On Solution of Nonlinear Abel–Volterra Integral Equation

Journal of Mathematical Analysis and Applications 229, 41]60 Ž1999. Article ID jmaa.1998.6139, available online at http:rrwww.idealibrary.com on

On Solution of Nonlinear Abel]Volterra Integral Equation Anatoly A. Kilbas Department of Mathematics and Mechanics, Belarusian State Uni¨ ersity, Minsk 220050, Belarus

and Megumi Saigo Department of Applied Mathematics, Fukuoka Uni¨ ersity, Fukuoka 814-80, Japan Submitted by Mimmo Iannelli Received March 28, 1996

This paper is devoted to studying the nonlinear Abel-Volterra integral equation of the form

w mŽ x. s

aŽ x . GŽ a .

wŽt.

x

H0

dt q f Ž x .

Ž x y t . 1y a

Ž0 - x - d F `.

with a ) 0 and m g R Ž m / 0, y1, y2, . . . .. The uniqueness of the solution is proved and the asymptotic behavior of w Ž x . as x ª 0 is obtained, provided that aŽ x . and f Ž x . have special power asymptotics near zero. The solution w Ž x . in closed form is given in some cases. Q 1999 Academic Press

1. INTRODUCTION The nonlinear Volterra integral equation

w mŽ x. s

aŽ x .

H GŽ a . 0

x

wŽ t.

Ž x y t.

1y a

dt q f Ž x .

Ž x ) 0.

Ž 1.1.

with a ) 0 and m ) 0 Ž m / 1. arises in heat theory w14x, in the nonlinear theory of wave propagation w17, 25x, and in water percolation w9, 22x. The equation is also given the generic name Abel’s type integral equation w11x, 41 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.

42

KILBAS AND SAIGO

and, so, it is called the Abel]Volterra integral equation w24x. Equation Ž1.1. and the similar equation of the form

w m Ž x . s aŽ x .

x

H0 k Ž x y t . r Ž t . dt q f Ž x .

Ž x ) 0.

Ž 1.2.

with the convolution kernel k Ž x y t . and m ) 0 were studied in w1, 4, 6, 15, 20, 21, 22x for aŽ x . s 1 and in w2, 3, 5, 7x for the general case. It is proved that the solvability of such equations is different in the cases m ) 1 and 0 - m - 1. In the former case the corresponding homogeneous equation w f Ž x . ' 0x can have a nontrivial solution, which was first proved in w25x, and the papers above were devoted to investigating problems concerning the existence and the uniqueness of a solution w Ž x . for the nonhomogeneous equation Ž1.2., the stability of such a solution, and the method of successive approximation to find its solution. The case 0 - m 1 is less studied. Some results were given in w1x, w4x, and w15x about the uniqueness of a solution for Eq. Ž1.2. with aŽ x . s 1 and a continuous kernel k Ž u. in the spaces of continuous and summable functions. The existence and the uniqueness for the homogeneous equation Ž1.2. with aŽ x . s 1, m - 0, and nonincreasing kernel k Ž u. in the class of almost decreasing functions was studied in w16x. Further, the same problem for Eq. Ž1.1. with the bounded function aŽ x . and m g R Ž m / 0, y1, y2, . . . . was investigated in w18x, where R is the real number field. The problems to solve Eqs. Ž1.1. and Ž1.2. in closed form and to find asymptotics of their solutions near zero and infinity, provided that such solutions exist, are also of importance. The asymptotic behavior of the solution w Ž x ., as x ª 0 and x ª `, of the Abel]Volterra integral equation of the form

wŽ x. s

1

H GŽ a . 0

x

f Ž t. y wmŽ t.

Ž x y t.

1y a

dt

Ž x ) 0.

Ž 1.3.

with a ) 0 and m ) 1 was studied in w14x and w23x for a s 1r2 and in w10x for any a ) 0 in the cases when f Ž t . has the general power asymptotics near zero and infinity Žsee also w12x., and several first terms of asymptotics of w Ž x . were found. It should be noted that the asymptotic behavior of solutions of nonlinear Volterra integral equations more general than Ž1.1. ] Ž1.3. was investigated by many authors Žsee results and bibliography in w13, Sects. 15 and 17]20x., but most of the results give only the first asymptotic terms of the solutions.

ABEL ] VOLTERRA INTEGRAL EQUATION

43

Our paper deals with the integral equation of the form

w mŽ x. s

aŽ x .

x

H GŽ a . 0

wŽ t.

Ž x y t.

1y a

dt q f Ž x .

Ž 0 - x - d F ` . Ž 1.4.

for a ) 0 and m g R Ž m / 0, y1, y2, . . . .. In Section 2 on the basis of the generalized Gronwall lemma we discuss the uniqueness of the solution w Ž x . of Eq. Ž1.4. in the spaces of continuous and measurable locally bounded functions. Section 3 is devoted to studying the asymptotic behavior of w Ž x ., as x ª 0, provided that aŽ x . and f Ž x . have the asymptotics `

aŽ x . ; x a p m

Ý

ak x a k

Ž x ª 0.

Ž 1.5.

fk x a k

Ž x ª 0.

Ž 1.6.

ksyl

with ayl / 0 and `

f Ž x . ; x a pm

Ý ksyn

with fyn / 0, where p g R and l, n g Z with Z being the set of all integers. We will show that under certain assumptions the solution w Ž x . of Eq. Ž1.4. has the asymptotic expansion

wŽ x. ;

`

Ý wk x a k

Ž x ª 0.

Ž 1.7.

kss

with s g Z Ž s ) y1ra ., where the coefficients w k are expressed in terms of a k and f k . In Section 4 we consider the simplest form of Eq. Ž1.4. with monomial aŽ x . s ax a pŽ myl . and quasimonomial f Ž x ., and find several first terms of the asymptotics of the solution w Ž x .. The obtained asymptotics allow us to solve in closed form Eq. Ž1.4. with the monomial power functions aŽ x . and f Ž x . and the corresponding homogeneous equation. We give such explicit solutions in Section 5.

2. UNIQUENESS OF THE SOLUTION For 0 - d F ` we denote by C Ž0, d . the space of continuous functions on Ž0, d . and by Bloc Ž0, d . the space of measurable and locally bounded Ž . Ž . Ž . functions. Let CqŽ 0, d . and Bq loc 0, d be subspaces of C 0, d and B loc 0, d Ž . consisting of nonnegative functions, and let C«qŽ 0, d . and Bq 0, d be loc, « Ž . Ž . subspaces of CqŽ 0, d . and Bq loc 0, d consisting of functions g x for which there exists a constant « ) 0 such that g Ž x . G « for all x g Ž0, d .. The

44

KILBAS AND SAIGO

following imbeddings are valid: C Ž 0, d . ; Bloc Ž 0, d . ,

Cq Ž 0, d . ; Bq loc Ž 0, d . ,

C«q Ž 0, d . ; Bq loc, « Ž 0, d . .

Ž 2.1.

a Let I0q w be the Riemann]Liouville fractional integral w24, Sect. 2.1x a w. Ž x. s Ž I0q

1

wŽ t.

x

H GŽ a . 0

Ž x y t.

1y a

Ž a ) 0, x ) 0 . , Ž 2.2.

dt

and let Ea Ž z . be the Mittag]Leffler function w8, Chap. 18.1x Ea Ž z . s

`

zk

Ý G Ž a k q 1.

Ž a ) 0, z g C . ,

Ž 2.3.

ks0

where C is the complex number field. The following assertion can be proved directly. Let a ) 0, A G 0, D ) 0, and d ) 0. If F Ž x . g Bloc Ž0, d .

LEMMA 1. and

a 0 F F Ž x . F A q D Ž I0q F. Ž x.

Ž 0 - x - d F `. ,

Ž 2.4.

Ž x ) 0. .

Ž 2.5.

then F Ž x . F AEa Ž Dx 1r a .

Remark 1. In particular, since E1Ž x . s e for a s 1, the latter relation takes the form z

F Ž x . F Ae D x ,

Ž 2.6.

which is the classical Gronwall inequality. Therefore by analogy with w13, Sect. 10.2.15x we call Ž2.5. the generalized Gronwall inequality. Lemma 1 allows us to investigate the uniqueness of a solution of Eq. Ž1.4.. THEOREM 1.

Let a ) 0, 0 - m F 1, and 0 - d F `.

Ži. If aŽ x ., f Ž x . g Bloc Ž0, d . and Eq. Ž1.4. has a solution in the space Bloc Ž0, d ., then the solution is unique. Žii. If aŽ x ., f Ž x . g C Ž0, d . and Eq. Ž1.4. has a solution in the space C Ž0, d ., then the solution is unique. Proof. Making the change of f Ž x . s w m Ž x . and setting m s 1rm Ž m G 1., we rewrite Eq. Ž1.4. in the form

fŽ x. s

aŽ x .

H GŽ a . 0

x

f mŽ t.

Ž x y t.

1y a

dt q f Ž x .

Ž 0 - x - d F ` . . Ž 2.7.

ABEL ] VOLTERRA INTEGRAL EQUATION

45

Since the mapping w Ž x . ¬ w m Ž x . is a homeomorphism in the spaces Bloc Ž0, d . and C Ž0, d ., Eqs. Ž1.4. and Ž2.7. are simultaneously solved in these spaces and their solutions are connected by f Ž x . s w m Ž x .. Ži. Let aŽ x ., f Ž x . g Bloc Ž0, d .. We suppose that Eq. Ž2.7. has two solutions f 1Ž x . and f 2 Ž x . in the space Bloc Ž0, d .. Then for any segment w d1 , d 2 x ; Ž0, d . there exist numbers K ) 0 and L ) 0 such that

f 1 Ž x . F L,

aŽ x . F K ,

f2 Ž x . F L

Ž 0 F d1 F x F d 2 - d . . Ž 2.8. As easily found, the estimate x m y y m F m x my1 Ž x y y .

Ž x ) y G 0, m G 1 .

Ž 2.9.

follows from the mean value theorem. From Ž2.7. in accordance with Ž2.2., Ž2.8., and Ž2.9. we have

f 1Ž x . y f 2 Ž x . F

aŽ x . GŽ a . =

x

H0 m max

f 1Ž t . y f 2 Ž t .

Ž x y t.

1y a

f 1Ž t . , f 2 Ž t .

4

my1

dt

a F K m L my1 Ž I0q f 1Ž t . y f 2 Ž t . . Ž x . .

Ž 2.10.

The function F Ž x . s < f 1Ž x . y f 2 Ž x .< satisfies the conditions of Lemma 1 with A s 0 and D s K m L my1, hence we have F Ž x . s 0 for x g w d1 , d 2 x. Since d1 and d 2 were chosen arbitrarily, we have f 1Ž x . s f 2 Ž x . for x g Ž0, d . and the uniqueness of the solution f Ž x . of Eq. Ž2.7. is proved. Žii. Let now aŽ x ., f Ž x . g C Ž0, d .. According to Ž2.1., aŽ x ., f Ž x . g Bloc Ž0, d . and by the above assertion, Eq. Ž2.7. has a unique solution f Ž x . g Bloc Ž0, d .. We prove that this solution f Ž x . belongs to the space C Ž0, d .. For any point x g Ž0, d . we choose the segment w d 3 , d 4 x ; Ž0, d . and d ) 0 such that x, x q d g w d 3 , d 4 x. Since f Ž x . g Bloc Ž0, d ., there exists M ) 0 such that

fŽ x. F M

Ž d3 F x F d4 . .

Ž 2.11.

46

KILBAS AND SAIGO

From Ž2.7. in accordance with Ž2.2., Ž2.9., and Ž2.11. we have aŽ x q d .

fŽ x q d . y fŽ x. F

GŽ a .

f mŽ t.

d

H0

Ž x q d y t.

aŽ x q d . y aŽ x .

q

GŽ a . aŽ x .

q

x

H GŽ a . 0

x

H0

1y a

dt

f mŽ t q d .

Ž x y t.

1y a

f mŽ t q d . y f m Ž t.

Ž x y t.

1y a

dt

dt

q f Ž x q d . y f Ž x. Mm

F

G Ž a q 1.

K v Ž x a , d . q d 4a v Ž a, d . q v Ž f , d .

a q K m M my1 Ž I0q f Ž t q d . y f Ž t . . Ž x . , Ž 2.12.

where

v Ž f , d . s sup

sup

0-t- d x , xqtg w d 3 , d 4 x

f Ž x q t. y f Ž x. .

Ž 2.13.

Applying Lemma 1 with A ' Ad s

Mm G Ž a q 1.

K v Ž x a , d . q d 4a v Ž a, d . q v Ž f , d . ,

D s K m M my1 ,

FŽ x. s f Ž x q d . y f Ž x. ,

we arrive at the estimate

f Ž x q d . y f Ž x . F Ad Ea Ž Dx 1r a . .

Ž 2.14.

From the assumption there hold the relations lim v Ž a, d . s 0,

dªq0

lim v Ž f , d . s 0,

d ªq0

lim v Ž x a , d . s 0.

d ªq0

Ž 2.15. Then we have lim d ªq0 Ad s 0 and, hence, lim f Ž x q d . s f Ž x .

dªq0

and f Ž x . g C Ž0, d .. This completes the proof of the theorem.

ABEL ] VOLTERRA INTEGRAL EQUATION

COROLLARY 1.1.

47

Let a ) 0, 0 - m F 1, and 0 - d F `.

Ži. If aŽ x ., f Ž x . g Bq Ž . Ž . loc 0, d and Eq. 1.4 has a solution in the space d ., then the solution is unique. Žii. If aŽ x ., f Ž x . g CqŽ 0, d . and Eq. Ž1.4. has a solution in the space qŽ C 0, d ., then the solution is unique. Ž Bq loc 0,

By using the inequality x m y y m F m y my1 Ž x y y .

Ž x ) y G 0, 0 - m - 1 .

Ž 2.16.

and the imbeddings in Ž2.1., the following statement is proved in a manner similar to Theorem 1. THEOREM 2.

Let a ) 0, m ) 1, and 0 - d F `.

q Ži. If aŽ x . g Bq Ž . Ž . Ž . Ž . loc 0, d , f x g B loc, « 0, d , and Eq. 1.4 has a soluq tion in the space Bloc, « Ž0, d ., then the solution is unique. Žii. If aŽ x . g CqŽ 0, d ., f Ž x . g Cq Ž . Ž . s 0, d , and Eq. 1.4 has a solution in the space C«qŽ 0, d ., then the solution is unique.

Remark 2. If the assumptions in Theorem 1, Corollary 1.1, and Theorem 2 hold, except the conditions on f Ž x ., then these statements stay true for the homogeneous equation corresponding to Ž1.4.:

w mŽ x. s

aŽ x .

wŽ t.

x

H GŽ a . 0

Ž x y t.

1y a

dt

Ž 0 - x - d F `. .

Ž 2.17.

Remark 3. The results in Theorem 1 and Corollary 1.1 are generalizations of those in w18x.

3. ASYMPTOTICS OF SOLUTION IN THE GENERAL CASE We suppose that Eq. Ž1.4. has the unique solution w Ž x . in one of the qw Ž . qw . q Ž . . spaces Bloc Ž0, d ., C w0, d ., Bq loc 0, d , C 0, d , B loc, 0 0, d , or C 0 0, d and study the asymptotic behavior of w Ž x ., as x ª 0, provided that aŽ x . and f Ž x . satisfy Ž1.5. and Ž1.6.. For this we need the following assertion, which is proved directly on the basis of the relation for m g R with m / 0, y1, y2, . . . , m

Ž1 y x. s

`

Ý ks0

m k x k

ž /

Ž < x < - 1. ,

Ž 3.1.

48

KILBAS AND SAIGO

where G Ž m q 1. m s k k!G Ž m y k q 1 .

ž /

are the binomial coefficients. LEMMA 2. Let p g Z, a g R, and  w k 4`p be a sequence of real numbers. If the function w Ž x . has the asymptotic relation

wŽ x. ;

`

Ý wk x a k

Ž x ª 0. ,

Ž 3.2.

ksp

then for m g R with m / 0, y1, y2, . . . , there holds the asymptotic `

w m Ž x . ; x a pm

Ý Fp , k x a k

Ž x ª 0. ,

Ž 3.3.

ks0

where the coefficients Fp, k are expressed in terms of the coefficients w k : Fp , 0 s Fp , 1 s Fp , 2 s Fp , 3 s Fp , 4 s

ž ž ž ž ž

m m wp , 0

/ /w w /w w ž /w w /w w ž / ž /w w w ž /w /w w ž /w w ž /w w ž / ž /w w w ž /w w ž / w w ž / ž / w ww w w w ž /w ž /w w ž /w w ž / ž /w w w ž /w w m 1

my 1 p pq1 ,

m 1

my 1 p pq2

q

m 2

my2 2 p pq1 ,

m 1

my 1 p pq3

q

m 2

2 1

m 1

my 1 p pq4

q

m 2

my2 p

q

Fp , 5 s

m 3

m 1

3 1

my 3 2 p pq1

my 1 p pq5

q

m 3

my 3 p

q

m 4

4 1

q 3 1

pq2

m 2

2 pq1

my 4 3 p pq1

my2 p pq1

q

2 1

m 4

my2 p

pq3

pq2

2 pq2

q

q

pq2

2 1

q

m 5

m 3

pq1

my3 3 wpq1 , p

Ž 3.4.

pq3

my4 4 p pq1 ,

pq1

3 2

q

pq4

pq1

q

pq2

pq3

x

2 pq2

my5 5 p pq1 ,

etc.

First we consider Eq. Ž1.4., where aŽ x . and f Ž x . have the asymptotics Ž1.5. and Ž1.6. in the case l y n y 1 ) y1ra . We will seek an asymptotic

ABEL ] VOLTERRA INTEGRAL EQUATION

49

solution w Ž x . of Eq. Ž1.4. in the form `

wŽ x. ;

Ý

wk x a k

Ž x ª 0. .

Ž 3.5.

kslyny1

By w24, Theorem 16.1x we have 1

wŽ t.

x

H GŽ a . 0

Ž x y t.

dt ;

1y a

`

G Ž a k q 1.

Ý

G Ž a k q a q 1.

kslyny1

w k x a kq a

Ž x ª 0. ,

and in view of the asymptotic Ž1.5. and general properties of asymptotic expansions, we have aŽ x .

wŽ t.

x

H GŽ a . 0

Ž x y t.

; x a pm

1y a

`

Ý ksyn

ž

dt lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

islyny1

a ky iy1 w i x a k

Ž x ª 0. .

/

Then, taking into account Lemma 2 and Ž1.6., we arrive at the asymptotic relation x a Ž lyny1. m

`

Ý F lyny1, k x a k ks0

; x a pm

`

Ý ksyn

q x a pm

ž

lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

islyny1

`

Ý

fk x a k

Ž x ª 0. ,

a ky iy1 w i x a k

/

Ž 3.6.

ksyn

where Fp, k are expressed in terms of w k by Ž3.4.. When the asymptotic expansion Ž3.5. of the solution w Ž x . of Eq. Ž1.4. satisfies the asymptotic relation Ž3.6., we say such an equation is asymptotically sol¨ able, and the asymptotic expansion Ž3.5. is called an asymptotic solution of Eq. Ž1.4. near zero. Let us now suppose that q s Ž l y n y p y 1. m is an integer for m g R Ž m / 0, y1, y2, . . . . such that q G yn. Then Ž3.6. is equivalent to the

50

KILBAS AND SAIGO

relation x a pm

`

Ý F lyny1, kyq x a k ksq `

; x a pm

Ý ksyn

ž

lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

islyny1

`

q x a pm

fk x a k

Ý

a ky iy1 w i x a k

/

Ž x ª 0. .

Ž 3.7.

ksyn

It follows from Ž3.7. that if the coefficients w k satisfy the conditions lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

islyny1

a ky iy1 w i q f k s 0

Ž k s yn, yn q 1, . . . , q y 1 . , Ž 3.8. F lyny1, kyq s

lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

islyny1

a kyiy1 w i q f k

Ž k s q, q q 1, . . . . , Ž 3.9. then Eq. Ž1.4. is asymptotically solvable and its asymptotic solution is given by Ž3.5., where if q s yn, relation Ž3.8. is excluded. Consequently we obtain the following theorem. THEOREM 3. Let a ) 0, p, m g R Ž m / 0, y1, y2, . . . . as well as l, n, q s Ž l y n y p y 1. m g Z such that l y n y 1 ) y1ra and q G yn. Let aŽ x . and f Ž x . ha¨ e the asymptotic expansions Ž1.5. and Ž1.6. and the coefficients w k satisfy relations Ž3.8. and Ž3.9.. Then Eq. Ž1.4. is asymptotically sol¨ able and its asymptotic solution near zero has the form Ž3.5.. Next we consider Eq. Ž1.4. in the case when aŽ x . and f Ž x . have the asymptotics Ž1.5. and Ž1.6. with n s l y p y 1 - 0. We will seek an asymptotic solution w Ž x . of Eq. Ž1.4. in the form

wŽ x. ;

`

Ý wk x a k

Ž x ª 0. ,

Ž 3.10.

ksq

where q is an unknown integer. If we suppose that mŽ q y p . is an integer, then applying the same arguments as in Theorem 3, we come to the

ABEL ] VOLTERRA INTEGRAL EQUATION

51

asymptotic relation x a pm

`

Fq , kymŽ qyp. x a k

Ý ksm Ž qyp .

`

; x a pm

Ý ksqylq1

lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

ž

isq

`

q x a pm

fx x a k

Ý

a ky iy1 w i x a k

Ž x ª 0. .

/

Ž 3.11.

kspylq1

Suppose that m ) 0 and Ž p y l q 1.rm is an integer and put q s p q Ž p y l q 1.rm with q ) y1ra . Then mŽ q y p . s p y l q 1 ) 0 and q ) p. Hence from Ž3.11. we obtain that if w k satisfy the conditions Fq , kqlypy1 s f k Fq , kqlypy1 s

Ž k s p y l q 1, p y l q 2, . . . , q y l . , Ž 3.12.

lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

isq

a kyiy1 w i q f k

Ž k s q y l q 1, q y l q 2, . . . . , Ž 3.13. then Eq. Ž1.4. is asymptotically solvable and its asymptotic solution near zero has the form Ž3.10.. Then we obtain the following result. THEOREM 4. Let a ) 0, m ) 0 and let p, l, n g Z be such that n s l y p y 1 - 0 and Ž p y l q 1.rm g Z and set q s p q Ž p y l q 1.rm ) y1ra . Let aŽ x . and f Ž x . ha¨ e the asymptotic expansions Ž1.5. and Ž1.6. and let the coefficients w k satisfy relations Ž3.12. and Ž3.13.. Then Eq. Ž1.4. is asymptotically sol¨ able and its asymptotic solution near zero has the form Ž3.10.. Setting p s y1 and replacing l by yl in Theorem 4, we have the following corollary. COROLLARY 4.1. Let a ) 0, m ) 0 and let l be a positi¨ e integer such that lrm g Z and set q s y1 q lrm. Let a Ž x . ; xya m

`

Ý ak x a k

Ž x ª 0. ,

Ž 3.14.

Ž x ª 0.

Ž 3.15.

ksl

f Ž x . ; xya m

`

Ý fk x a k ksl

52

KILBAS AND SAIGO

with a l / 0, f l / 0, and let the coefficients w k satisfy the relations Fq , kyl s f k Fq , kyl s

Ž k s l, l q 1, . . . , l q q . ,

kyly1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

isq

Ž 3.16.

a kyiy1 w i q f k

Ž k s l q q q 1, l q q q 2, . . . . . Ž 3.17. Then Eq. Ž1.4. is asymptotically sol¨ able and its asymptotic solution near zero has the form Ž3.10.. Now we suppose that m - 1 Ž m / 0, y1, y2, . . . . and Ž p y l q 1.r Ž1 y m. is a positive integer and put q s p y Ž p y l q 1.rŽ1 y m. ) y1r a . Then mŽ q y p . s q y l q 1 is an integer and q - p. From Ž3.11. we finally obtain that if w k satisfy the conditions lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

Fq , kqlyqy1 s

isq

a kyiy1 w i

Ž k s q y l q 1, q y l q 2, . . . , p y l . , Ž 3.18. lqky1

G Ž a i q 1.

Ý

G Ž a i q a q 1.

Fq , kqlyqy1 s

isq

a kyiy1 w i q f k

Ž k s p y l q 1, p y l q 2, . . . . , Ž 3.19. then Ž3.10. is also the asymptotic solution of Eq. Ž1.4.. Therefore we arrive at the result: THEOREM 5. Let a ) 0, m - 1 Ž m / 0, y1, y2, . . . ., let p, l, n g Z be such that n s l y p y 1 - 0 and Ž p y l q 1.rŽ1 y m. g Z and let q s p y Ž p y l q 1.rŽ1 y m. ) y1ra . Let aŽ x . and f Ž x . ha¨ e the asymptotic expansions Ž1.5. and Ž1.6. and the coefficients w k satisfy relations Ž3.18. and Ž3.19.. Then Eq. Ž1.4. is asymptotically sol¨ able and its asymptotic solution near zero has the form Ž3.10.. Remark 4. The condition l y n y 1 ) y1ra in Theorem 3 or q ) y1ra in Theorems 4 and 5 can be considered as two cases when l y n y 1 or q is nonnegative and negative but greater than y1ra . They correspond to the solution w Ž x . of Eq. Ž1.4. being bounded and integrable near zero, respectively. The latter case shows that the order a of Eq. Ž1.4. satisfies

a-y

1 lyny1

or

a-y

1 q

.

Ž 3.20.

ABEL ] VOLTERRA INTEGRAL EQUATION

53

If, for example, l s n or q s y1, then in Theorems 3]5, Eq. Ž1.4. is considered with 0 - a - 1. If yŽ l y n y 1. or yq is a sufficiently large positive integer, then a will be a sufficiently small positive number. Remark 5. The results in Theorems 3]5 allow us to find an asymptotic solution of the linear Abel]Volterra integral equation Ž1.4. Žfor m s 1. provided that aŽ x . and f Ž x . have the asymptotics Ž1.5. and Ž1.6. with m s 1. Such solutions in the particular case p s 0, l s 1, and n s 0 or n s 1 were obtained by the authors w19x. Theorems 3 and 5 also give the asymptotic solution w Ž x . of Eq. Ž1.4. with negative m Ž m / y1, y2, . . . ..

4. ASYMPTOTICS OF THE SOLUTION IN A SPECIAL CASE We consider Eq. Ž1.4. in the special case when aŽ x . s ax a Ž p myl . and Ž f x . is a quasipolynomial,

w mŽ x. s

ax a Ž p myl . GŽ a .

x

H0

wŽ t.

Ž x y t.

1y a

N

dt y x a p m

fk x a k

Ý ksyn

Ž 0 - x - d F `.

Ž 4.1.

with a / 0 and N G yn. Theorem 3 implies the next theorem: THEOREM 6. Let a ) 0, p, m g R Ž m / 0, y1, y2, . . . . and let l, n, Ž l y n y p y 1. m g Z be such that l y n y 1 ) y1ra and Ž l y n y p y 1. m G yn. Let w k satisfy the following conditions: Ži. When Ž l y n y p y 1. m ) N, assume that

wk s 0

Ž k s N q l, N q l q 1, . . . , Ž l y n y p y 1 . m q l y 2 . , Ž 4.2. F lynyl , kylq1yŽ lynypy1. m s

aG Ž a k q 1 . G Ž a k q a q 1.

wk

Ž k s Ž l y n y p y 1. m q l y 1, Ž l y n y p y 1. m q l, . . . . . Ž 4.3. Žii. When yn - Ž l y n y p y 1. m F N, assume that F lyny1, kylq1yŽ lynypy1. m s

aG Ž a k q 1 . G Ž a k q a q 1.

Ž k s Ž l y n y p y 1. m q l y 1, Ž l y n y p y 1 . m q l, . . . , N q l y 1 . ,

w k y f kylq1

Ž 4.4.

54

KILBAS AND SAIGO

F lyny1, kylq1yŽ lynypy1. m s

aG Ž a k q 1 . G Ž a k q a q 1.

wk

Ž k s N q l, N q l q 1, . . . . . Ž 4.5. Žiii. When Ž l y n y p y 1. m s yn, assume that aG Ž a k q 1 .

F lyny1, kylq1qn s

G Ž a k q a q 1.

w k y f kylq1

Ž k s yn q l y 1, yn q l, . . . , N q l y 1 . , Ž 4.6. aG Ž a k q 1 .

F lyny1, kylq1qn s

G Ž a k q a q 1.

wk

Ž k s N q l, N q l q 1, . . . . . Ž 4.7.

Then Eq. Ž4.1. is asymptotically sol¨ able and its asymptotic solution near zero has the form lqNy1

G Ž a k q a q 1.

Ý

aG Ž a k q 1 .

wŽ x. ;

kslyny1

`

q

f ky lq1 x a k

wk x a k

Ý

Ž x ª 0.

Ž 4.8.

ks Ž lynypy1 . mqly1

in case Ži., Ž lynypy1 . mqly2

G Ž a k q a q 1.

Ý

aG Ž a k q 1 .

wŽ x. ;

kslyny1 `

q

wk x a k

Ý

f ky lq1 x a k

Ž x ª 0.

Ž 4.9.

ks Ž lynypy1 . mqly1

in case Ž ii ., and Ž3.5. in case Ž iii .. COROLLARY 6.1. Under the assumptions of Theorem 6Ži. the solution w Ž x . of Eq. Ž4.1. has the asymptotic

wŽ x. s

lqNy1

G Ž a k q a q 1.

Ý

aG Ž a k q 1 .

kslyny1

f ky lq1 x a k

q wŽ lynypy1. mqly1 x aŽ lynypy1. mqly14 q O Ž x aŽ lynypy1. mql4 .

Ž x ª 0. ,

Ž 4.10.

ABEL ] VOLTERRA INTEGRAL EQUATION

55

where

wŽ lynypy1. mqly1 s

G Ž a  Ž l y n y p y 1. m q l 4 q 1. aG Ž a  Ž l y n y p y 1 . m q l y 1 4 q 1 . =

ž

m

G Ž a  l y n4 q 1 . fyn aG Ž a  l y n y 1 4 q 1 .

/

.

Ž 4.11.

Furthermore, if N ) yn, we ha¨ e

wŽ x. s

lqNy1

G Ž a k q a q 1.

Ý

aG Ž a k q 1 .

kslyny1

f ky lq1 x a k

q wŽ lynypy1. mqly1 x aŽ lynypy1. mqly14 q wŽ lynypy1. mql x aŽ lynypy1. mql4 q O Ž x aŽ lynypy1. mqlq14 .

Ž x ª 0 . , Ž 4.12. where wŽ lynypy1. mqly1 is gi¨ en by Ž4.11. and

wŽ lynypy1. mql s

mG Ž a  Ž l y n y p y 1 . m q l q 1 4 q 1 . aG Ž a  Ž l y n y p y 1 . m q l 4 q 1 . = =

ž

G Ž a  l y n4 q 1 . fyn aG Ž a  l y n y 1 4 q 1 .

my 1

/

G Ž a  l y n q 1 4 q 1 . fyn q1 aG Ž a  l y n4 q 1 .

.

Ž 4.13.

COROLLARY 6.2. Under the assumptions of Theorem 6Žii. the solution w Ž x . of Eq. Ž4.1. has the asymptotic

wŽ x. s

Ž lynypy1 . mqly2

G Ž a k q a q 1.

Ý

aG Ž a k q 1 .

kslyny1

f ky lq1 x a k

q wŽ lynypy1. mqly1 x aŽ lynypy1. mqly14 q O Ž x aŽ lynypy1. mql4 .

Ž x ª 0 . , Ž 4.14.

56

KILBAS AND SAIGO

where

wŽ lynypy1. mqly1 s

G Ž a  Ž l y n y p y 1. m q l 4 q 1. aG Ž a  Ž l y n y p y 1 . m q l y 1 4 q 1 .

½

= fŽ lynypy1. m q

ž

m

G Ž a  l y n4 q 1 . fyn aG Ž a  l y n y 1 4 q 1 .

/5

.

Ž 4.15. Furthermore, if N G Ž l y n y p y 1. m ) yn q 1, we ha¨ e Ž lynypy1 . mqly2

G Ž a k q a q 1.

Ý

aG Ž a k q 1 .

wŽ x. s

kslyny1

f ky lq1 x a k

q wŽ lynypy1. mqly1 x aŽ lynypy1. mqly14 q wŽ lynypy1. mql x aŽ lynypy1. mql4 q O Ž x aŽ lynypy1. mqlq14 .

Ž x ª 0 . , Ž 4.16. where wŽ lynypy1. mqly1 is gi¨ en by Ž4.15. and

wŽ lynypy1. mql s

mG Ž a  Ž l y n y p y 1 . m q l q 1 4 q 1 . aG Ž a  Ž l y n y p y 1 . m q l 4 q 1 .

½

= fŽ lynypy1. mq1 q

=

ž

G Ž a  l y n4 q 1 . fyn aG Ž a  l y n y 1 4 q 1 .

G Ž a  l y n q 1 4 q 1 . fyn q1 aG Ž a  l y n4 q 1 .

my 1

/

5

. Ž 4.17.

COROLLARY 6.3. Under the assumptions of Theorem 6Žiii., the solution w Ž x . of Eq. Ž4.1. has the asymptotic

w Ž x . s Ax a Ž lyny1. q O Ž x a Ž lyn. .

Ž x ª 0. ,

Ž 4.18.

where j s A is a solution of the equation

jmy

aG Ž a  l y n y 1 4 q 1 . G Ž a  l y n4 q 1 .

j q fyn s 0.

Ž 4.19.

ABEL ] VOLTERRA INTEGRAL EQUATION

57

Furthermore, if N G yn q 1 and aG Ž a  l y n4 q 1 . G Ž a  l y n q 14 q 1.

/ mAmy 1 ,

Ž 4.20.

then we ha¨ e

w Ž x . s Ax a Ž lyny1. q Bx a Ž lyn. q O Ž x a Ž lynq1. .

Ž x ª 0 . , Ž 4.21.

where Bs

aG Ž a  l y n4 q 1 . G Ž a  l y n q 14 q 1.

y1

y mA

my 1

fyn q1 .

Ž 4.22.

5. SOLUTION IN CLOSED FORM We consider the particular case of Eq. Ž4.1. when f Ž x . is a monomial Ž N s yn.,

w mŽ x. s

ax a Ž p myl . GŽ a .

x

H0

wŽ t.

Ž x y t.

1y a

dt y bx a Ž p myn.

Ž 0 - x - d F `. Ž 5.1.

with a ) 0, m g R Ž m / 0, y1, y2, . . . ., p, l, n g Z, and a / 0. We show that such an equation in case Žiii. above can be solved in closed form. If l y n y 1 ) y1ra and Ž l y n y p y 1. m q n s 0, then it follows from Corollary 6.3 that the solution w Ž x . of Eq. Ž5.1. has the asymptotic Ž4.18. near zero, where j s A is a solution of the equation

jmy

aG Ž a  l y n y 1 4 q 1 . G Ž a  l y n4 q 1 .

j q b s 0.

Ž 5.2.

It is directly verified that the first term Ax a Ž lyny1. of the asymptotic Ž4.18. gives the exact solution of Eq. Ž5.1.. Moreover, it is also the solution for a ) 0, p, l, n g R such that a Ž l y n y 1. ) y1. We replace a Ž pm y l . by l, a Ž pm y n. by n, and b by yb. If we choose a ) 0 and l, n, m g R Ž m / 0, 1. such that l q a y n - 1 Ž l q a y n / 0. and Ž n y l y a . m s n, and suppose that the equation

jmy

aG Ž n y l y a q 1 . G Ž n y l q 1.

jybs0

Ž 5.3.

58

KILBAS AND SAIGO

with a, b g R Ž a / 0. is solvable, for indeed j s c is its solution, then we can say that the nonlinear integral equation

w mŽ x. s

ax l

wŽ t.

x

H GŽ a . 0

dt q bx n

Ž 0 - x - d F ` . Ž 5.4.

w Ž x . s cx ny ly a .

Ž 5.5.

Ž x y t.

1y a

has the solution

We set, again, n y l y a s b . Then we arrive at the following statement: THEOREM 7. Let a ) 0, b ) y1 Ž b / 0., and l g R with l / ya and l / ya y b . For a, b g R Ž a / 0. let the equation

j 1q Ž lq a .r b y

aG Ž b q 1 . G Ž a q b q 1.

jybs0

Ž 5.6.

be sol¨ able and let j s c be its solution. Then the nonlinear integral equation

w 1q Ž lq a .r b Ž x . s

ax l

wŽ t.

x

H GŽ a . 0

Ž x y t.

dt q bx aq bql

1y a

Ž 0 - x - d F `. Ž 5.7.

is sol¨ able and its solution is gi¨ en by

w Ž x . s cx b .

Ž 5.8.

Remembering Theorems 1 and 2, we have the following theorem. THEOREM 8. Let the assumptions of Theorem 7 be satisfied and let j s c be the unique solution of Eq. Ž5.6.. If y1 - Ž l q a .rb - 0, then Ž5.8. is the unique solution of Eq. Ž5.7. in the space C Ž0, d .. If additionally a ) 0, b ) 0, and c ) 0, then this solution belongs to the space CqŽ 0, d .. If b - 0, a q l - 0, a ) 0, b ) 0, and c ) 0, then Ž5.8. is the unique solution of Eq. Ž5.7. with 0 - d - ` in the space C«qŽ 0, d ., « s d b. Now we consider the homogeneous nonlinear integral equation

w 1q Ž lq a .r b Ž x . s

ax l

H GŽ a . 0

x

wŽ t.

Ž x y t.

1y a

dt

Ž 0 - x - d F ` . Ž 5.9.

ABEL ] VOLTERRA INTEGRAL EQUATION

59

corresponding to Eq. Ž5.7.. A direct calculation proves that the function

wŽ x. s

aG Ž b q 1 . G Ž a q b q 1.

b r Ž lq a .

xb

Ž 5.10.

is the exact solution of Eq. Ž5.9.. Taking into account Remark 2, we come to the following result: THEOREM 9. If the assumptions of Theorem 8 are satisfied, then the homogeneous nonlinear Eq. Ž5.9. with a g R Ž a / 0. is sol¨ able and its solution has the form Ž5.10.. If y1 - Ž l q a .rb - 0, then Ž5.10. is the unique solution of Eq. Ž5.9. in the space Bloc Ž0, d .. If additionally a ) 0, then this solution belongs to the Ž . space Bq loc 0, d . If b - 0, a q l - 0, and a ) 0, then Ž5.10. is the unique solution of Eq. Ž5.9. with 0 - d - ` in the space C«qŽ 0, d ., « s d b. Remark 6. In the particular case when l s 0 and a s G Ž a q b q 1.rG Ž b q 1., the solution of Eq. Ž5.9. with b ) 0, which appears in applications, was obtained in w25x. In the case when l s 0, a s G Ž a ., and b - yar2 Žand hence 1 q arb - y1., the solution of Eq. Ž5.9. was given in w16x, and the uniqueness of this solution in a certain subspace of continuous nondecreasing positive functions was proved.

REFERENCES 1. S. N. Askhabov, Integral equations of convolution type with power nonlinearity, Colloq. Math. 62 Ž1991., 49]65. 2. S. N. Askhabov and M. A. Betilgiriev, Nonlinear convolution type equations, ‘‘Seminar Analysis,’’ pp. 1]30, Karl-Weierstrass-Institute of Mathematics, Berlin, 1990. 3. S. N. Askhabov and M. A. Betilgiriev, Nonlinear integral equations of convolution type with almost increasing kernels in cones, Differential Equations 27 Ž1991., 234]242. 4. S. N. Askhabov, N. K. Karapetyants, and A. Ya. Yakubov, A nonlinear equation of convolution type, Differentsial’nye Ura¨ neniya 22 Ž1986., 1606]1609 Žin Russian.. 5. S. N. Askhabov, N. K. Karapetyants, and A. Ya. Yakubov, Integral equations of convolution type with power nonlinearity and systems of such equations, So¨ iet Math. Dokl. 41 Ž1990., 323]327. 6. P. J. Bushell, On a class of Volterra and Fredholm non-linear integral equations, Math. Proc. Cambridge Philos. Soc. 79 Ž1976., 329]335. 7. P. J. Bushell and W. Okrasinski, Nonlinear Volterra integral equations and the Apery ´ identities, Bull. London Math. Soc. 24 Ž1992., 478]484. 8. A. Erdelyi, ´ W. Magnus, F. Oberhettinger, and F. S. Tricomi, ‘‘Higher Transcendental Functions,’’ Vol. 3, McGraw-Hill, New York, 1954. 9. J. Goncerzewicz, H. Marcinkowska, W. Okrasinski, and K. Tabisz, On the percolation of ´ water from a cylindrical reservoir into the surrounding soil, Zastos. Mat. 16 Ž1978r79., 249]261.

60

KILBAS AND SAIGO

10. R. Gorenflo and A. A. Kilbas, Asymptotic solution of a nonlinear Abel]Volterra integral equation of second kind, J. Fract. Calc. 8 Ž1995., 103]117. 11. R. Gorenflo and S. Vessela, ‘‘Abel Integral Equations, Analysis and Applications,’’ Springer-Verlag, Berlin, 1991. 12. G. Gripenberg, Asymptotic solutions of some nonlinear Volterra integral equations, SIAM J. Math. Anal. 12 Ž1981., 595]602. 13. G. Gripenberg, S.-O. Londen, and O. Staffans, ‘‘Volterra Integral and Functional Equations,’’ Encyclopedia of Mathematics and its Applications, Vol. 34, Cambridge Univ. Press, Cambridge, UK, 1990. 14. R. A. Handelsman and W. E. Olmstead, Asymptotic solution to a class of nonlinear Volterra integral equations, SIAM J. Appl. Math. 22 Ž1972., 373]384. 15. N. K. Karapetyants, On a class of nonlinear equations of convolution types, in ‘‘Scientific Proceedings of the Jubilee Seminar on Boundary Value Problems,’’ Minsk, 1985, pp. 158]161 Žin Russian.. 16. N. K. Karapetyants and A. Ya. Yakubov, A convolution equation with power nonlinearity of negative order, So¨ iet Math. Dokl. 44 Ž1992., 517]520. 17. J. J. Keller, Propagation of simple nonlinear waves in gas filled tubes with friction, Z. Angew. Math. Phys. 32 Ž1981., 170]181. 18. A. A. Kilbas, A nonlinear Volterra integral equation with a weak singularity, Proc. Russian People’s Friendship Uni¨ . Math. Ser. 2 Ž1995., 47]61. 19. A. A. Kilbas and M. Saigo, On solution of integral equation of Abel]Volterra type, Differential Integral Equations 8 Ž1995., 993]1011. 20. W. Okrasinski, On the existence and uniqueness of nonnegative solutions of a certain ´ nonlinear convolution equation, Ann. Polon. Math. 36 Ž1979., 61]72. 21. W. Okrasinski, On a nonlinear convolution equation occurring in the theory of water ´ percolation, Ann. Polon. Math. 37 Ž1980., 223]229. 22. W. Okrasinski, Nonlinear Volterra equations and physical applications, Extracta Math. 4 ´ Ž1989., 51]80. 23. W. E. Olmstead and R. A. Handelsman, Asymptotic solution to a class of nonlinear Volterra integral equation. II, SIAM J. Appl. Math. 30 Ž1976., 180]189. 24. S. G. Samko, A. A. Kilbas, and O. I. Marichev, ‘‘Fractional Integrals and Derivatives. Theory and Applications,’’ Gordon and Breach, New York, 1993. 25. W. R. Schneider, The general solution of a nonlinear integral equation of convolution type, Z. Angew. Math. Phys. 33 Ž1982., 140]142.