UNIQUENESS AND STABILITY RESULTS FOR CAPUTO FRACTIONAL VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-3-313-325 УДК 517.9

Uniqueness and Stability Results for Caputo Fractional Volterra-Fredholm Integro-Differential Equations

Ahmed A. Hamoud*

Department of Mathematics Taiz University Taiz, Yemen

Received 10.08.2020, received in revised form 10.01.2021, accepted 20.03.2021

Abstract. In this paper, we established some new results concerning the uniqueness and Ulam's stability results of the solutions of iterative nonlinear Volterra-Fredholm integro-differential equations subject to the boundary conditions. The fractional derivatives are considered in the Caputo sense. These new results are obtained by applying the Gronwall-Bellman's inequality and the Banach contraction fixed point theorem. An illustrative example is included to demonstrate the efficiency and reliability of our results.

Keywords: Volterra-Fredholm integro-differential equation, Caputo sense, Gronwall-Bellman's inequality, Banach contraction fixed point theorem.

Citation: A.A. Hamoud, Uniqueness and Stability Results for Caputo Fractional Volterra-Fredholm Integro-Differential Equations, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 313-325. DOI: 10.17516/1997-1397-2021-14-3-313-325.

Introduction

In recent years, there has been a growing interest in the linear and nonlinear integro-differential equations which are a combination of differential and integral equations [3,16,18,21]. The nonlinear integro-differential equations play an important role in many branches of nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, electrostatics, biology, chemistry and economics [13] and signal processing [25].

The challenging work is to find the solution while dealing with Volterra-Fredholm fractional integro-differential equations. Therefore, many researchers have tried their best to use different techniques to find the analytical and numerical solutions of these problems [1,2,4,6-8,10,14,22, 23,29].

The study of iterative differential and integro-differential equations is linked to the wide applications of calculus in mathematical sciences. These equations are vital in the study of infection models. Many papers have dealt with the existence, uniqueness and other properties of solutions of special forms of the iterative differential equations and integro-differential equations [12,15,16,19,20].

Recently, Cheng et al. (2009), in [5,20] investigated analytic and exact solutions of an iterative functional differential equation

u' (x) = f (x,u(h(x) + g(u(x)))),

u(xo) = Xq.

*drahmedselwi985<8hotmail.com https://orcid.org/0000-0002-8877-7337 © Siberian Federal University. All rights reserved

Lauran (2011) [19], investigated the existence and uniqueness results for first order differential and iterative differential equations with deviating argument of the type

u'(t) = f (t,u(t),u(u(t)),u(Xu(t))), u(to) = xo.

In [15], Ibrahim (2013) investigated the existence and uniqueness of solution for iterative differential equations of the type

Dau(t) = f (t,u(u(t))), u(0) = uo.

Kendre et al. (2015), [16] investigated the existence of solution for iterative integro-differential equations of the type

u'(t) = f (t,u(u(t)), k(t, s)u(u(s))ds),

J to

u(to) = xo.

Unhale and Kendre (2019), in [28] established the existence and uniqueness of solution for iterative integro-differential equations of the type

Dau(t) = f(t)+ i h(t,s)u(Xu(s))ds,

o

u(0) = uo.

Motivated by these problems, in this paper, we discuss new uniqueness and stability results for nonlinear fractional Volterra-Fredholm integro-differential equation with deviating argument of the type

'au(x) = f (x) +

oo with the boundary condition

r /

Dau(x) = f (x)+ h(x, s)u(u(s))ds + / k(x, s)u(u(s))ds, x,s £ J := [0, T], (1)

oo

on

au(0) + bu(T) = c, a, b, c £ R, a + b = 0, (2)

where Da(.), 0 < a < 1, is the Caputo fractional derivative, f (t), h(x,s) and k(x,s) are given continuous functions, u(x) is the unknown function to be determined.

The main objective of the present paper is to study the new uniqueness and stability results for iterative nonlinear fractional Volterra-Fredholm integro-differential equation with deviating argument.

The rest of the paper is organized as follows: In Section 1, some essential notations, definitions and Lemmas related to fractional calculus are recalled. In Section 2, the new uniqueness and stability results of the solution for nonlinear fractional Volterra-Fredholm integro-differential equation have been proved. In Section 3, we investigate the Ulam-Hyers stability and generalized Ulam-Hyers stability for the problem (1)-(2). In Section 4, focuses on an example to illustrate the theory. Finally, we will give a report on our paper and a brief conclusion.

1. Preliminaries

The mathematical definitions of fractional derivative and fractional integration are the subject of several different approaches. The most frequently used definitions of the fractional calculus involves the Riemann-Liouville fractional derivative, Caputo derivative, etc. The following observations are taken from [7,9-11,17,18,24,26,29].

Definition 1.1 ([16]). The Riemann-Liouville fractional integral of order a > 0 of a function f is defined as

1 rx

J°f (x) = t^ (x - t)a-1f (t)dt, x> 0, a e R+,

r(a) Jo (3)

J0f (x) = f (x),

where R+ is the set of positive real numbers.

Definition 1.2 ([16]). The Riemann-Liouville derivative of order a with the lower limit zero for a function f : [0,1) —>• R can be written as

Df= m-a dX [xxBrdt x>00<a<1 (4)

Definition 1.3 ([24]). The Caputo derivative of order a for a function f : [0,1) —>• R can be written as

1 rx f'(t)

Da f (x) = —-- / ( ) dt, x> 0, 0 < a < 1.

' r(1 - a) Jo (x - t)a ' '

Definition 1.4 ([26]). The fractional derivative of f (x) in the Caputo sense is defined by

cDaf (x) = Jn-aDnf (x) =

1 (x - t)n-a-1 ^^^dt, n - 1 < a < n,

V ' Jj-n ' '

r(n — a) Jo dt

dnf(x)

(5)

dxn ' '

where the parameter a is the order of the derivative, in general it is real or even complex.

Definition 1.5 ([26]). The Riemann-Liouville fractional derivative of order a > 0 is normally defined as

Daf (x) = Dm Jm-af (x), m — 1 < a < m. (6)

Lemma 1.1 ([24], Gronwall-Bellman's Inequality). Let u(x) and f (x) be nonnegative continuous functions defined on J =[a,a + h] and c be a nonnegative constant. If

then

f X

u(x) ^ c + f (s)u(s)ds, x £ J, J a

u(x) ^ c exp ^ J f (s)ds^j , x £ J.

Theorem 1.1 ([26], Banach contraction principle). Let (X,d) be a complete metric space, then each contraction mapping T : X —>• X has a unique fixed point x of T in X i.e. Tx = x.

2. Main results

In this section, we shall give an existence and uniqueness results of Eq. (1), with the boundary condition (2). Let B = C(J, J) be the Banach space equipped with the norm

||u|| = maxx£[oiT] \u(x)\. For convenience, we are listing the following hypotheses used in our further discussion: (A1) There exist two constants and jk such that

Ph = sup{\h(t, s)\ : 0 < s < t < T}. Pk = sup{|fc(t, s)\ :0 < s < t < T}.

(A2) There exists a constant M > 0 such that

\u(t1) — u(t2)\ < M\t1 — t2\a, for u G B, t1,t2 G J, t1 < t2.

(A3) There exists a constant L > 0 such that L = sup{\f (t)\ : 0 < t < T}.

(A4) Let p :=

T a(L + T + Pk)) r(a + 1)

1 +

\b\

\a + b\

+

M

\a + b\

< T < M.

Lemma 2.1. If a function u £ C[0,T] satisfies (1)-(2) in the closed interval [0, T], then the problems (1)-(2) are equivalent to the problem of finding a continuous solution of the integral equation

u(x) = i (—— -(f (t) + i h(t, s)u(u(s))ds + f k(t, s)u(u(s))ds^)dt —

J 0 r(a) v Jo J 0 1

a + b

b(T — t)

1

r(a)

f (t) + j h(t, s)u(u(s))ds + j k(t, s)u(u(s))ds^Jdt — c

Theorem 2.1. Suppose that the hypotheses (A1)-(A4) are satisfied and

T a+1(ph + Pk )(M + 1)f1+ \b\

r(a +1)

\a + b\

< 1.

Then there is a unique solution to the problems (1)-(2).

Proof. Let B(p) = {u £ B : 0 < u < p, \u(t1) - u(t2)\ < M\ti - t2\a}.

To apply Banach contraction principle, we define an operator ^ : B(p) —> B(p) by

($u)(x)

rx (x — t)a-1 o r(a)

f (t) + J h(t, s)u(u(s))ds + j k(t, s)u(u(s))ds^Jdt—

a+b

1b(T — t)

1

0

r(a)

f(t) + 0 h(t, s)u(u(s))ds + 0 k(t, s)u(u(s))ds dt — c

1

0

1

So, we have 0 < |^u| =

(x -1)

a— 1

h(t, s)u(u(s))ds + k(t, s)u(u(s))ds)dt-

o r(a) v Jo Jo 1

" T b(T - t)a—1 ( ft fT \

———--if (t) + h(t, s)u(u(s))ds + k(t, s)u(u(s))ds)dt - c

r(a) v Jo Jo 1

T

1

a + b

I o

<

' (x -1)

a—1

+

Jo r(a) J0

1 [T lbl(T - t)a—1 la + blJo Via)

o Jo

+ [ |h(t, s)||u(u(s))|ds + / |k(t, s)||u(u(s))|ds)dt+

<

/0

+ J |h(t, s)||u(u(s))|ds + J k(t, s)||u(u(s))|dsjdt+

+

<

|a + b|

<

x

(x -1)

+

Jo r(a)

c

<

—1 1 — (L + (¡3h + pk )T 3)ds +

' m(T - t)a—1

<

a+b

T a(L + T 3(ph + Pk))

r(a +1)

1 +

|b|

a+bi

+

a+b

a+b|

T(a)

-(L +(ph + Pk)T 3)dt+

-P.

Also, for each 0 ^ x1 ^ x2 ^ T, we have

\^u(x2) - ^u(x1 )| <

rxi (x2 - t)a—1 - (X1 - t)a — 1 lo r(a)

rx2 (x t)a—1 ( rt pT ,

' v ' f (t) + j h(t, s)u(u(s))ds + J k(t, s)u(u(s))dsjdt

<

+

+ 0 h(t, s)u(u(s))ds + 0 k(t, s)u(u(s))ds dt

+

r(a)

<

<

1

rx 1

r(a) Jo

(X1 - t)a—1 - (X2 - t)

a-1

t

+ |h(t,s)| |u(u(s))|ds +

/0

ia— 1

+ j0 |k(t,s)||u(u(s))|ds)dt + ^0-) jj (X2 - ta1 ^f (t)| + ^ |h(t,s)||u(u(s))|ds +

+ J k(t, s)| |u(u(s))|ds)dt.

Hence,

^u^) - ^u^x^ <

T(a)

(X1 - t)a—1 - (X2 - t)a—1 [L + T3(Ph + Pk)]dt +

1 ix2

+ (X2 - t)a—1[L + T3(ph + Pk)]dt <

r(a) Jxi

<

<

r(aK xi

[L + T 3(ph + Pk)] r(a +1)

2[L + T 3(ph + Pk)] r(a + 1)

xa - xa + 2(x2 - x1)a X - X1 |a.

<

o

1

0

This shows that ^ maps from B(p) —> B(p). Now, for all u,v £ B(p), we have \^u(x) - ^v(x)\ <

< —^ [ (x - t)a-1( i \h(t, s)\\u(u(s)) - v(v(s))\ds +[ \k(t,s)\\u(u(s)) - v(v(s))\ds]dt+

r(a) Jo \Jo Jo 1

+ \a + lb\r(a) .[ (T - t)a-1(fQ h(t, s)\\u(u(s)) - v(v(s))\ds+ ,-T

+ J^ \k(t,s)\ \u(u(s)) — v(v(s))\d^jdt <

< (Phr+aPk) £ (— — t)a-1( fo \u(u(s)) — u(v(s))\ + \u(v(s)) — v(v(s))\ds+

+ J \u(u(s)) — u(v(s)) \ + \u(v(s)) — v(v(s))\d^jdt+

+ mttpkUT (T — t)a-1(f4 \u(u(s)) — u(v(s))\ + \u(v(s)) — v(v(s))\ds+ \a + b\i(a) Jo \ Jo

+ J \u(u(s)) — u(v(s))\ + \u(v(s)) — v(v(s))\d^jdt <

J (x — t)a-1(Jo (M \u(s) — v(s)\ + \u(s) — v(s)\)d^jdt+

< (Pl + Pk ) I (x — tr-1

r(a) Jo x J o

+ ^mk) £ (T — t)a-1U!(M \u(s) — v(s)\ + \u(s) — v(s)\)ds)dt <

^ (phr + l3k) J (x — t)a-1^j^ ((M +1) \u(s) — v(s)\)ds^dt+

o yJo

+ \l+mo) £ (T — t)a-1( /'(M + 1) Ks) — v(s)!)ds)dt <

r(a)

b\(Pf

a + b\r(a) Jo V Jo

^ T(Ph + Pk)(M +1) fx( 1 + \b\T(Ph + Pk)(M + 1) fT( 1 ^

^ -f^j- № — v\\ (x — t) dt + -1 , ,|p/ N- № — v\\ (T — t) dt ^

1(a) Jo \a + b\l(a) Jo

< Ta+1 (Ph + Pk)(M + 1) + \b\TTa+1(Ph + Pk)(M + 1) ^

^ --\u — v\ +-a + b\r(a + 1)— \u — v\ ^

Ta+1(Ph + Pk )(M + 1) fx+ \b\

\ u — v\ .

r(a + 1) V \a + b\.

Since

\Ta+1(Ph + Pk)(m + 1) (i + jb^j < 1

<

r(a +1) V \a + b\,

by the Banach contraction principle, ^ has a unique fixed point. This means that the problems (1)-(2) has unique solution. □

The above theorem shows that there exists a unique solution to the problems (1)-(2). However, it does not tell us how to find this solution. To find the solution of the problems (1)-(2), we will define the following sequence

rx (x — t)a-1 ( f* fT )

un+1 (x) = J —ra-f (t) + J h(t,s)un(un(s))ds + J k(t,s)un(un(s))dsjdt —

1

a+b

io r(a) Jo Jo

rT b(T — t)a-1 ( f * fT )

-r(0)-f (t) + J h(t,s)un(un(s))ds + j k(t,s)un(un(s))dsjdt — c

where n = 0,1,2,... and u0(x) is fixed functions of the class C1 mapping [0, T] —> [0, T] such that |u0(x)| < T. For this, we have the following theorem.

Theorem 2.2. If the assumptions of the Theorem 2.1 are satisfied then the sequences defined in (7) converges uniformly to the unique solution of the problems (1)-(2).

Proof. Let Uk = maxxeJ [uk(x) - uk—1(x)Y Then (x - t)a—1

U1 = max |ul(x) - uo(x)| =

xEJ

max

xeJ

1

a+ b

f (t) + h(t,s)uo(uo(s))ds + k(t,s)uo(uo(s))ds)dt-0 r(a) v Jo Jo '

T b(T- t)a—1f ft fT \

f(t)+ h(t, s)uo(uo(s))ds + k(t, s)uo(uo(s))ds \dt- c

|c

r(a)

- uo(x)

<

^ Ta(L + T3(Ph + Pk))f1 + b

r(a +1)

+

|a + b| |a + b|

<

< T.

Since uo : [0,T] —> [0,T], we have U1 < T. U2 = max |u2(x) - Ul(x)| =

xEJ

max

xeJ

(x -1)

a—1

. . . f(t)+ h(t, s)u1(u1(s))ds + k(t, s)u1(u1(s))ds)dt-

'0 r(a) v Jo Jo 1

rT b(T - t)a—1 ( ft fT \

-——r-[f(t)+ h(t, s)u1(u1(s))ds + k(t, s)u1(u1(s))ds)dt - c

Jo r(a) v Jo Jo 1

a—1 - ft cT

T

a+b

(x -1)

a+b

^ max

x^J I Jo

.f (t)+ h(t, s)uo(uo(s))ds + k(t,s)uo(uo(s))ds)dt-

r(a) v Jo Jo 1

rT b(T - t)a—1 ( ft fT \

' -^^-[f(t)+ h(t,s)uo(uo(s))ds + k(t,s)uo(uo(s))ds)dt - c

0 r(a) v Jo Jo 1

fx (x - t)a—1 '

dt

<

tv \ (t)| + Ut,s)Uu1(u1(s)) - uo(uo(s))|ds+ r(a) v Jo

fT m(T - t)a—1

+ j |k(t, s)||u1 (u1(s)) - uo(uo(s))|ds^Jdt -

a+b

o

r(a)

x(lf (t)| + J |h(t, s)| |u1 (u1(s)) - uo(uo(s))|ds + J |k(t, s)| |u1(u1(s)) - uo(uo(s))|dsjdt j <

< TU1 < T2.

Assume that result is true for n i.e. Un < TUn—1 < Tn. Now, we show that result holds for n + 1

Un+1 = max |un+l(x) - u^x^ =

xEJ

= max

xeJ

x

(x -1)'

1

a + b

I s (f(t)+ h(t,s)u1(un(s))ds + k(t,s)un(un(s))ds)dt-

lo r(a) v Jo Jo 1

rT b(T - t)a—1 ( ft fT \

--If (t)+ h(t,s)un(un(s))ds + k(t,s)un(un(s))ds)dt - c

Jo r(a) v Jo Jo 1

o

1

x

o

1

1

(x — t)

a—1

1

a + b f-T

f (t) + J h(t, s)un-i(un-i(s))ds + J k(t,s)un-i(un-i(s))ds^dt-

'T b(T — t)a-i f if'

f (t) + J h(t,s)un-i(un-i(s))ds+

r(a)

+ J k(t,s)un-i(un-i(s))ds^Jdt — c dt^j ^

(x — t)a-i f fl

\f (t)| + |h(t, s)| |u„(u„(s)) — un-i(un-i(s))\ds+ Jo

^ max

l Jo

T

r(a)

m(T — t)

+ [ ^(t^^^n (un(s)) — un-i (un-i (s^ds^dt — -—i— /

Jo 1 ^ + b Jo r(a)

X (\f (t)| + J ht, s^^n^n^)) — un-i(un-i(s))ds+ + J ^(t^^^n (un(s)) — un-i (un-i (s))|ds)dtj <

a-i

< TUn < Tn+i.

Thus by induction, we have Uk < TSince

T a(L + T 3(ph + fa)) r(a +1)

1 +

|b|

a+b\_

+

a+b

< T < 1.

Hence Uk tends to zero as k tends to infinity. Since the family {Uk} is the Arzela-Ascoli family thus for every subsequence {ukj} of {Uk} there exists a subsequence {ukj} uniformly convergent and the limit needs to be a solution of the problem (1)-(2). Thus, the sequence {Uk} tends uniformly to the unique solution of the problem (1)-(2). □

3. Stability results

In this section, we investigate the Ulam-Hyers stability and generalized Ulam-Hyers stability for the problem (1)-(2).

Definition 3.1 ([27]). The Eq. (1) is Ulam-Hyers stable if there exists a real number Q > 0 such that for each e > 0 and for each solution v G Ci(J,, J) of the inequality

f x rT

Dav(x) — f (x) — h(x, s)v(v(s))ds — k(x, s)v(v(s))ds

oo

there exists a solution u G Ci(J,J) of Eq. (1) with

^(x) — u(x^ ^ Qe.

^ e, x G J,

(7)

(8)

Definition 3.2 ( [27]). The Eq. (1) is generalized Ulam-Hyers stable if there exists © G C(R+,R+), ©(0) = 0 such that for each e > 0 and for each solution v G Ci(J,J) of the

inequality

f x fT

Dav(x) — f (x) — h(x, s)v(v(s))ds — k(x, s)v(v(s))ds

oo

^ e, x G J,

(9)

there exists a solution u G Ci(J,J) of Eq. (1) with

^(x) — u(x^ < ©(e).

(10)

x

o

x

c

Theorem 3.1. If the assumptions of the Theorem 2.1 are satisfied, then the problem (1)-(2) is Ulam-Hyers stable.

Proof. Let e > 0 and let function v e C 1(J,J) which satisfies the inequality

r r

Dav(x) - f (x) - h(x, s)v(v(s))ds - k(x, s)v(v(s))ds ^ e, (11)

oo

and let u e C(J, J) be the unique solution of the following problem

r l'T

Dau(x) = f (x)+ h(x, s)u(u(s))ds + k(x, s)u(u(s))ds,

oo u(0) = v(0), u(T )= v(T).

from Lemma 2.1, we obtain

u(x) = i (—- \-(f (t) + i h(t, s)u(u(s))ds + f k(t, s)u(u(s))ds^)dt -

Jo r(a) V Jo Jo 1

fT b(T - t)a—1 - rt rT

a + b

+ j h(t, s)u(u(s))ds + j k(t, s)u(u(s))ds^Jdt — c

/ (x — t)

Au + ' K '

0 r(a) Jo Jo

x { +\a-1 . r t r T

Jo r(a)

where

f (t) + j h(t, s)u(u(s))ds + j k(t, s)u(u(s))ds^Jdt,

1

Au = ~rb

a + b

T urn 1 , ft i-T

c

b(T — t)a

+ h(t, s)u(u(s))ds + k(t, s)u(u(s))ds)dt

o r(a) v Jo Jo

Let A

1 r iT b(T — t)a-1 ' ^ rT

a + b

c — -f (t)+ h(t,s)v(v(s))ds + k(t,s)v(v(s))ds^dt

o r(a) ^ Jo Jo

On the other hand, if u(0) = v(0), u(T) = v(T), then Au = Av and

u(x) = Av + J -f (t) + J h(t, s)u(u(s))ds + J k(t, s)u(u(s))ds^Jdt.

From inequality (11) we have

-e < Dav(x) - f (x) -I h(x,s)v(v(s))ds -I k(x, s)v(v(s))ds < e. (12)

oo

If we integrate each term of the above inequality and appling the boundary conditions, then we have

v(x) — Av — J ^ f(t) + J h(t, s)v(v(s))ds + J k(t,s)v(v(s))ds^dt ■

For any x G J, we have \v(x) — u(x)\ ^

<

(x) — Av — J ^Tt)— f (t)+ J h(t, s)v(v(s))ds + J k(t,s)v(v(s))ds)dt

v(x) — Av — I ———- f (t)+ h(t, s)v(v(s))ds + k(t,s)v(v

r(a) Jo

+

+

<

fx (x _ t)a-1 f rT \

' ---!-- h(t, s)\v(v(s)) - u(u(s))\ds + k(t, s)\v(v(s)) - u(u(s))\ds)dt <

o r(a) \J° J° /

rx (x _ t)a-1 / r t + 1 ( Ißh [\v(v(s)) - v(u(s))\ + \v(u(s)) - u(u(s))\]ds +

r(a +l~) Jo r(a) y jq +ßk [\v(v(s)) - v(u(s))\ + \v(u(s)) - u(u(s))\]ds)dt <

eT -- +

a ßh

r(a + ^ r(a) jo jq

x f-T

1

i f (x - t)a-1(M + 1)\v(s) - u(s)\dsdt +

Jo Jo

+ i i (x - t)a-1(M + 1)\v(s) - u(s)\dsdt <

r(a) J o Jo

eT a

<

+

T a(ßh + ßk )(M +1)

r(a + 1) r(a +1)

Using Gronwall's inequality, we get

eT a

v(x) - u(x)

<

r(a + 1)

1+

\v(s) - u(s)\ds.

ÇT a(ßh + ßk )(M + 1)

r(a + 1)

Qe,

where £ = £(a) a constant, which completes the proof.

Moreover, if we set ©(e) = Qe, ©(0) = 0, then boundary value problem (1)-(2) is generalized Ulam-Hyers stable. □

4. An example

We consider the nonlinear iterative fractional integro-differential equation (1)-(2) with a = 0.5, T = 0.5, L = 0.2, M = 0.4, /3h = /3k =0.5, a = b = 1, and c = 0. New, we have

|c| " " ^

T"(L + T3(ßh + ßk)) (x+ \b\

r(a +1)

+

0.5° (0.2 + 0.53(0.5 + 0.5)) f 1

\a + b\J \a + b\

r(0.5 + 1)

0.2298098, N (1.5)

1+2)+0

Also,

T a+1(M + 1)(ßh + ßk ) r(a + 1)

1+

\b\

\a + b\

r(1.5) _ 0.3447145 _ 0.886227 _ 0.38897 < 0.5 _ T.

0.5°'5+1(0.4 + 1)(0.5 + 0.5) f 1 r(0.5+ 1) V +2

_ 0.494975

_ 0.886227

_ 0.8378

< 1.

(1.5)

Since all the hypotheses of Theorem 2.1 are fulfilled, then there exists a unique solution of the given equation.

o

x

o

Conclusion

The main purpose of this paper was to present new existence and uniqueness results as well as the Ulam-Hyers stability and generalized Ulam-Hyers stability results of the solution for Caputo fractional iterative Volterra-Fredholm integro-differential. The techniques used to prove our results are a variety of tools such as the Gronwall-Bellman's inequality, some properties of fractional calculus and the Banach contraction fixed point theorem. Moreover, the results of references [15,16,28] appear as a special case of our results.

The author would like to thank the referees for their valuable suggestions and comments that improved this paper.

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Результаты единственности и устойчивости для Капуто дробных интегро-дифференциальных уравнений Вольтерра-Фредгольма

Ахмед А. Хамуд

Кафедра математики Университет Таиз Таиз, Йемен

Аннотация. В этой статье мы установили некоторые новые результаты, касающиеся единственности и устойчивости Улама решений итерационных нелинейных интегро-дифференциальных уравнений Вольтерра-Фредгольма с граничными условиями. Дробные производные рассматриваются в смысле Капуто. Эти новые результаты получены путем применения неравенства Гронуолла-Беллмана и теоремы Банаха о сжатии неподвижной точки. Включен наглядный пример, чтобы продемонстрировать эффективность и надежность результатов.

Ключевые слова: интегро-дифференциальное уравнение Вольтерра-Фредгольма, смысл Капуто, неравенство Гронуолла-Беллмана, теорема Банаха о сжатии неподвижной точки.