Analysis of Abel-type nonlinear integral equations with weakly singular kernels
© Wang et al.; licensee Springer. 2014
Received: 14 October 2013
Accepted: 19 December 2013
Published: 17 January 2014
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.
KeywordsAbel-type nonlinear integral equations weakly singular kernels existence numerical solutions
Abel-type integral equations are associated with a wide range of physical problems such as heat transfer , nonlinear diffusion , propagation of nonlinear waves , 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).
Many analysis techniques are used to construct the suitable order interval (see Lemma 2, ) and the spaces with suitable weighted norms.
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 , 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 [, Theorem 5]. So even for equation (1) (or (5)) we get a new result.
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.
Let ℳ be the set with the supremum-norm . Clearly, the set is a closed subspace of Banach space . Thus, is a Banach space.
where and . Note that the similar inequality (5) of  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.
where , and .
for all and for all .
, and , for all .
For and , we set(8)
For , we set and(9)
Similarly for and .
We note  an important estimate on the function K, which will be used in the sequel.
Proof We only check the case of with , since the other cases are trivial.
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.
is the well-known Beta function.
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.
where and are defined in equation (8) or (9).
Remark 3.1 Note that and . Next, .
The following result is clear.
then for all . Consequently, the order interval is well defined.
From now on, we suppose that all above assumptions hold: equations (3), (4), (i)-(iii), and (11).
Lemma 3.4 Any solution of equation (2), with for all , satisfies .
Proof Step 1: We prove that for a solution x of equation (2).
Since implies , estimate (13) is an improvement of equation (12).
Hence we can complete the proof. □
Lemma 3.5 The operator maps the order interval into itself.
Consequently, is well defined and . The proof is completed. □
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 .
where we set .
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. □
Hence condition (19) is satisfied and then we get a uniqueness result by Theorem 3.7. Note there is gap in the proof of [, 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.
This completes the proof. □
Now we are ready to present our main result of this section.
5 Illustrative examples
In this section, we pay our attention to show three numerical performance results.
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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