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

JinRong Wang12, Chun Zhu2 and Michal Fečkan34*

Author Affiliations

1 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, P.R. China

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

3 Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, Bratislava, 842 48, Slovakia

4 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

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Boundary Value Problems 2014, 2014:20  doi:10.1186/1687-2770-2014-20

Dedicated to Professor Ivan Kiguradze

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/20

 Received: 14 October 2013 Accepted: 19 December 2013 Published: 17 January 2014

© 2014 Wang et al.; licensee Springer.

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

### 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.

##### Keywords:
Abel-type nonlinear integral equations; weakly singular kernels; existence; numerical solutions

### 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 [4-16] 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 [17-24].

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 [25-33] 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:

(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:

(2)

where are given functions for some , h is increasing, g is nondecreasing such that

(3)

for some positive constants , , , , , , and , the function is non-negative and it has either the form or for some function , specified later. is the Gamma function. Of course, we suppose

(4)

It is obvious that equation (1) or

(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 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 ):

(6)

where , , and and the symbol denotes the Erdélyi-Kober-type fractional integrals [19] of the function , which is given by

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 ℳ be the set with the supremum-norm . Clearly, the set is a closed subspace of Banach space . Thus, is a Banach space.

Let q be a continuous function on with for all and let be the set

with the weighted norm

(7)

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

where and . Note that the similar inequality (5) of [13] is incorrect.

Consider the cone in ℳ. The so-called partial ordering induced by the cone is given by for all and all . In general [34,35], a set is called an order interval where E is an ordered Banach space. We know that every order interval is closed. Moreover, if for all with , then every order interval is bounded.

We introduce some conditions on the functions K, , as follows:

(i) where , and .

(ii) for all and for all .

(iii) , and , for all .

• For and , we set

(8)

• For , we set and

(9)

Similarly for and .

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

Lemma 2.2The functionhas the following estimate:

(10)

Proof We only check the case of with , since the other cases are trivial.

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

from 0 to t and using we immediately derive

Replacing t by , 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 2.3Letλ, γ, μ, andνbe constants such that, , , and. Then

and

where

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 , , we derive

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 , we introduce the following functions:

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

Remark 3.1 Note that and . Next, .

The following result is clear.

Lemma 3.2If

(11)

thenfor all. Consequently, the order intervalis well defined.

Remark 3.3 If , , and then equation (11) reads

which is satisfied, since equation (3) implies , (see equation (4)) and clearly . This case occurs for instance when and with and , .

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

Lemma 3.4Any solutionof equation (2), withfor all, satisfies.

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

Set . Then we obtain

which implies that

(12)

Next we set

Then we have

and so

hence

thus

consequently

(13)

Since implies , estimate (13) is an improvement of equation (12).

Step 2: We prove that . Fix and set

for . Then like above, for , we get

which implies

Hence

and so

Consequently, we arrive at

Since is arbitrarily, we have

Hence we can complete the proof. □

To solve equation (2), we introduce an operator by

(14)

Lemma 3.5The operatormaps the order intervalinto itself.

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

(15)

(16)

First we show equation (15):

Secondly we derive equation (16):

Since obviously, the operator S is strictly increasing in and if then , . Hence

so

Consequently, is well defined and . The proof is completed. □

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

Theorem 3.6Equation (2) has a solution in. Moreover,

are fixed points ofwith

Now we are ready to state the following uniqueness result. But first we note that the above considerations can be repeated for any , so we get , , , , , and as continuous functions of . Note is nonincreasing, is nondecreasing, and , can be continuously extended to . Then . We still keep the notation , , , and .

Theorem 3.7If there are constantsψ, χand continuous functionsandonsuch that

(17)

for all, , then equation (2) has a unique solution inprovided we have

(18)

and

(19)

where we set.

Proof For any we set and . Clearly, we have

for with and specified below, so . Then for any , we derive

which implies

consequently, we obtain

(20)

with

Since (note equation (18))

and as , we see that

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

for any . Furthermore, for , we have (note equation (18))

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

Summarizing we see that there is and so that

This shows that is a contraction with respect to the norm 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 . Then , , , and . Moreover, Remark 3.3 can be applied to get an existence result. If in addition then , and it is not difficult to see that , , , and

Then , so equation (18) holds. Next, we derive

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 4.1Letand. Then

Proof Set . By using Lemma 2.3, we have

This completes the proof. □

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

Theorem 4.2Let, , , and. Then equation (6) is solvable and its solutioncan be written as

(21)

where the constantCsatisfies the following equation:

(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

(23)

First, Theorem 4.2 gives the exact solution of equation (23). Next, by changing we get

(24)

Of course, we get a solution . In equation (5) for (24), we set , , , , , , , and . After some computation, we find that

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

Figure 1. Solution of equation (23) and the boundariesFandGfor Example 5.1 coincide with the unique solution.

Example 5.2 In equation (5), we set , , , , , , and . Now, we turn to consider the following homogeneous Abel-type integral equation with weakly singular kernels and power-law nonlinearity:

(25)

After some computation, we find that

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

Figure 2. Solution (black line) of equation (25) and the boundariesF(red line) andG(blue line) for Example 5.2.

Example 5.3 In equation (2), we set , , , , , , , , , and . Now, we turn to considering the following homogeneous Abel-type integral equation with weakly singular kernels and polynomial law nonlinearity:

(26)

It is clear that now , , and , so equation (4) holds. After some computation, we find that

Since now , obviously, all the assumptions in Theorem 3.6 are satisfied. Then, the problem (5.3) has a solution in . Numerical results are given in Figure 3.

Figure 3. Solution (black line) of equation (26) and the boundariesF(red line) andG(blue line) for Example 5.3.

### 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.

### 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.

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