is Ea-Ulam-Hyers stable.
Proof. Let e > 0, z e C2-Y;y [a, b] be a function satisfying the inequality 4.1 and let x e C2-r,y [a, b] be the unique solution of the following problem
( HDaaf'yx(t) = Xx(t) + f (t, x(t)), t e (a, b], \ x(a+) = z(a+), x(b)= z(b). Now, by using Lemma 2.2, we have
x(t) = Az + /' Nya-1 (t, s)Ea,aQy(t, s)f (s, x(s))ds, t e (a, b].
a
Hence, from (H4) and Lemmas 4.2, 1.3, for each t e (a, b], we have
\z(t) - x(t)| <
z(t) -A - i N?-1(t, s)EaiaQ<y(t, s)f (s, z(s))ds
J a
f Nva-1 (t, s)EaaQ"v(t, s) (f (s, z(s)) - f (s, x(s))) ds
a
<
< £
+
f 1
f^X \Si\ Qy(Ti, a)EazZ+1Qy(Ti, a) + + lj EaQay(b, a)
Lf
T(a)
i Nva-1(t, s) \z(t) - x(t)\ ds.
a
(4.2)
Using Lemma 4.1, we obtain \z(t) - x(t)\ <
^ £
YYK £ ^ Qy(T, a)Ea,z+iQy(Ti, a) + + lj EaQy(b, a)
For all t € (a, b], we have
Ea(LfQay(t, a)).
Wz - x||2-r, y < £
£ \Si\ QZy(Ti, a)Eaz+lQ"y(Ti, a) + (+ lj EaQay(b, a)
x Q- 2(t, a)Ea(LfQy(t, a)) 1
< £
1
Take CEa = £ get
YYK X ^ Qy(Ti, a)Ea,z+iQy(Ti, a) + ^^ + V EQy^, a) x Q--2(b, a)Ea(LfQay(t, a)).
1 m r / 1
fYKx ^Qy(Ti, a)EaZ+iQy(Ti, a)+ + 1 ) EaQay(b, a)
Q-- 2(b, a), we
Hz - x^ y < CEaeEa(LfQy (t, a)). Thus, Eq. (0.3) is Ea-Ulam-Hyers stable.
□
x
x
5. An example
In this section, one example is given to illustrate our theory results Example 6.1 Consider the following problem
h - 2et H n 2 ' 3'e
1
e
D0?' y(t) = - 2 y(t) + Y+e sin y(t), r > 0 t € J :=(0,1]., 2 i 1,
310++y(
y(0) = 0, y(1) = -10+ y(-).
(5.1)
3 2 5 1 2 1
Here a = -, ß = 3, - = a + 2ß - aß = -, m = 1, t = -, 81 = -, Z = 2, (a, b] = (0,1],
2 3 6 2 3 2
1 ...... e-rt
y(t) = e'., X = -- and f (t,y(t)) = —- siny(t). Then 2 1 + et
\ f (t, y(t ))\ =
Let z, y € R and t € [0,1]. Then
1 + e
- siny(t)
e-rt
z
1 + e
\ f (t, z(t)) - f (t, y(t ))\ z
1 + el
(sinz(t) - siny(t))
z^ \z(t) - y(t)\
We note that V = W e L± [0,1]. Thus, for t e [0,1] and Choosing suitable q' e (0,1), we
2 q'
can arrive at the following inequality
e - 1 p +(e - 1)2 q o
r( - )K r(2) r( 2)
L_L [0,1] q'
< 1.
Then all the assumptions in Theorem 3.2 are satisfied, the problem (5.1) has a unique solution in C7 e [0,1]. Also we see that the inequality
'Dir y(t) + 1 y(t) -
1 + e
siny(t)
z £e- (et -1) 2, t € [0,1]
is satisfied. For z, x e Ci [0,1], we have
(5.2)
|z - x|2—y Z CE1 £E2
2 (e' - 1) 2
t € [0,1], z, x € Ci [0,1]
where
Ce 1 =
13 --E- -(e2 - 1)2 +
r(6) \K\ 3 ' \T(6)K
Thus the Eq. (5.1) is Ea-Ulam-Hyers stable.
+ 1 E-((e - 1)2)
(e - 1) --2 > 0.
6. Concluding remarks
We can conclude that the main results of this article have been successfully achieved, through some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we have investigated the existence and uniqueness of the solutions of nonlinear Cauchy problem for y-Hilfer fractional differential equation with constant coefficient. Further, we discussed Ea-Ulam-Hyers stability of solutions to such equations in the weighted space C2-%y [a, b].
e
e
—rt
e
1
1
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К теории у-гильферовской нелокальной задачи Коши
Мохаммед А. Альмалахи Сатиш К. Панчал
Кафедра математики Университет доктора Бабасахеба Амбедкара Маратвады
Аурангабад, Индия
Аннотация. В данной статье мы выводим формулу представления решения дробно-дифференциального уравнения у-Гильфера с постоянным коэффициентом в виде функции Миттаг-Леффлера с использованием последовательного приближения Пикара. Более того, используя некоторые свойства функции Миттаг-Леффлера и теоремы о неподвижной точке, такие как Банаха и Шефера, мы вводим новые результаты о некоторых качественных свойствах решения, таких как существование и единственность. Обобщенная лемма о неравенстве Гронуолла используется при анализе устойчивости Еа-Улама-Хайерса. Наконец, дан один пример, иллюстрирующий полученные результаты.
Ключевые слова: дробные дифференциальные уравнения, дробные производные, Еа-устойчи-вость Улама-Хайерса, теорема о неподвижной точке.
