Abstract
In this paper, we investigate Abeltype 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 63:8896, 2005). As an extension, solutions in closed form of some ErdélyiKobertype fractional integral equations are given. Finally theoretical results with three illustrative examples are presented.
MSC: 26A33, 45E10, 45G05.
Keywords:
Abeltype nonlinear integral equations; weakly singular kernels; existence; numerical solutions1 Introduction
Abeltype 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 Abeltype 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 Darbouxtype 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 Abeltype nonlinear integral equation with powerlaw nonlinearity on an order interval as follows:
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 Abeltype nonlinear integral equation with weakly singular kernels:
where are given functions for some , h is increasing, g is nondecreasing such that
for some positive constants , , , , , , and , the function is nonnegative and it has either the form or for some function , specified later. is the Gamma function. Of course, we suppose
It is obvious that equation (1) or
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 ErdélyiKobertype fractional integral equations (the special case of equation (5) if ):
where , , and and the symbol denotes the ErdélyiKobertype 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 ErdélyiKobertype 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 supremumnorm . 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
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 socalled 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:
We note [34] an important estimate on the function K, which will be used in the sequel.
Lemma 2.2The functionhas the following estimate:
Proof We only check the case of with , since the other cases are trivial.
Integrating n times stepbystep 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 wellknown 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
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).
which implies that
Next we set
Then we have
and so
hence
thus
consequently
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
Hence we can complete the proof. □
To solve equation (2), we introduce an operator by
Lemma 3.5The operatormaps the order intervalinto itself.
Proof To achieve our aim, we only need to verify that and :
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 ArzelaAscoli 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,
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
for all, , then equation (2) has a unique solution inprovided we have
and
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
with
Since (note equation (18))
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 ErdélyiKobertype integral equations
This section is devoted to a derivation of explicit solutions of some ErdélyiKobertype integral equations. In order to establish this, we introduce the following useful result.
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
where the constantCsatisfies the following equation:
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
First, Theorem 4.2 gives the exact solution of equation (23). Next, by changing we get
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 Abeltype integral equation with weakly singular kernels and powerlaw nonlinearity:
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 Abeltype integral equation with weakly singular kernels and polynomial law nonlinearity:
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 VEGAMS 1/0071/14, VEGASAV 2/0029/13 and APVV013410.
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