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Abstract and Applied Analysis
Volume 2015, Article ID 759378, 10 pages
http://dx.doi.org/10.1155/2015/759378
Research Article

Positive Solutions for Nonlinear -Fractional Difference Eigenvalue Problem with Nonlocal Conditions

Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia

Received 16 August 2015; Revised 5 November 2015; Accepted 26 November 2015

Academic Editor: Svatoslav Staněk

Copyright © 2015 Wafa Shammakh and Maryam Al-Yami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of positive solutions for nonlinear -fractional difference eigenvalue problem with nonlocal boundary conditions is investigated. Based on the fixed point index theory in cones, sufficient existence of positive solutions conditions is derived for the problem.

1. Introduction

The fractional -calculus is the -extension of ordinary fractional calculus. It has been used by many researchers to adequately describe the evolution of a variety of engineering, economical, physical, and biological processes.

We consider a nonlinear -fractional difference eigenvalue problem with nonlocal boundary conditions given by where denote the fractional -derivative of the Caputo type, is a parameter, and is given by a Riemann-Stieltjes integral

This type of BC includes, as particular cases, multipoint problems when , (see [1]) and a continuously distributed case when (see [24]).

More recently, many people pay attention to BVPs involving nonlinear -difference equations [512].

In [13], Yuan and Yang dealt with some existence and uniqueness results for nonlinear boundary value problems for delayed -fractional difference systems based on a contraction mapping principle and Krasnoselskii’s fixed-point theorem.

In [14], Yang investigated the sufficient conditions for the existence and nonexistence positive solutions for BVP involving nonlinear -fractional difference equations.

Ferreira [4] studied the existence of positive solutions to the nonlinear -fractional BVPs by means of Krasnoselskii’s fixed point theorem in cones.

In this paper, we obtain the results on the existence of one and two positive solutions by utilizing the results of Webb and Lan [15] involving comparison with the principle characteristic value of a related linear problem to the -fractional case. We then use the theory worked out by Webb and Infante in [1619] to study the general nonlocal BCs.

2. Preliminaries

In this section, we will present some definitions and lemmas that will be used in the proof of our main results.

Let defined by [20] The -analogue of the power function with is More generally, if , then Note that if then The -gamma function is defined by and satisfies

The -derivative of a function is here defined by and -derivatives of higher order are defined byThe -integral of a function defined in the interval is given byIf and is defined in the interval , its integral from to is defined by Similarly as done for derivatives, it can be defined an operator ; namely, The fundamental theorem of calculus applies to these operators and ; that is, and if is continuous at , then Basic properties of the two operators can be found in the book [20]. We now point out four formulas that will be used later:where denotes the -derivative with respect to variable [21].

Remark 1 (see [21]). We note that if and , then .

Definition 2 (see [22]). Let and let be a function defined on The fractional -integral of the Riemann-Liouville type is and

Definition 3 (see [22]). The fractional -derivative of the Riemann-Liouville type of order is defined by andwhere is the smallest integer greater than or equal to

Definition 4 (see [22]). The fractional -derivative of the Caputo type of order is defined by

Lemma 5 (see [22]). Let and let be a function defined on Then, the next formulas hold:(1),(2).

Lemma 6 (see [22]). Let , . Then, the next formulas hold:

Theorem 7 (see [23]). Let and Then, the following equality holds:

Lemma 8 (see [24]). Suppose is a completely continuous operator and has no fixed points on . Then the following are true: (i)If for all , then , where is the fixed point index on (ii)If for all , then .

Lemma 9 (see [24]). Let be a cone in Banach space . Suppose that is a completely continuous operator. There exists such that for any and .

Lemma 10 (see [24]). Let be a cone in Banach space . Suppose that is a completely continuous operator. If for any and , then .

Lemma 11. Let be a given function and , then is a solution of BVP (1)-(2) if and only if is a solution of the integral equation where

Proof. Assume that is a solution of BVP (1)-(2).
Applying Theorem 7, (1) can be reduced to an equivalent integral equation: By (2), we obtain Therefore, we obtainConversely, if is a solution of the integral equation (20), using Lemmas 5 and 6, we haveA simple computation shows .

Remark 12. is Green’s function for the local BVP

Lemma 13. Function defined in (20) satisfies the following conditions:(H1) is continuous and for all ;(H2) for all , where

Proof. It is obvious that is nonnegative and continuous.
(H1) For ,and for ,and it is clear that and Therefore
For fixed and we havethat is, is an increasing function of . Obviously, , is increasing in ; therefore is an increasing function of for fixed .
Thus, (H1) holds.
(H2) Suppose now that :On the other hand, if , then we haveand this finished the proof of (H2).

Defining , Green’s function for nonlocal BVP (1)-(2) is given by Throughout the paper we assume the following:

(H3) A is a function of bounded variation, and satisfies for almost every . Note that exists for almost every by (H1).

(H4) The functions satisfy almost everywhere, , and

(H5) satisfies Caratheodory conditions; that is, is measurable for each fixed and is continuous for almost every , and for each , there exists such that for all and almost all

(H6) One has the following:

Lemma 14. If satisfies (H1), (H2), then satisfies (H1), (H2) for a function , the same interval , and the same constant , where satisfies (H4) and

Proof. We have and for Note that because has finite variation and
Thus, Green’s function satisfies (H1), (H2) for a function and the constant .

3. Main Result

Set as a Banach space with the norm . Let denote the standard cone of nonnegative functions. Define where is some subset of .

Note that so . For any , let , , , , and and is bounded.

Define a nonlinear operator and a linear operator by

Lemma 15 (see [18]). Under hypotheses (H1)–(H6) the maps defined in (38) are compact.

Theorem 16. Under hypotheses (H1)–(H6) the maps are .

Proof. For and we haveHence,Also, for , we haveSimilar to the proofs of Lemma 15 and Theorem 16, is compact and maps into .

We will use the Krein-Rutman theorem. We recall that is an eigenvalue of with corresponding eigenfunction if and . The reciprocals of eigenvalues are called characteristic values of . The radius of the spectrum of , denoted by , is given by the well-known spectral radius formula .

Theorem 17 (see [15]). Let be a total cone in a real Banach space and let be a compact linear operator with . If then there is such that .

Thus is an eigenvalue of , the largest possible real eigenvalue, and is the smallest positive characteristic value.

Lemma 18 (see [15]). Assume that (H1)–(H3) hold and let be as defined in (39). Then

Theorem 19 (see [15]). When (H1)–(H3) hold, is an eigenvalue of with eigenfunction in .

Theorem 20 (see [15]). Let and be a corresponding eigenfunction in of norm 1. Then , where If for and for , the first inequality is strict unless is constant for . If for , the second inequality is strict unless is constant for .

Proof (for the local BVP (1)-(2) if ). We now compute the constant and the optimal value of ; that is, we determine so that is minimal.
For , we have by direct integrationFor ,Then we haveAnd the maximum of this expression occurs when ; henceThen
For , we have by direct integrationThen The sign of derivative shows that this is an increasing function of so the minimum occurs at . LetThe minimal value of corresponds to the maximal value of ConsiderThe quantity is an increasing function of so its maximum occurs when LetThen the maximum of occurs when ConsiderHence the minimal value of is

4. The Existence of at Least One Positive Solution

For convenience, we introduce the following notations:Under hypotheses (H1)–(H4) let be defined byThen is a compact linear operator and .

Hence is an eigenvalue of with an eigenfunction in . Let Note that ; hence the condition in the following theorem is more stringent compared with the case if could be used.

Theorem 21. Assume that (A1),(A2).Then (1)-(2) had at least one positive solution.

Proof. Let be such that Then there exists such thatLet We prove that which implies the result. In fact, if (58) does not hold, then there exist and such that
This implies Thus, we have shown This gives And by iterating Thereforeand we have a contradiction. It follows that Let , be chosen so that for all , as in (H2), and almost all
We claim that for all and when . Note that with .
We have for all .
Now, if our claim is false, then we have Therefore,From (66) we firstly deduce that on
Then we have Inserting this into (66) we obtain for
Repeating this process gives Since is strictly positive on this is a contradiction; then By (64) and (69), one hasTherefore, has at least one fixed point , and is a positive solution of BVP (1)-(2).

Theorem 22. Assume that(A3),(A4)Then (1)-(2) had at least one positive solution.

Proof. Let satisfy Then there exists such that For any we have by (71) thatLet be the positive eigenfunction of corresponding to ; that is, . We may suppose that has no fixed point on ; otherwise, the proof is finished. In the following we will show that If (73) is not true, then there is and such that . It is clear that and .
Set Obviously, . It follows from that and using this and (72), we have which contradicts (74). Thus, (73) holds.
By Lemma 9, we haveOn the other hand, let satisfy . Then there exists such thatBy (H5) there exists an function such thatHence, we have Since is the radius of the spectrum of , exists.
LetWe prove that, for each , In fact, if not, there exist and such that .
This together with (80) implies This implies Therefore, we have , a contradiction. Taking , it follows from (74) and properties of index that Now (77) and (85) combined implyTherefore, has at least one fixed point , and is a positive solution of BVP (1)-(2).

5. The Existence of Two Positive Solutions

Theorem 23. Suppose (A2), (A3), and(A5) for some Then (1)-(2) had at least two positive solutions.

Proof. By (A5), we haveso that , for all . Now Lemma 8 yields On the other hand, in view of (A2), we may take so that (69) holds (see the proof of Theorem 21). From (A3), we may take so that (77) holds (see the proof of Theorem 22).
Combining (88), (69), and (77), we arrive at Consequently, has at least two fixed points, with one on and the other on Therefore, (1)-(2) had at least two positive solutions.

Theorem 24. Suppose (A1), (A4), and (A6) for some .Then (1)-(2) had at least two positive solutions.

Proof. By (A6), we haveso that , for all , and by Lemma 8 this yields On the other hand, in view of (A1), we may take so that (64) holds (see the proof of Theorem 21). In addition, from (A4), we may take so that (85) holds (see the proof of Theorem 22).
Combining (91), (64), and (85), we arrive at Hence, has at least two fixed points, with one on and the other on Therefore, (1)-(2) had at least two positive solutions.

We illustrate the applicability of these results with some examples.

Example 25. Consider the problem Here we have , , and
It is readily shown that , , .
Also, for By calculation, we find , and the smallest calculated is . We find Hence, by Theorem 21, there is at least one positive solution if and ; that is, there is a positive solution if
By Theorem 22, there does not exist a positive solution if either or ; that is, if or no positive solution exists.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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