Skip to main content

Analysis of Abel-type nonlinear integral equations with weakly singular kernels

Abstract

In this paper, we investigate Abel-type nonlinear integral equations with weakly singular kernels. Existence and uniqueness of nontrivial solution are presented in an order interval of a cone by using fixed point methods. As a byproduct of our method, we improve a gap in the proof of Theorem 5 in Buckwar (Nonlinear Anal. TMA 63:88-96, 2005). As an extension, solutions in closed form of some Erdélyi-Kober-type fractional integral equations are given. Finally theoretical results with three illustrative examples are presented.

MSC:26A33, 45E10, 45G05.

1 Introduction

Abel-type integral equations are associated with a wide range of physical problems such as heat transfer [1], nonlinear diffusion [2], propagation of nonlinear waves [3], and they can also be applied in the theory of neutron transport and in traffic theory. In the past 70 years, many researchers investigated the existence and uniqueness of nontrivial solutions for a large number of Abel-type integral equations by using various analysis methods (see [416] and references therein).

Fractional calculus provides a powerful tool for the description of hereditary properties of various materials and memory processes. In particular, integral equations involving fractional integral operators (which can be regarded as an extension of Abel integral equations) appear naturally in the fields of biophysics, viscoelasticity, electrical circuits, and etc. There are some remarkable monographs that provide the main theoretical tools for the qualitative analysis of fractional order differential equations, and at the same time, show the interconnection as well as the contrast between integer order differential models and fractional order differential models [1724].

It is remarkable that many researchers pay attention to the study of the existence and attractiveness of solutions for fractional integral equations by using functional analysis methods such as the contraction principle, the Schauder fixed point theorem and a Darboux-type fixed point theorem involving a measure of noncompactness (see [2533] and references therein).

A completely different approach is given in Buckwar [13] to discussing the existence and uniqueness of nontrivial solutions for Abel-type nonlinear integral equation with power-law nonlinearity on an order interval as follows:

x p (t)= 1 Γ ( α ) 0 t [ K ( t , s ) ( t s ) 1 α ] x(s)ds,t[0,T].
(1)

Many analysis techniques are used to construct the suitable order interval (see Lemma 2, [13]) and the spaces with suitable weighted norms.

Motivated by [6, 11, 13, 33], we extend to study the following Abel-type nonlinear integral equation with weakly singular kernels:

h ( x ( t ) ) = β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( x ( s ) ) ds,t[0,T],
(2)

where h,gC([0,M),[0,+)) are given functions for some M(0,+], h is increasing, g is nondecreasing such that

a x p h(x) a + x p + , b x q g(x) b + x q + ,0x<M,
(3)

for some positive constants a ± , b ± , p ± , q ± , 0<α<1, γβ>0, and 0< q + q < p + p , the function K(t,s) is non-negative and it has either the form K(t,s)= k 1 ( t β s β ) or K(t,s)= k 2 (t,s) for some function k 1 , k 2 specified later. Γ() is the Gamma function. Of course, we suppose

a M p p + a + , b M q q + b + .
(4)

It is obvious that equation (1) or

x p (t)= β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] x q (s)ds,t[0,T],
(5)

are special cases of equation (2), which of course all have trivial solutions.

Thus, the main purpose of this paper is to prove the existence and uniqueness of nontrivial solutions for equation (2). The key difficult comes from the weakly singular kernels ( t β s β ) 1 α and nonlinear terms in equation (2). Although we are motivated by [13], we have to introduce novel techniques and results to overcome the difficult from the weakly singular kernels and nonlinear terms h and g. For example, the first important step is how to construct a suitable order interval to help us to apply the fixed point theorem in such an order interval. More details of the novel techniques and results will be found in the proof. As a byproduct of our method, we improve a gap in the proof of [[13], Theorem 5]. So even for equation (1) (or (5)) we get a new result.

As an extension, we find general solutions in closed form of some Erdélyi-Kober-type fractional integral equations (the special case of equation (5) if b=0):

φ m (x)=a x β ( m N ) N ( E K I 0 + ; σ , η α φ N )(x)+b x β m N ,x>0,
(6)

where α,b,σ0, N0, and ηR and the symbol I 0 + ; σ , η α E K φ N denotes the Erdélyi-Kober-type fractional integrals [19] of the function φ N , which is given by

( E K I 0 + ; σ , η α φ N )(x):= σ x σ ( α + η ) Γ ( α ) 0 x t σ η + σ 1 φ N ( t ) d t ( x σ t σ ) 1 α ,x>0.

The plan of this paper is as follows. In Section 2, some notation and preparation results are given. Existence and uniqueness results of a nontrivial solution of equation (2) in an order interval are given in Section 3. In Section 4, we find general solutions in closed form of some Erdélyi-Kober-type fractional integral equations, and finally theoretical results with three illustrate examples are presented in Section 5.

2 Preliminary

Let be the set M:={fC[0,T]:f(0)=0} with the supremum-norm f M := sup 0 < t T {|f(t)|}. Clearly, the set (M, M ) is a closed subspace of Banach space (C[0,T], C ). Thus, (M, M ) is a Banach space.

Let q be a continuous function on [0,T] with q(t)>0 for all t>0 and let M q be the set

M q := { f M : sup 0 < t T | f ( t ) | q ( t ) < }

with the weighted norm

f q := sup 0 < t T { | f ( t ) | q ( t ) } .
(7)

Remark 2.1 If q(0)>0, then the set M q is the same as the set , but with an equivalent norm, and the constants can be determined in the following inequality:

c 2 u v g u v M c 1 u v g ,

where c 1 = max t [ 0 , T ] {g(t)} and c 2 = min t [ 0 , T ] {g(t)}. Note that the similar inequality (5) of [13] is incorrect.

Consider the cone P M :={uM:u(t)0,t[0,T]} in . The so-called partial ordering induced by the cone P M is given by uvu(t)v(t) for all u,vM and all t[0,T]. In general [34, 35], a set [f,g]={hE:fhg} is called an order interval where E is an ordered Banach space. We know that every order interval [f,g] is closed. Moreover, if f E g E for all f,gE with 0fg, then every order interval [f,g] is bounded.

We introduce some conditions on the functions K, k i , i=1,2 as follows:

  1. (i)

    k 1 C ( n ) [0,T] where n{0,1,2,}, and 0 k 2 C [ 0 , T ] 2 .

  2. (ii)

    k 1 (t)>0 for all t(0,T] and k 2 (t,s)0 for all 0stT.

  3. (iii)

    k 1 (0)= k 1 ( 1 ) (0)== k 1 ( n 1 ) (0)=0, and k 1 ( n ) (t) k 1 ( n ) (0)>0, for all t(0,T].

  • For K(t,s)= k 1 ( t β s β ) and n1, we set

    K low = k 1 ( n ) (0), K up = max t [ 0 , T ] { k 1 ( n ) ( t ) } .
    (8)
  • For K(t,s)= k 2 (t,s), we set n=0 and

    K low = K min := min 0 s t T { k 2 ( t , s ) } , K up = K max := max 0 s t T { k 2 ( t , s ) } .
    (9)

Similarly for K(t,s)= k 1 ( t β s β ) and n=0.

We note [34] an important estimate on the function K, which will be used in the sequel.

Lemma 2.2 The function K(,) has the following estimate:

1 n ! ( t β s β ) n K low K(t,s) 1 n ! ( t β s β ) n K up ,0stT.
(10)

Proof We only check the case of K(t,s)= k 1 ( t β s β ) with n1, since the other cases are trivial.

Integrating n times step-by-step all sides of the inequality

k 1 ( n ) (0) k 1 ( n ) (t) K up

from 0 to t and using k 1 (0)= k 1 ( 1 ) (0)== k 1 ( n 1 ) (0)=0 we immediately derive

t n n ! k 1 ( n ) (0) k 1 (t) t n n ! K up .

Replacing t by ( t β s β ), we obtain the desired result. □

To end this section, we collect the following basic facts, which will be used several times in the next section.

Lemma 2.3 Let λ, γ, μ, and ν be constants such that λ>0, Re(γ)>0, Re(μ)>0, and Re(ν)>0. Then

0 t ( t λ s λ ) ν 1 s μ 1 ds= t λ ( ν 1 ) + μ λ B ( μ λ , ν ) ,t[0,+),

and

a t ( t λ s λ ) ν 1 s γ 1 ( s λ a λ ) μ d s ( t λ a λ ) ν + γ λ + μ 1 λ B ( γ λ + μ , ν ) , t [ a , + ) , a 0 ,

where

B(ξ,η)= 0 1 s ξ 1 ( 1 s ) η 1 ds ( Re ( ξ ) > 0 , Re ( η ) > 0 )

is the well-known Beta function.

Proof The first result have been reported in [36] or [[37], Formula 3.251]. We only verify the second inequality. In fact, for any t[a,+), a0, we derive

a t ( t λ s λ ) ν 1 s γ 1 ( s λ a λ ) μ d s = 1 λ a λ t λ ( t λ u ) ν 1 u γ 1 λ ( u a λ ) μ u 1 λ 1 d u ( set  u = s λ ) = 1 λ a λ t λ ( t λ u ) ν 1 u γ λ 1 ( u a λ ) μ d u = 1 λ 0 t λ a λ ( t λ a λ z ) ν 1 ( a λ + z ) γ λ 1 z μ d u ( set  u = a λ + z ) 1 λ 0 t λ a λ ( t λ a λ z ) ν 1 z γ λ + μ 1 d u = ( t λ a λ ) ν + γ λ + μ 1 λ B ( γ λ + μ , ν ) .

The proof is completed. □

3 Existence and uniqueness of nontrivial solution in an order interval

In this section, we will use the fixed point method to prove the existence and uniqueness of nontrivial solution for equation (2) in an order interval.

For all t[0,T], we introduce the following functions:

F ( t ) = A t τ , G ( t ) = B t τ + , A = ( b β α K low a + Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) ) 1 p + q , B = ( b + β α K up a Γ ( α ) n ! B ( β ( n + α 1 ) q + + γ p β ( p q + ) , n + α ) ) 1 p q + , τ = β ( n + α 1 ) + γ p + q , τ + = β ( n + α 1 ) + γ p q + ,

where K low and K up are defined in equation (8) or (9).

Remark 3.1 Note that β(n+α1) q +γ p ± β(α1) q +β p ± =β( p ± q )+αβ q >0 and β(n+α1)+γβ(α1)+β=αβ>0. Next, τ τ + .

The following result is clear.

Lemma 3.2 If

A T τ τ + B<M T τ +
(11)

then F(t)G(t)<M for all t[0,T]. Consequently, the order interval [F,G] P M is well defined.

Remark 3.3 If p + = p , q + = q , and M=+ then equation (11) reads

a + b + a b K low K up

which is satisfied, since equation (3) implies a + a >0, b + b >0 (see equation (4)) and clearly K low K up 1. This case occurs for instance when h(x)= x p h ˜ (x) and g(x)= x q g ˜ (x) with p>q and 0< inf R h(x) sup R h(x)<, 0< inf R g(x) sup R g(x)<.

From now on, we suppose that all above assumptions hold: equations (3), (4), (i)-(iii), and (11).

Lemma 3.4 Any solution x P M of equation (2), with M>x(t)>0 for all t(0,T], satisfies x[F,G].

Proof Step 1: We prove that xG for a solution x of equation (2).

Set x + (t)= max s [ 0 , t ] x(s)=x( s t ). Then we obtain

a x p ( t ) h ( x ( t ) ) = β 1 α Γ ( α ) 0 t K ( t , s ) s γ 1 ( t β s β ) 1 α g ( x ( s ) ) d s b + x + q + ( t ) β 1 α Γ ( α ) 0 s t K ( s t , s ) s γ 1 ( s t β s β ) 1 α d s b + x + q + ( t ) β 1 α K up Γ ( α ) n ! 0 s t ( s t β s β ) n + α 1 s γ 1 d s = b + x + q + ( t ) β α K up Γ ( α ) n ! B ( γ β , n + α ) s t β ( n + α 1 ) + γ b + x + q + ( t ) β α K up Γ ( α ) n ! B ( γ β , n + α ) t β ( n + α 1 ) + γ ,

which implies that

x(t) x + (t) ( b + β α K up a Γ ( α ) n ! B ( γ β , n + α ) ) 1 p q + t β ( n + α 1 ) + γ p q + .
(12)

Next we set

Ξ:= sup t ( 0 , T ] x ( t ) t τ + ( b + β α K up a Γ ( α ) n ! B ( γ β , n + α ) ) 1 p q + .

Then we have

a x p ( t ) h ( x ( t ) ) = β 1 α Γ ( α ) 0 t K ( t , s ) s γ 1 ( t β s β ) 1 α g ( x ( s ) ) d s b + Ξ q + β 1 α K up Γ ( α ) n ! 0 t ( t β s β ) n + α 1 s γ 1 s q + τ + d s = b + Ξ q + β α K up Γ ( α ) n ! B ( γ + q + τ + β , n + α ) t β ( n + α 1 ) + γ + q + τ + b + Ξ q + β α K up Γ ( α ) n ! B ( β ( n + α 1 ) q + + γ p β ( p q + ) , n + α ) t p τ + ,

and so

x ( t ) t τ + Ξ q + p ( b + β α K up a Γ ( α ) n ! B ( β ( n + α 1 ) q + + γ p β ( p q + ) , n + α ) ) 1 p ,

hence

Ξ Ξ q + p ( b + β α K up a Γ ( α ) n ! B ( β ( n + α 1 ) q + + γ p β ( p q + ) , n + α ) ) 1 p ,

thus

Ξ ( b + β α K up a Γ ( α ) n ! B ( β ( n + α 1 ) q + + γ p β ( p q + ) , n + α ) ) 1 p q + ,

consequently

x(t) ( b + β α K up a Γ ( α ) n ! B ( β ( n + α 1 ) q + + γ p β ( p q + ) , n + α ) ) 1 p q + t τ + =G(t).
(13)

Since β ( n + α 1 ) q + + γ p β ( p q + ) > γ β implies B( β ( n + α 1 ) q + + γ p β ( p q + ) ,n+α)<B( γ β ,n+α), estimate (13) is an improvement of equation (12).

Step 2: We prove that xF. Fix a(0,T) and set

ϒ a := inf t ( a , T ] x ( t ) ( t β a β ) Θ >0

for Θ:= β ( n + α 1 ) + γ β ( p + q ) >0. Then like above, for t(a,T], we get

a + x p + ( t ) h ( x ( t ) ) = β 1 α Γ ( α ) 0 t K ( t , s ) s γ 1 ( t β s β ) 1 α g ( x ( s ) ) d s b ϒ a q β 1 α Γ ( α ) a t K ( t , s ) s γ 1 ( t β s β ) 1 α ( s β a β ) q Θ d s b ϒ a q β 1 α K low Γ ( α ) n ! a t ( t β s β ) n + α 1 s γ 1 ( s β a β ) q Θ d s b ϒ a q β α K low Γ ( α ) n ! B ( γ β + q Θ , n + α ) ( t β a β ) n + α 1 + γ β + q Θ = b ϒ a q β α K low Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) ( t β a β ) p + Θ ,

which implies

( x ( t ) ( t β a β ) Θ ) p + ϒ a q b β α K low a + Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) .

Hence

ϒ a p + ϒ a q b β α K low a + Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) ,

and so

ϒ a ( b β α K low a + Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) ) 1 p + q .

Consequently, we arrive at

x ( t ) ϒ a ( t β a β ) Θ ( b β α K low a + Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) ) 1 p + q ( t β a β ) Θ .

Since a(0,T) is arbitrarily, we have

x(t) ( b β α K low a + Γ ( α ) n ! B ( β ( n + α 1 ) q + γ p + β ( p + q ) , n + α ) ) 1 p + q t β Θ =F(t).

Hence we can complete the proof. □

To solve equation (2), we introduce an operator S h , g :[F,G] P M C[0,T] by

S h , g (x)(t)= h 1 ( β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( x ( s ) ) d s ) ,t[0,T].
(14)

Lemma 3.5 The operator S h , g maps the order interval [F,G] into itself.

Proof To achieve our aim, we only need to verify that SFF and SGG:

h ( F ( t ) ) β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( F ( s ) ) ds,t[0,T],
(15)
h ( G ( t ) ) β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( G ( s ) ) ds,t[0,T].
(16)

First we show equation (15):

β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( F ( s ) ) d s b β 1 α A q K low Γ ( α ) n ! 0 t ( t β s β ) n + α 1 s q τ + γ 1 d s = b β α A q K low Γ ( α ) n ! B ( q τ + γ β , n + α ) t β ( n + α 1 ) + q τ + γ = a + A p + t p + τ h ( F ( t ) ) .

Secondly we derive equation (16):

β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( G ( s ) ) d s b + β 1 α B q + K up Γ ( α ) n ! 0 t ( t β s β ) n + α 1 s q + τ + + γ 1 d s , = b + β α B q + K up Γ ( α ) n ! B ( q + b + γ β , n + α ) t β ( n + α 1 ) + q + τ + + γ = a B p t p τ + h ( G ( t ) ) .

Since obviously, the operator S is strictly increasing in [F,G] and if x[F,G] then F(t)x(t)G(t)<M, t[0,T]. Hence

h ( F ( t ) ) β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( F ( s ) ) d s β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( x ( s ) ) d s β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( G ( s ) ) d s h ( G ( t ) ) ,

so

β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] g ( x ( s ) ) ds [ 0 , h ( G ( t ) ) ] =h ( [ 0 , G ( t ) ] ) .

Consequently, S h , g is well defined and S h , g ([F,G])[F,G]. The proof is completed. □

From the Arzela-Ascoli theorem and since S h , g :[F,G][F,G] is nondecreasing, it follows that S h , g is compact, so the Schauder fixed point theorem implies the following existence result [35, 38, 39].

Theorem 3.6 Equation (2) has a solution in [F,G]. Moreover,

lim n S h , g n (F)= x and lim n S h , g n (G)= x +

are fixed points of S h , g with

F x x + G.

Now we are ready to state the following uniqueness result. But first we note that the above considerations can be repeated for any 0< T 1 T, so we get K low ( T 1 ), K up ( T 1 ), A( T 1 ), B( T 1 ), F T 1 , and G T 1 as continuous functions of T 1 . Note K low ( T 1 ) is nonincreasing, K up ( T 1 ) is nondecreasing, and K low ( T 1 ), K up ( T 1 ) can be continuously extended to T 1 =0. Then K low (0)= K up (0). We still keep the notation K low = K low (T), K up = K up (T), F= F T , and G= G T .

Theorem 3.7 If there are constants ψ, χ and continuous functions a g (t)>0 and a h (t)>0 on [0,T] such that

a h ( T 1 ) t ψ | x ( t ) y ( t ) | | h ( x ( t ) ) h ( y ( t ) ) | , | g ( x ( t ) ) g ( y ( t ) ) | a g ( T 1 ) t χ | x ( t ) y ( t ) | ,
(17)

for all T 1 (0,T], t(0, T 1 ], x,y[ F T 1 , G T 1 ] then equation (2) has a unique solution in [F,G] provided we have

β(n+α1)+χ+γψ,
(18)

and

Λ:= a g ( 0 ) β α K up ( 0 ) a h ( 0 ) Γ ( α ) n ! B ( χ + γ + τ + β , n + α ) 0 β ( n + α 1 ) + χ + γ ψ <1,
(19)

where we set 0 0 =1.

Proof For any x,y[F,G] we set x 1 = S h , g (x) and y 1 = S h , g (y). Clearly, we have

x y q := sup t ( 0 , T ] | x ( t ) y ( t ) | t τ + ( 1 + ι t ϖ ) 2B

for q(t)= t τ + (1+ι t ϖ ) with ϖ>0 and ι>0 specified below, so [F,G] M q . Then for any t(0,T], we derive

a h ( t ) t ψ | x 1 ( t ) y 1 ( t ) | | h ( x 1 ( t ) ) h ( y 1 ( t ) ) | β 1 α Γ ( α ) 0 t [ K ( t , s ) s γ 1 ( t β s β ) 1 α ] | g ( x ( s ) ) g ( y ( s ) ) | d s a g ( t ) β 1 α K up ( t ) Γ ( α ) n ! 0 t ( t β s β ) n + α 1 s χ + γ 1 | x ( s ) y ( s ) | d s a g ( t ) β 1 α K up ( t ) Γ ( α ) n ! [ 0 t ( t β s β ) n + α 1 s χ + γ + τ + 1 d s + ι 0 t ( t β s β ) n + α 1 s χ + γ + τ + + ϖ 1 d s ] x y q = a g ( t ) β 1 α K up ( t ) Γ ( α ) n ! [ B ( χ + γ + τ + β , n + α ) t β ( n + α 1 ) + χ + γ + τ + + ι B ( χ + γ + τ + + ϖ β , n + α ) t β ( n + α 1 ) + χ + γ + τ + + ϖ ] x y q ,

which implies

| x 1 ( t ) y 1 ( t ) | t τ + ( 1 + ι t ϖ ) a g ( t ) β 1 α K up ( t ) a h ( t ) Γ ( α ) n ! ( 1 + ι t ϖ ) [ B ( χ + γ + τ + β , n + α ) t β ( n + α 1 ) + χ + γ ψ + ι B ( χ + γ + τ + + ϖ β , n + α ) t β ( n + α 1 ) + χ + γ + ϖ ψ ] x y q ,

consequently, we obtain

S h , g ( x ) S h , g ( y ) q L x y q x,y[F,G]
(20)

with

L : = sup t ( 0 , T ] L ( t ) , L ( t ) = L 1 ( t ) + L 2 ( t ) , L 1 ( t ) : = a g ( t ) β 1 α K up ( t ) a h ( t ) Γ ( α ) n ! ( 1 + ι t ϖ ) B ( χ + γ + τ + β , n + α ) t β ( n + α 1 ) + χ + γ ψ , L 2 ( t ) : = ι a g ( t ) β 1 α K up ( t ) a h ( t ) Γ ( α ) n ! ( 1 + ι t ϖ ) B ( χ + γ + τ + + ϖ β , n + α ) t β ( n + α 1 ) + χ + γ + ϖ ψ .

Since (note equation (18))

L 2 ( t ) ι t ϖ a g ( t ) β 1 α K up a h ( t ) Γ ( α ) n ! ( 1 + ι t ϖ ) B ( χ + γ + τ + + ϖ β , n + α ) T β ( n + α 1 ) + χ + γ ψ a g ( t ) β 1 α K up a h ( t ) Γ ( α ) n ! B ( χ + γ + τ + + ϖ β , n + α ) T β ( n + α 1 ) + χ + γ ψ

and B( χ + γ + τ + + ϖ β ,n+α)0 as ϖ+, we see that

sup t ( 0 , T ] L 2 (t)< 1 Λ 4

for any ϖ>0 sufficiently large uniformly for any ι>0. So we take and fix such a ϖ. Next, by equation (19) there is a t 0 (0,T] so that

L 1 (t) a g ( t ) β 1 α K up ( t ) a h ( t ) Γ ( α ) n ! B ( χ + γ + τ + β , n + α ) t β ( n + α 1 ) + χ + γ ψ < 1 + Λ 2 <1

for any t(0, t 0 ]. Furthermore, for t[ t 0 ,T], we have (note equation (18))

L 1 ( t ) max t [ t 0 , T ] a g ( t ) β 1 α K up a h ( t ) Γ ( α ) n ! B ( χ + γ + τ + β , n + α ) T β ( n + α 1 ) + χ + γ ψ 1 + ι t 0 ϖ 1 + Λ 2 < 1

for any ι>0 sufficiently large, so we fix such ι>0. Consequently we get

sup t ( 0 , T ] L 1 (t) 1 + Λ 2 .

Summarizing we see that there is ϖ>0 and ι>0 so that

L 1 + Λ 2 + 1 Λ 4 = 3 + Λ 4 <1.

This shows that S h , g :[F,G][F,G] is a contraction with respect to the norm q with a constant L. By the contraction mapping principle, one can obtain the result immediately. □

Remark 3.8 Consider equation (5). Of course, we can suppose p>1=q. Then p ± =p, q ± =1, a ± = b ± =1, and τ ± =τ:= β ( n + α 1 ) + γ p 1 . Moreover, Remark 3.3 can be applied to get an existence result. If in addition K low >0 then BA>0, and it is not difficult to see that ψ=(p1)τ, χ=0, a g ( T 1 )=1, and

a h ( T 1 )=p A p 1 ( T 1 )=p β α K low ( T 1 ) Γ ( α ) n ! B ( β ( n + α 1 ) + γ p β ( p 1 ) , n + α ) .

Then β(n+α1)+χ+γ=ψ, so equation (18) holds. Next, we derive

Λ= K up ( 0 ) p K low ( 0 ) B ( γ + τ β , n + α ) B ( β ( n + α 1 ) + γ p β ( p 1 ) , n + α ) = K up ( 0 ) p K low ( 0 ) = 1 p <1.

Hence condition (19) is satisfied and then we get a uniqueness result by Theorem 3.7. Note there is gap in the proof of [[13], Theorem 5]. So here we give its correct proof.

4 General solutions of Erdélyi-Kober-type integral equations

This section is devoted to a derivation of explicit solutions of some Erdélyi-Kober-type integral equations. In order to establish this, we introduce the following useful result.

Lemma 4.1 Let ση+β>σ and α,σ>0. Then

( E K I 0 + ; σ , η α t β )(x)= Γ ( η + 1 + β σ ) Γ ( η + 1 + α + β σ ) x β .

Proof Set t=xy. By using Lemma 2.3, we have

( E K I 0 + ; σ , η α t β ) ( x ) = σ x σ ( α + η ) Γ ( α ) 0 x t σ η + σ 1 t β d t ( x σ t σ ) 1 α = σ x β Γ ( α ) 1 σ B ( σ η + σ + β σ , α ) = Γ ( η + 1 + β σ ) Γ ( η + 1 + α + β σ ) x β .

This completes the proof. □

Now we are ready to present our main result of this section.

Theorem 4.2 Let α>0, σ>0, β σ +η+1>0, m,b,β R , and a,N,m0. Then equation (6) is solvable and its solution φ(x) can be written as

φ(x)= C 1 N x β N ,
(21)

where the constant C satisfies the following equation:

C m N =aC Γ ( η + 1 + β σ ) Γ ( η + 1 + α + β σ ) +b.
(22)

Proof With the help of Lemma 4.1, substituting equation (21) into (6), we find that C satisfies equation (22) which completes the proof. □

5 Illustrative examples

In this section, we pay our attention to show three numerical performance results.

Example 5.1 We consider the problem

x 2 (t)= ( 1 4 ) 1 2 Γ ( 1 2 ) 0 t [ t 2 s 1 2 ( t 1 4 s 1 4 ) 1 2 ] x(s)ds,t[0,1].
(23)

First, Theorem 4.2 gives the exact solution x(t)= 88 , 179 π t 19 8 262 , 144 0.596211 t 2.375 of equation (23). Next, by changing x(t)=z(t)t we get

z 2 (t)= ( 1 4 ) 1 2 Γ ( 1 2 ) 0 t [ s 1 2 ( t 1 4 s 1 4 ) 1 2 ] z(s)ds,t[0,1].
(24)

Of course, we get a solution z(t)= 88 , 179 π t 11 8 262 , 144 . In equation (5) for (24), we set K(t,s)=1, α= 1 2 , γ= 3 2 , β= 1 4 , n=0, T=1, p=2, and q=1. After some computation, we find that

F(t)=G(t)= 2 Γ ( 1 2 ) B ( 23 2 , 1 2 ) t 11 8 = 88 , 179 π t 11 8 262 , 144 .

Obviously, all the assumptions in Theorem 3.7 are satisfied. Numerical result is given in Figure 1.

Figure 1
figure 1

Solution of equation ( 23 ) and the boundaries F and G for Example 5.1 coincide with the unique solution.

Example 5.2 In equation (5), we set K(t,s)= e 4 t , α= 3 4 , β=γ= 1 2 , n=0, T=1, p=2, and q=1. Now, we turn to consider the following homogeneous Abel-type integral equation with weakly singular kernels and power-law nonlinearity:

x 2 (t)= ( 1 2 ) 1 4 Γ ( 3 4 ) 0 t [ e 4 t s 1 2 ( t 1 2 s 1 2 ) 1 4 ] x(s)ds,t[0,1].
(25)

After some computation, we find that

F ( t ) = ( 1 2 ) 3 4 Γ ( 3 4 ) B ( 7 4 , 3 4 ) t 3 8 = 4 2 3 4 Γ ( 7 4 ) 3 π t 3 8 1.16274 t 0.375 , G ( t ) = e 4 ( 1 2 ) 3 4 Γ ( 3 4 ) B ( 1 , 3 4 ) t 3 8 = 42 3 4 e 4 3 Γ ( 3 4 ) t 3 8 99.9092 t 0.375 .

Obviously, all the assumptions in Theorem 3.7 are satisfied. Then, the problem (5.2) has a unique solution in [F,G]. Numerical results are given in Figure 2.

Figure 2
figure 2

Solution (black line) of equation ( 25 ) and the boundaries F (red line) and G (blue line) for Example 5.2.

Example 5.3 In equation (2), we set K(t,s)= e 4 t , α= 3 4 , β=γ= 1 2 , n=0, T=1, p ± =2, q ± =1, h(x)= x 2 (1 1 300 x), g(x)=x, and M=200. Now, we turn to considering the following homogeneous Abel-type integral equation with weakly singular kernels and polynomial law nonlinearity:

x 2 (t) ( 1 1 300 x ( t ) ) = ( 1 2 ) 1 4 Γ ( 3 4 ) 0 t [ e 4 t s 1 2 ( t 1 2 s 1 2 ) 1 4 ] x(s)ds,t[0,1].
(26)

It is clear that now a + =1, a = 1 3 , and b ± =1, so equation (4) holds. After some computation, we find that

F ( t ) = ( 1 2 ) 3 4 Γ ( 3 4 ) B ( 7 4 , 3 4 ) t 3 8 = 4 2 3 4 Γ ( 7 4 ) 3 π t 3 8 1.16274 t 0.375 , G ( t ) = 3 e 4 ( 1 2 ) 3 4 Γ ( 3 4 ) B ( 7 4 , 3 4 ) t 3 8 = 42 3 4 e 4 Γ ( 7 4 ) π t 3 8 190.45 t 0.375 .

Since now A1.16274<190.45B<M=200, obviously, all the assumptions in Theorem 3.6 are satisfied. Then, the problem (5.3) has a solution in [F,G]. Numerical results are given in Figure 3.

Figure 3
figure 3

Solution (black line) of equation ( 26 ) and the boundaries F (red line) and G (blue line) for Example 5.3.

References

  1. Mann WR, Wolf F: Heat transfer between solids and gases under nonlinear boundary conditions. Q. Appl. Math. 1951, 9: 163-184.

    MATH  MathSciNet  Google Scholar 

  2. Goncerzewicz J, Marcinkowska H, Okrasinski W, Tabisz K: On percolation of water from a cylindrical reservoir into the surrounding soil. Zastos. Mat. 1978, 16: 249-261.

    MATH  MathSciNet  Google Scholar 

  3. Keller JJ: Propagation of simple nonlinear waves in gas filled tubes with friction. Z. Angew. Math. Phys. 1981, 32: 170-181. 10.1007/BF00946746

    MATH  Article  Google Scholar 

  4. Atkinson KE: An existence theorem for Abel integral equations. SIAM J. Math. Anal. 1974, 5: 729-736. 10.1137/0505071

    MATH  MathSciNet  Article  Google Scholar 

  5. Bushell PJ, Okrasinski W: Nonlinear Volterra integral equations with convolution kernel. J. Lond. Math. Soc. 1990, 41: 503-510.

    MATH  MathSciNet  Article  Google Scholar 

  6. Gorenflo R, Vessella S Lecture Notes in Mathematics 1461. In Abel Integral Equations. Springer, Berlin; 1991.

    Google Scholar 

  7. Okrasinski W: Nontrivial solutions to nonlinear Volterra integral equations. SIAM J. Math. Anal. 1991, 22: 1007-1015. 10.1137/0522065

    MATH  MathSciNet  Article  Google Scholar 

  8. Gripenberg G: On the uniqueness of solutions of Volterra equations. J. Integral Equ. Appl. 1990, 2: 421-430. 10.1216/jiea/1181075572

    MATH  MathSciNet  Article  Google Scholar 

  9. Mydlarczyk W: The existence of nontrivial solutions of Volterra equations. Math. Scand. 1991, 68: 83-88.

    MATH  MathSciNet  Google Scholar 

  10. Kilbas AA, Saigo M: On solution of nonlinear Abel-Volterra integral equation. J. Math. Anal. Appl. 1999, 229: 41-60. 10.1006/jmaa.1998.6139

    MATH  MathSciNet  Article  Google Scholar 

  11. Karapetyants NK, Kilbas AA, Saigo M: Upper and lower bounds for solutions of nonlinear Volterra convolution integral equations with power nonlinearity. J. Integral Equ. Appl. 2000, 8: 421-448.

    MathSciNet  Article  Google Scholar 

  12. Diogo T, Lima P: Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods. J. Comput. Appl. Math. 2002, 140: 537-557. 10.1016/S0377-0427(01)00408-3

    MATH  MathSciNet  Article  Google Scholar 

  13. Buckwar E: Existence and uniqueness of solutions of Abel integral equations with power-law non-linearities. Nonlinear Anal. TMA 2005, 63: 88-96. 10.1016/j.na.2005.05.004

    MATH  MathSciNet  Article  Google Scholar 

  14. Cima A, Gasull A, Mañosas F: Periodic orbits in complex Abel equations. J. Differ. Equ. 2007, 232: 314-328. 10.1016/j.jde.2006.09.002

    MATH  Article  Google Scholar 

  15. Giné J, Santallusia X: Abel differential equations admitting a certain first integral. J. Math. Anal. Appl. 2010, 370: 187-199. 10.1016/j.jmaa.2010.04.046

    MATH  MathSciNet  Article  Google Scholar 

  16. Gasull A, Li C, Torregrosa J: A new Chebyshev family with applications to Abel equations. J. Differ. Equ. 2012, 252: 1635-1641. 10.1016/j.jde.2011.06.010

    MATH  MathSciNet  Article  Google Scholar 

  17. Baleanu D, Machado JAT, Luo AC-J: Fractional Dynamics and Control. Springer, Berlin; 2012.

    MATH  Book  Google Scholar 

  18. Diethelm K Lecture Notes in Mathematics. The Analysis of Fractional Differential Equations 2010.

    Chapter  Google Scholar 

  19. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

    MATH  Google Scholar 

  20. Lakshmikantham V, Leela S, Devi JV: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge; 2009.

    MATH  Google Scholar 

  21. Miller KS, Ross B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York; 1993.

    MATH  Google Scholar 

  22. Michalski MW Dissertationes Mathematicae CCCXXVIII. In Derivatives of Noninteger Order and Their Applications. Inst. Math., Polish Acad. Sci., Warszawa; 1993.

    Google Scholar 

  23. Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.

    MATH  Google Scholar 

  24. Tarasov VE: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin; 2011.

    Google Scholar 

  25. Balachandran K, Park JY, Julie MD: On local attractivity of solutions of a functional integral equation of fractional order with deviating arguments. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 2809-2817. 10.1016/j.cnsns.2009.11.023

    MATH  MathSciNet  Article  Google Scholar 

  26. Banás J, O’Regan D: On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. J. Math. Anal. Appl. 2008, 345: 573-582. 10.1016/j.jmaa.2008.04.050

    MATH  MathSciNet  Article  Google Scholar 

  27. Banaś J, Rzepka B: Monotonic solutions of a quadratic integral equation of fractional order. J. Math. Anal. Appl. 2007, 322: 1371-1379.

    Google Scholar 

  28. Banaś J, Zajac T: Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity. Nonlinear Anal. TMA 2009, 71: 5491-5500. 10.1016/j.na.2009.04.037

    MATH  Article  Google Scholar 

  29. Banaś J, Zajac T: A new approach to the theory of functional integral equations of fractional order. J. Math. Anal. Appl. 2011, 375: 375-387. 10.1016/j.jmaa.2010.09.004

    MATH  MathSciNet  Article  Google Scholar 

  30. Banaś J, Rzepka B: The technique of Volterra-Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders. Comput. Math. Appl. 2012, 64: 3108-3116. 10.1016/j.camwa.2012.03.006

    MathSciNet  Article  Google Scholar 

  31. Darwish MA: On quadratic integral equation of fractional orders. J. Math. Anal. Appl. 2005, 311: 112-119. 10.1016/j.jmaa.2005.02.012

    MATH  MathSciNet  Article  Google Scholar 

  32. Wang J, Dong X, Zhou Y: Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 545-554. 10.1016/j.cnsns.2011.05.034

    MATH  MathSciNet  Article  Google Scholar 

  33. Wang J, Dong X, Zhou Y: Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 3129-3139. 10.1016/j.cnsns.2011.12.002

    MATH  MathSciNet  Article  Google Scholar 

  34. Bohl E: Monotonie: Löbarkeit und Numerik bei Operatorgleichungen. Springer, Berlin; 1974.

    Book  Google Scholar 

  35. Zeidler E: Applied Functional Analysis: Applications to Mathematical Physics. Springer, New York; 1995.

    Google Scholar 

  36. Prudnikov AP, Brychkov YA, Marichev OI 1. In Integral and Series: Elementary Functions. Nauka, Moscow; 1981.

    Google Scholar 

  37. Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. 7th edition. Academic Press, Amsterdam; 2007.

    MATH  Google Scholar 

  38. Okrasinski W: Nonlinear Volterra equations and physical applications. Extr. Math. 1989, 4: 51-80.

    MathSciNet  Google Scholar 

  39. Schneider W: The general solution of a nonlinear integral equation of convolution type. Z. Angew. Math. Phys. 1982, 33: 140-142. 10.1007/BF00948318

    MATH  MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The first and second authors acknowledge the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169), Key Support Subject (Applied Mathematics), Key Project on the Reforms of Teaching Contents and Course System of Guizhou Normal College and Doctor Project of Guizhou Normal College (13BS010). The third author acknowledges the support by Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0029/13 and APVV-0134-10.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michal Fečkan.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.

Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

Authors’ original file for figure 1

Authors’ original file for figure 2

Authors’ original file for figure 3

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Wang, J., Zhu, C. & Fečkan, M. Analysis of Abel-type nonlinear integral equations with weakly singular kernels. Bound Value Probl 2014, 20 (2014). https://doi.org/10.1186/1687-2770-2014-20

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2014-20

Keywords

  • Abel-type nonlinear integral equations
  • weakly singular kernels
  • existence
  • numerical solutions