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
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).
A completely different approach is given in Buckwar  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:
Many analysis techniques are used to construct the suitable order interval (see Lemma 2, ) and the spaces with suitable weighted norms.
where are given functions for some , h is increasing, g is nondecreasing such that
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
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 , 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.
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 ):
where , , and and the symbol denotes the Erdélyi-Kober-type fractional integrals  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.
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
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  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
• For , we set and
Similarly for and .
We note  an important estimate on the function K, which will be used in the sequel.
Lemma 2.2The function has the following estimate:
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
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.
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.
then for all . Consequently, the order interval is 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 solution of equation (2), with for all , satisfies .
Proof Step 1: We prove that for a solution x of equation (2).
Set . Then we obtain
which implies that
Next we set
Then we have
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
Consequently, we arrive at
Since is arbitrarily, we have
Hence we can complete the proof. □
To solve equation (2), we introduce an operator by
Lemma 3.5The operator maps the order interval into 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
Consequently, is well defined and . The proof is completed. □
Theorem 3.6Equation (2) has a solution in . Moreover,
are fixed points of with
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 functions and on such that
for all , , then equation (2) has a unique solution in provided we have
where we set .
Proof For any we set and . Clearly, we have
for with and specified below, so . Then for any , we derive
consequently, we obtain
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 [, 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 4.1Let and . 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 solution can 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 Abel-type integral equation with weakly singular kernels and power-law 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 Abel-type 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.
The authors declare that they have no competing interests.
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.
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.
Keller, JJ: Propagation of simple nonlinear waves in gas filled tubes with friction. Z. Angew. Math. Phys.. 32, 170–181 (1981). Publisher Full Text
Atkinson, KE: An existence theorem for Abel integral equations. SIAM J. Math. Anal.. 5, 729–736 (1974). Publisher Full Text
Okrasinski, W: Nontrivial solutions to nonlinear Volterra integral equations. SIAM J. Math. Anal.. 22, 1007–1015 (1991). Publisher Full Text
Gripenberg, G: On the uniqueness of solutions of Volterra equations. J. Integral Equ. Appl.. 2, 421–430 (1990). Publisher Full Text
Kilbas, AA, Saigo, M: On solution of nonlinear Abel-Volterra integral equation. J. Math. Anal. Appl.. 229, 41–60 (1999). Publisher Full Text
Diogo, T, Lima, P: Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods. J. Comput. Appl. Math.. 140, 537–557 (2002). Publisher Full Text
Buckwar, E: Existence and uniqueness of solutions of Abel integral equations with power-law non-linearities. Nonlinear Anal. TMA. 63, 88–96 (2005). Publisher Full Text
Cima, A, Gasull, A, Mañosas, F: Periodic orbits in complex Abel equations. J. Differ. Equ.. 232, 314–328 (2007). Publisher Full Text
Giné, J, Santallusia, X: Abel differential equations admitting a certain first integral. J. Math. Anal. Appl.. 370, 187–199 (2010). Publisher Full Text
Gasull, A, Li, C, Torregrosa, J: A new Chebyshev family with applications to Abel equations. J. Differ. Equ.. 252, 1635–1641 (2012). Publisher Full Text
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.. 15, 2809–2817 (2010). Publisher Full Text
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.. 345, 573–582 (2008). Publisher Full Text
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. 71, 5491–5500 (2009). Publisher Full Text
Banaś, J, Zajac, T: A new approach to the theory of functional integral equations of fractional order. J. Math. Anal. Appl.. 375, 375–387 (2011). Publisher Full Text
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.. 64, 3108–3116 (2012). Publisher Full Text
Darwish, MA: On quadratic integral equation of fractional orders. J. Math. Anal. Appl.. 311, 112–119 (2005). Publisher Full Text
Wang, J, Dong, X, Zhou, Y: Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations. Commun. Nonlinear Sci. Numer. Simul.. 17, 545–554 (2012). Publisher Full Text
Wang, J, Dong, X, Zhou, Y: Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator. Commun. Nonlinear Sci. Numer. Simul.. 17, 3129–3139 (2012). Publisher Full Text
Schneider, W: The general solution of a nonlinear integral equation of convolution type. Z. Angew. Math. Phys.. 33, 140–142 (1982). Publisher Full Text