Multiple Positive Solutions for Quadratic Integral Equations of Fractional Order

Abstract

In this paper, the existence of multiple positive solutions for a class of quadratic integral equation of fractional order is obtained, by utilizing Avery-Henderson and Leggett-Williams multiple fixed point theorems on cones. An example is given to illustrate the applicability of our results. We believe that this is a first result concerning the existence of multiple solutions for such quadratic integral equation of fractional order.

1. Introduction and Preliminaries

Recently, there has been great interest for many authors to study quadratic functional integral equations, which has become one of the most attractive and interesting research areas of integral equations and functional integral equations. There is large literature on this topic. We refer the reader to [1–6] for some of very recent results. In fact, as noted in some earlier literature (see, e.g., [5] and references therein), the nonlinear quadratic functional integral equations have been applied to, for example, the theory of radiative transfer, kinetic theory of gases, the theory of neutron transport, the traffic theory, plasma physics, and numerous branches of mathematical physics.

On the other hand, due to the fact that fractional differential and integral equations have recently been extensively applied in various areas of engineering, mathematics, physics and bioengineering, and other applied sciences, there has been a significant development in fractional integral equations in recent years. Many authors especially have focused on the existence and qualitative properties of solutions for quadratic integral equations of fractional order such asand several important variants of (1), and obtain substantial results on this topic (cf. [1–3, 5–9]). However, to the best of our knowledge, it seems that there are no results concerning the existence of multiple solutions for (1) and its variants. That is the main goal of this work. It seems that this is a first result concerning the existence of multiple solutions for such quadratic integral equation of fractional order.

Stimulated by the above works, we aim to investigate the existence of multiple positive solutions for the following fractional order quadratic integral equation:See Section 2 for the hypotheses on the involved functions.

Throughout the rest of this paper, if there is no special statement, we denote by the set of real numbers, and by the set of all functions satisfying that there exists a constant such that

Next, let us recall some notations about cones and two fixed point theorem. For more details, we refer the reader to [10, 11].

Let be a real Banach space. A closed convex set in is called a cone if the following conditions are satisfied:(i)If , then for any ;(ii)If and , then . A nonnegative continuous functional is said to be a concave on if is continuous and Letting be three positive constants and be a nonnegative continuous functional on , we denote In addition, we say that is increasing on if for all with .

The following two theorems are the well-known Avery-Henderson multiple fixed point theorem and Leggett-Williams multiple fixed point theorem, respectively.

Lemma 1 (see [10]). Let be a cone in a real Banach space , and be two increasing, nonnegative, and continuous functionals on , and be a nonnegative continuous functional on with such that, for some and , Moreover, suppose that there exists a completely continuous operator and such that and(i), for all ;(ii), for all ;(iii), and , for all Then has at least two fixed points belonging to such that

Lemma 2 (see [11]). Let be a cone in a real Banach space ,   be a positive constant, be a completely continuous mapping, and be a concave nonnegative continuous functional on with for all . Suppose that there exist three constants with such that (i), and for all ;(ii) for all ;(iii) for all with Then has at least three fixed points in . Furthermore, , and

2. Main Results

Firstly, we list some assumptions:(H1) and .(H2). Moreover, and .(H3), , , and (H4)There exist such that  By a solution of (2) we mean a function satisfying the equation, where Now, we are ready to present our main result.

Theorem 3. Let (H1)–(H4) hold. Then, there exists such that (2) has at least two nonnegative solutions provided that and .

Proof. By using (H3), we can choose satisfyingLet It is not difficult to verify that is a cone in . Let Obviously, ,  , and are increasing, nonnegative, and continuous functionals on with . Moreover, we have for all and . We divide the remaining proof by three steps.
Step  1. Let For every , define an operator on by It is easy to see that .
For and , there holds which yields that Let Then, noting that ,   has a unique fixed point provided that .
Step  2. Now, define an operator on by Noting that, for every and , by (12), we have which yields that .
Next, let us show that is completely continuous. Let in . For , we have which yields thatwhere since . Combining (24) with the fact is uniformly continuous on and uniformly converges to on , we conclude that in . It suffices to show is precompact. It follows from the above proof that is uniformly bounded on . For every and with , we have On the other hand, we have By Lebesgue’s dominated convergence theorem, as , Combining this with the fact that are uniformly continuous on compact sets, we conclude that, for every , there exists such that, for all with and , and , there hold Thus, we conclude that, for every , holds for all with and , and , which yields that, for all with and , there holds Then, we have where are similarly defined. Noting that for all , we conclude and so we have Thus, is equicontinuous on . Then, it is not difficult to obtain that is equicontinuous on . This proves that is precompact, and thus is completely continuous.
Step  3. It remains to verify the assumptions (i)–(iii) of Lemma 1. For every , noting that and for all , by (H4), we have For every , noting that for all , again by (H4), we have In addition, letting it is easy to see that the assumption (iii) of Lemma 1 holds.
Then, by Lemma 1, we conclude that has at least two fixed points . Thus, that is, are two nonnegative solutions for (2).

Remark 4. Due to the influence of fractional term in (2), it becomes more difficult to verify the assumptions of Lemma 1. In addition, we have tried to consider the case of , where it seems that the approach used here can not be applied to study (2). We leave it for further research.

By using Lemma 2, we can obtain the existence of three nonnegative solutions to (2).

Corollary 5. Let (H1)–(H4) hold and the first inequality in (H4) is strengthened toThen, there exists such that (2) has at least three nonnegative solutions provided that and .

Proof. By using (H3), we can choose such thatLet be the same as in Theorem 3 and Moreover, denote and () be the same as in Theorem 3, and Then, by a similar proof to Step of Theorem 3, we can prove that has a unique fixed point provided that , where Now, we can define Analogously to the proof of Theorem 3, we can also show that is completely continuous. Let Obviously, is a concave nonnegative continuous functional on and . For every , by (42), we have which means that maps to . Similarly, by (H4), we have that is, the condition (ii) of Lemma 2 holds. Moreover, the condition (iii) of Lemma 2 follows from the definition of and . It remains to show that the condition (i) of Lemma 2 holds. For every , noting that and thus for all , by (40), we have Then, by Lemma 2, has at least three fixed points in , and thus (2) has at least three nonnegative solutions.

Next, we present an example to show the applicability of our results.

Example 6. For and , let : where are two constants.
It is not difficult to show that (H1) holds with and . Moreover, it is easy to verify that (H2) and (H3) hold. Letting and , we have provided that , and provided that . Then, by Corollary 5, (2) has at least three nonnegative solutions provided that and is sufficiently small.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

H.-S. Ding acknowledges support from NSFC (11461034), the NSF of Jiangxi Province (20143ACB21001), and the Foundation of Jiangxi Provincial Education Department (GJJ150342). M. M. Liu acknowledges support from the Graduate Innovation Fund of Jiangxi Province (YC2016-S157). The research of J. J. Nieto has been partially supported by AEI and the Ministerio de Economia y Competitividad of Spain under Grant MTM2016-75140-P and XUNTA de Galicia under Grant GRC2015-004 and cofinanced by the European Community fund FEDER.