Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
Multiple Positive Solutions of a Singular Semipositone Integral Boundary Value Problem for Fractional -Derivatives Equation
By using the fixed point index theorem, this paper investigates a class of singular semipositone integral boundary value problem for fractional -derivatives equations and obtains sufficient conditions for the existence of at least two and at least three positive solutions. Further, an example is given to illustrate the applications of our main results.
Studies on -difference equations appeared already at the beginning of the 20th century in intensive works especially by Jackson , Carmichael , and other authors such as Poincare, Picard and, Ramanujan . Up to date, it has evolved into a multidisciplinary subject, for example, see [4–7] and the references therein. For some recent work on -difference equations, we refer the reader to the papers [8–21], and the basic definitions and properties of -difference calculus can be found in the book [3, 22]. On the other hand, fractional differential equations have gained importance due to their numerous applications in many fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, and probability . Many researchers studied the existence of solutions to fractional boundary value problems, for example, [24–35] and the references therein.
The fractional -difference calculus had its origin in the works by Al-Salam  and Agarwal . More recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional -difference calculus were made, specifically, -analogues of the integral and differential fractional operators properties such as the Mittag-Leffler function, the -Laplace transform, and -Taylor’s formula [3, 13, 22, 38], just to mention some.
However, the theory of boundary value problems for nonlinear -difference equations is still in the initial stage and many aspects of this theory need to be explored.
In , Ferreira considered a Dirichlet type nonlinear -difference boundary value problem as follows: By applying a fixed point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
In other paper, Ferreira  studied the existence of positive solutions to nonlinear -difference boundary value problem as follows:
By using a fixed point theorem in a cone, El-Shahed and Al-Askar  were concerned with the existence of positive solutions to nonlinear -difference equation: where and is the fractional -derivatives of the Caputo type.
Recently, Liang and Zhang  discussed the following nonlinear -fractional three-point boundary value problem: By using a fixed point theorem in partially ordered sets, the authors obtained sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.
In , Graef and Kong investigated the following boundary value problem with fractional -derivatives: where is a parameter, and the uniqueness, existence, and nonexistence of positive solutions are considered in terms of different ranges of .
Furthermore, Ahmad et al.  studied the following nonlinear fractional -difference equation with nonlocal boundary conditions where is the fractional -derivative of the Caputo type, and . The existence of solutions for the problem is shown by applying some well-known tools of fixed point theory such as Banach’s contraction principle, Krasnoselskii’s fixed point theorem, and the Leray-Schauder nonlinear alternative.
It is known that the fractional -derivative of Riemann-Liouville type played an important role in the development of the theory of fractional -derivatives and -integrals and for its applications in pure mathematics, and is a useful tool in the description of nonconservative models. The details can be found in .
Since -calculus has a tremendous potential for applications [3, 22], we find it pertinent to investigate problems in this field. Motivated by the papers [20, 21, 30], we will deal with the following integral boundary value problem of nonlinear fractional -derivatives equation: subject to the boundary conditions where , is parameter with , is the -derivative of Riemann-Liouville type of order , is continuous and semipositone and may be singular at , in which . In the present work, we investigate the existence of positive solutions for fractional -derivatives integral boundary value problem (7) and (8) involving the Riemann-Liouville’s fractional derivative, which is different from . We gave the corresponding Green’s function of the boundary value problem (7) and (8), gave some properties of Green’s function, and constructed a cone by properties of Green’s function. Moreover the existence of at least two and three positive solutions to the boundary value problem (7) and (8) is enunciated.
2. Preliminaries on -Calculus and Lemmas
Let and define The -analogue of the power function with is More generally, if , then Clearly, if then . The -gamma function is defined by and satisfies .
The -derivative of a function is defined by and the -derivatives of higher order by The -integral of a function defined in the interval is given by If and is defined in the interval , then its integral from to is defined by Similar to that for derivatives, an operator is given by The fundamental theorem of calculus applies to these operators and , that is, and if is continuous at , then The following formulas will be used later, namely, the integration by parts formula: where denotes the derivative with respect to the variable .
Definition 1. Let and be a function defined on . The fractional -integral of Riemann-Liouville type is and
The fractional -derivative of order is defined by and for , where is the smallest integer greater than or equal to .
Remark 2. Let and be two functions defined on , then .
Lemma 3. Assume that and , then .
Lemma 4. Let and be a function defined in . Then, the following formulas hold: (1), (2).
Lemma 5 (see ). Let and be a positive integer. Then, the following equality holds:
Lemma 6. Let . Then the unique solution of the equation subject to BC (8) is given by where
Proof. Let us put . In view of Definition 1 and Lemma 4, we see that Then, it follows from Lemma 5 that the solution of (25) and BC (8) is given by for some constants . Since , we have . Differentiating both sides of (29) and with the help of (20) and (22), we obtain Then by the boundary conditions , we get . Thus, (29) reduces to Using the boundary condition , we get Hence, we have Integrate the above equation (33) from to , and using (11), (19) and (20), we obtain then Combining this with (29) and (31) yields This completes the proof of the lemma.
Lemma 8. The function defined by (27) satisfies the following conditions: (i) is a continuous function on , and , for ;(ii), for , where
Proof. The continuity of is easily checked. On the other hand, when , in view of Lemma 3, we have
Further, since we have we get When , since , we have This completes the proof of the lemma.
Remark 9. If we let , then According to , we may take .
Lemma 10. Let . Then the boundary value problem
has a unique solution with
From [39, Theorem 2.3.1], one has the following definition. Let be a retract of real Banach space , be a relatively bounded open subset of be completely continuous operator. The integer be defined by where is an arbitrary retraction and such that . Then the integer is called the fixed point index of on with respect to .
Lemma 11 (see ). Let be a cone in a Banach space . Let be an open bounded subset of with and . Assume that is a compact map such that for . Then (i)if for , then ;(ii)if there exists such that for all and , then .
3. The Main Results
In order to abbreviate our discussion, we give the following assumptions. There exists , such that for all , where , , , and is nonincreasing with respect to , and , are nondecreasing with respect to , where . .
Setting and for any , , we consider the following singular nonlinear boundary value problem: where .
Lemma 12. For any , let be the operator defined by Then is completely continuous.
Proof. For any , Lemma 8 implies that on , and
On the other hand,
Then , which leads to . Thus .
It follows from the nonnegativeness and continuity of and that the operator is continuous. Suppose is any bounded set; then, for any , there is a constant number such that . Let for all , by Lemma 8, we have Hence, is bounded.
On the other hand, for any , according to , there is a constant such that From the property of continuity of , there exists with such that for any , and , when we have where . By means of the Arzela-Ascoli Theorem, is completely continuous.
Theorem 13. Suppose and hold. In addition, assume that the following conditions are satisfied. There exists a constant such that
where . There exist constants with such that
Then BVP (7) and (8) has at least two positive solutions with .
Proof. First, we prove that
To see this, let . Then and for . Now for , we get So, for any , we get It follows from , (61) and Lemma 8 that, for any , This together with (56) yields . From the (i) of Lemma 11, (59) is satisfied.
Let us choose such that Then for the above , according to and , there exists such that, for any ,
Take then .
Now let and . Then, for any , we have This together with (56) yields , which implies that
In the following, let It is easy to see that are bounded sets and satisfy For any , by Remark 9 and (61), we have Thus This means that , for . Thus, it follows from the (i) of Lemma 11 that Similarly, we can prove that, for any , Thus, using (59), (67), and (72), we obtain which implies that has at least one fixed point and satisfies and .
Obviously, is continuous for any . Also, it can be seen that has uniform lower and upper bounds. This directly comes from . Hence, in order to pass the solution of the problem (48) to that of the original problem (7) and (8), we need the following fact:
As in the proof of Lemma 12, we can prove that the sequence is equicontinuous on . Now the Arzela-Ascoli Theorem guarantees that the sequence has a subsequence , converging uniformly on to . As for (61) and the fact , we obtain that Moreover, satisfies the following integral equation: Letting , we have Let , then is a positive solution of BVP (7) and (8).
From (73), has at least one fixed point . Similar to (76), there exists a subsequence such that , and . Let , then is also a positive solution of BVP (7) and (8).
Since , we have . This implies that are two different positive solutions of BVP (7) and (8).
Theorem 14. Suppose , , , and hold. In addition, assume that the following conditions are satisfied. There exists with as given in such that Then BVP (7) and (8) has at least three different positive solutions.
Proof. It can be seen that condition is equivalent to condition . As a consequence we obtain that the BVP (7) and (8) has at least two different positive solutions , with .
On the other hand, choose a real number such that
By , there exists such that Choose and let .
In the following, we will prove that Suppose that (85) is false; then there exists , such that
For and for any , we have It follows from (83) and (85) that we have which is a contradiction. Hence, (85) is ture; from the (ii) of Lemma 11, we get Combining this with (67) yields This implies that has at least one fixed point with . Similar to (76), there exists a subsequence such that , and . Let , then is also a positive solution of BVP (7) and (8). So the proof is complete.
Corollary 15. Suppose , , , and hold. In addition, there exist constants , , with such that