JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/418569 418569 Research Article Positive Solutions for a Class of Singular Boundary Value Problems with Fractional q -Difference Equations Wang Jufang http://orcid.org/0000-0003-1166-7938 Yu Changlong Guo Yanping Sadarangani Kishin College of Sciences Hebei University of Science and Technology Shijiazhuang Hebei 050018 China hebust.edu.cn 2015 642015 2015 06 07 2014 21 09 2014 642015 2015 Copyright © 2015 Jufang Wang et al. 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.

We discuss a class of singular boundary value problems of fractional q -difference equations. Some existence and uniqueness results are obtained by a fixed point theorem in partially ordered sets. Finally, we give an example to illustrate the results.

1. Introduction

In recent years, many papers on fractional differential equations have appeared, because of their demonstrated applications in various fields of science and engineering; see  and the references therein. Based on the increasingly extensive application of discrete fractional calculus and the development of q -difference calculus or quantum calculus (see  and the references therein), fractional q -difference equations have attracted the attention of researchers for the numerous applications in a number of fields such as physics, chemistry, aerodynamics, biology, economics, control theory, mechanics, electricity, signal and image processing, biophysics, blood flow phenomena, aerodynamics, and fitting of experimental data; see . Some recent work on the existence theory of fractional q -difference equations can be found in . However, the study of singular boundary value problems (BVPs) with fractional q -difference equations is at its infancy and lots of work on the topic should be done.

Recently, in , Ferreira has investigated the existence of positive solution for the following fractional q -difference equations BVP (1) D q α y x + f x , y t = 0 , 0 < x < 1 , y 0 = D q y 0 = 0 , D q y 1 = β 0 , by applying a fixed point theorem in cones.

More recently, in , Caballero et al. have studied positive solutions for the following BVP: (2) D 0 + α u t + f t , u t = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 , by fixed point theorem in partially ordered sets.

Motivated by the work above, in this paper, we discuss the existence and uniqueness of solutions for the singular BVPs of factional q -difference equations given by (3) ( D q α u ) ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 , ( D q u ) ( 0 ) = 0 , where 2 < α 3 and f : [ 0,1 ] × [ 0 , ) [ 0 , ) is continuous with lim t 0 + f ( t , · ) = (i.e., f is singular at t = 0 ).

2. Preliminary Results

For convenience, we present some definitions and lemmas which will be used in the proofs of our results.

Let q ( 0,1 ) and define (4) a q = 1 - q a 1 - q , a R .

The q -analogue of the power function ( a - b ) n with n N 0 is (5) a - b 0 = 1 , a - b n = k = 0 n - 1 a - b q k , n N , a , b R . More generally, if α R , then (6) a - b ( α ) = a α n = 0 n - 1 a - b q n a - b q α + n . Note that if b = 0 then a ( α ) = a α . The q -gamma function is define by (7) Γ q x = 1 - q ( x - 1 ) 1 - q x - 1 , x R { 0 , - 1 , - 2 , } , and it satisfies Γ q ( x + 1 ) = [ x ] q Γ q ( x ) .

Following, let us recall some basic concepts of q -calculus .

Definition 1.

For 0 < q < 1 , we define the q -derivative of a real-value function f as (8) D q f x = f x - f q x 1 - q x , D q f 0 = lim x 0 D q f x . Note that lim q 1 - D q f ( x ) = f ( x ) .

Definition 2.

The higher order q -derivatives are defined inductively as (9) ( D q 0 f ) ( x ) = f ( x ) , ( D q n f ) ( t ) = D q ( D q n - 1 f ) ( t ) , n N .

For example, D q ( t k ) = [ k ] q t k - 1 , where k is a positive integer and the bracket [ k ] q = ( q k - 1 ) / ( q - 1 ) . In particular, D q ( t 2 ) = ( 1 + q ) t .

Definition 3.

The q -integral of a function f in the interval [ 0 , b ] is given by (10) I q f x = 0 x f t d q t = x 1 - q n = 0 f x q n q n , x 0 , b . If a [ 0 , b ] and f is defined in the interval [ 0 , b ] , its integral from a to b is define by (11) a b f ( t ) d q t = 0 b f ( t ) d q t - 0 a f ( t ) d q t . Similarly as done for derivatives, an operator I q n can be define, namely, (12) ( I q 0 f ) ( x ) = f ( x ) , ( I q n f ) ( x ) = I q ( I q n - 1 f ) ( x ) , n N .

Observe that (13) D q I q f ( x ) = f ( x ) , and if f is continuous at x = 0 , then I q D q f ( x ) = f ( x ) - f ( 0 ) .

We now point out three formulas ( i D q denotes the derivative with respect to variable i ) (14) a t - s ( α ) = a α t - s ( α ) , (15)    t D q t - s ( α ) = α q t - s ( α - 1 ) , D x q 0 x f x , t d q t = 0 x x D q f x , t d q t + f q x , x .

Remark 4.

We note that if α 0 and a b t , then ( t - a ) ( α ) ( t - b ) ( α ) .

Definition 5 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Let α 0 and f be a function defined on [ 0,1 ] . The fractional q -integral of the Riemann-Liuville type is ( I q 0 f ) ( x ) = f ( x ) and (16) I q α f x = 1 Γ q α 0 x x - q t α - 1 f t d q t , α > 0 , x 0,1 .

Definition 6 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

The fractional q -derivative of the Riemann-Liuville type of α 0 is defined by ( D q 0 f ) ( x ) = f ( x ) and (17) ( D q α f ) ( x ) = ( D q m I q m - α f ) ( x ) , α > 0 , where m is the smallest integer greater than or equal to α .

Lemma 7 (see [<xref ref-type="bibr" rid="B21">21</xref>, <xref ref-type="bibr" rid="B23">23</xref>]).

Let α , β 0 and let f be a function define on [ 0,1 ] . Then, the next formulas hold:

( I q β I q α f ) ( x ) = ( I q α + β f ) ( x ) ,

( D q α I q α f ) ( x ) = f ( x ) .

Lemma 8 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

Let α > 0 and let p be a positive integer. Then, the following equality holds: (18) I q α D q p f x = D q p I q α f x - k = 0 p - 1 x α - p + k Γ q α + k - p + 1 D q k f 0 .

Lemma 9.

Let y ( t ) C [ 0,1 ] L 1 [ 0,1 ] and 2 < α 3 ; then the BVP (19) D q α u t + y t = 0 , 0 < t < 1 , u 0 = u 1 = 0 , D q u 0 = 0 , has a unique solution (20) u ( t ) = 0 1 G ( t , q s ) y ( s ) d q s , where (21) G t , s = 1 Γ q ( α ) 1 - s ( α - 1 ) t α - 1 - t - s ( α - 1 ) , 0 s t 1 , 1 - s ( α - 1 ) t α - 1 , 0 t s 1 .

Proof.

By Lemmas 7 and 8, we see that (22) D q α u t = - y t D q α u t I q α D q 3 I q 3 - α u t = - I q α y t D q α u t u t = c 1 t α - 1 + c 2 t α - 2 D q α u t u t = + c 3 t α - 3 - 1 Γ q α 0 t t - q s α - 1 D q α u t u t = c 3 t α - 3 1 Γ q α × y s d q s , where c 1 , c 2 , and c 3 are some constants to be determined. Since u ( 0 ) = 0 , we must have c 3 = 0 . Now, differentiating both sides of (22) and using (15), we obtain (23) D q u t = α - 1 q c 1 t α - 2 + α - 2 q c 2 t α - 3 D q u t - 1 Γ q ( α ) 0 t α - 1 q t - q s ( α - 2 ) y ( s ) d q s . Using ( D q u ) ( 0 ) = 0 and u ( 1 ) = 0 , we must set c 2 = 0 , and (24) c 1 = 1 Γ q ( α ) 0 1 1 - q s ( α - 1 ) y ( s ) d q s .

Finally, we obtain (25) u t = t α - 1 Γ q α 0 1 1 - q s α - 1 y s d q s u t - 1 Γ q α 0 t t - q s α - 1 y s d q s u t = 0 1 G ( t , q s ) y ( s ) d q s . The proof is complete.

Lemma 10.

Function G defined above satisfies the following conditions:

G ( t , q s ) is a continuous function on [ 0,1 ] × [ 0,1 ] ;

G ( t , q s ) 0 for t , s [ 0,1 ] .

Proof.

(i) Obviously, G ( t , q s ) is continuous on [ 0,1 ] × [ 0,1 ] .

(ii) Let (26) g 1 t , s = 1 - s α - 1 t α - 1 - t - s α - 1 , 0 s t 1 , g 2 ( t , s ) = 1 - s ( α - 1 ) t α - 1 , 0 t s 1 . It is clear that g 2 ( t , q s ) 0 , for t , s [ 0,1 ] . Now, in view of Remark 4, for t 0 (27) g 1 t , q s = 1 - q s α - 1 t α - 1 - t - q s α - 1 g 1 t , q s = t α - 1 1 - q s α - 1 - 1 - q s t α - 1 g 1 t , q s t α - 1 1 - q s α - 1 - 1 - q s α - 1 = 0 . Therefore, G ( t , q s ) 0 . This proof is complete.

By J we denote the class of those functions β : [ 0 , ) [ 0,1 ) satisfying the following condition; β ( t n ) 1 implies t n 0 .

Theorem 11 (see [<xref ref-type="bibr" rid="B31">31</xref>]).

Let ( X , ) be a partially ordered set and suppose that there exists a metric d in X such that ( X , d ) is a complete metric space. Let T : X X be a nondecreasing mapping such that there exists an element x 0 X with x 0 T x 0 . Suppose that there exists β J such that (28) d T x , T y β d x , y · d x , y f o r x , y X w i t h x y . Assume that either T is continuous or X is such that (29) i f    x n    i s a n o n d e c r e a s i n g s e q u e n c e i n    X s u c h t h a t    x n x t h e n    x n x n N . Besides if (30) f o r e a c h    x , y X    t h e r e e x i s t s    z X w h i c h i s c o m p a r a b l e t o    x    a n d    y , then T has a unique fixed point.

Let C [ 0,1 ] = { x : [ 0,1 ] R , continuous } be the Banach space with the classic metric given by d ( x , y ) = sup 0 t 1 { | x ( t ) - y ( t ) | } .

Notice that this space can be equipped with a partial order given by (31) x , y C [ 0,1 ] , x y x ( t ) y ( t ) , for    t [ 0,1 ] . In , it is proved that ( C [ 0,1 ] , ) satisfies condition (29) of Theorem 11. Moreover, for x , y C [ 0,1 ] , as the function max ( x , y ) C [ 0,1 ] , ( C [ 0,1 ] , ) satisfies condition (30).

3. Main Result

In this section, we will consider the question of positive solutions for BVP (3). At first, we prove some lemmas required for the main result.

Lemma 12.

Let 0 < σ < 1 , 2 < α 3 and F : ( 0,1 ] R is a continuous function with lim t 0 + F ( t ) = . Suppose that t σ F ( t ) is a continuous function on [ 0,1 ] . Then the function defined by (32) H ( t ) = 0 1 G ( t , q s ) F ( s ) d q s is continuous on [ 0,1 ] , where G ( t , s ) is Green function be given in Lemma 9.

Proof.

We will divide the proof into three parts.

Case 1 ( t 0 = 0 ). First, H ( 0 ) = 0 . Since t σ F ( t ) is continuous on [ 0,1 ] , we can find a positive constant M such that | t σ F ( t ) | M for any t [ 0,1 ] . Thus, (33) H t - H 0 = H t = 0 1 G t , q s F s d q s = 0 1 G t , q s s - σ s σ F s d q s = 0 t 1 - q s α - 1 t α - 1 - t - q s α - 1 Γ q α s - σ s σ F ( s ) d q s + t 1 1 - q s α - 1 t α - 1 Γ q α s - σ s σ F ( s ) d q s = 0 1 1 - q s α - 1 t α - 1 Γ q α s - σ s σ F ( s ) d q s - 0 t t - q s α - 1 Γ q α s - σ s σ F ( s ) d q s M 0 1 1 - q s α - 1 t α - 1 Γ q α s - σ d q s + M 0 t t - q s α - 1 Γ q α s - σ d q s = M t α - 1 Γ q α 0 1 1 - q s α - 1 s - σ d q s + 0 t 1 - q s t ( α - 1 ) s - σ d q s . For 0 t ( 1 - ( q s / t ) ) ( α - 1 ) s - σ d q s , let u = s / t ; then we obtain (34) 0 t 1 - q s t ( α - 1 ) s - σ d q s = t 1 - σ 0 1 1 - q u ( α - 1 ) u - σ d q u . Hence, (35) H t M t α - 1 Γ q α 0 1 1 - q s α - 1 s - σ d q s H t + M t α - σ Γ q α 0 1 1 - q u α - 1 u - σ d q u H t = M t α - 1 Γ q α + M t α - σ Γ q α β q ( 1 - σ , α ) , where β q denotes the q -beta function.

When t 0 , we see that H ( t ) H ( 0 ) ; that is H ( t ) is continuous at t 0 = 0 .

Case 2 ( t 0 ( 0,1 ) ). We should prove H ( t n ) H ( t 0 ) when t n t 0 . Without loss of generality, we consider t n > t 0 (it is the same argument for t n < t 0 ). In fact, (36) H t n - H t 0 = 0 1 t n α - 1 1 - q s α - 1 Γ q α s - σ s σ F ( s ) d q s - 0 t n t n - q s α - 1 Γ q α s - σ s σ F s d q s - 0 1 t 0 α - 1 1 - q s α - 1 Γ q α s - σ s σ F s d q s + 0 t 0 t 0 - q s α - 1 Γ q α s - σ s σ F ( s ) d q s = 0 1 ( t n α - 1 - t 0 α - 1 ) 1 - q s α - 1 Γ q α s - σ s σ F ( s ) d q s - 0 t 0 t n - q s α - 1 - t 0 - q s α - 1 Γ q α s - σ s σ F s d q s - t 0 t n t n - q s α - 1 Γ q α s - σ s σ F ( s ) d q s M t n α - 1 - t 0 α - 1 Γ q α 0 1 1 - q s α - 1 s - σ d q s + M Γ q α 0 t 0 t n - q s α - 1 - t 0 - q s α - 1 s - σ d q s + M Γ q α t 0 t n t n - q s α - 1 s - σ d q s = M Γ q ( α ) β q ( 1 - σ , α ) ( t n α - 1 - t 0 α - 1 ) + M Γ q ( α ) ( a n + b n ) , where (37) a n = 0 t 0 t n - q s α - 1 - t 0 - q s α - 1 s - σ d q s , b n = t 0 t n t n - q s α - 1 s - σ d q s . When n , we verify a n 0 .

As t n t 0 , then ( ( t n - q s ) ( α - 1 ) - ( t 0 - q s ) ( α - 1 ) ) s - σ 0 , when n . Moreover, (38) t n - q s α - 1 - t 0 - q s α - 1 s - σ 2 s - σ , 0 1 2 s - σ d q s = 2 1 - σ q s 1 - σ 0 1 = 2 1 - σ q < .

We have ( ( t n - q s ) ( α - 1 ) - ( t 0 - q s ) ( α - 1 ) ) s - σ converges pointwise to the zero function and | ( t n - q s ) ( α - 1 ) - ( t 0 - q s ) ( α - 1 ) | s - σ is bounded by a function belonging to L 1 [ 0,1 ] , by Lebesgue’s dominated convergence theorem a n 0 when n .

Now, we prove b n 0 when n .

In fact, as (39) b n = t 0 t n t n - q s α - 1 s - σ d q s b n t 0 t n s - σ d q s = s 1 - σ 1 - σ q t 0 t n b n = 1 1 - σ q t n 1 - σ - t 0 1 - σ , and taking into account that t n t 0 , we get b n 0 when n .

Above all, we obtain | H ( t n ) - H ( t 0 ) | 0 , when n .

Case 3 ( t 0 = 1 ). It is easy to check that H ( 1 ) = 0 and H ( t ) is continuous at t 0 = 1 . The proof is the same as the proof of Case 1.

Lemma 13.

Suppose that 0 < σ < 1 . Then, (40) max 0 t 1 0 1 G ( t , q s ) s - σ d q s = A α - 1 - A α - σ Γ q ( α ) β q ( 1 - σ , α ) , where A = ( ( α - 1 ) / ( α - σ ) ) 1 / ( 1 - σ ) .

Proof.

(41) 0 1 G t , q s s - σ d q s = 0 t 1 - q s α - 1 t α - 1 - t - q s α - 1 Γ q α s - σ d q s + t 1 1 - q s α - 1 t α - 1 Γ q α s - σ d q s = 0 1 1 - q s α - 1 t α - 1 Γ q α s - σ d q s - 0 t t - q s α - 1 Γ q α s - σ d q s = t α - 1 Γ q α 0 1 1 - q s α - 1 s - σ d q s - t α - σ Γ q α 0 1 1 - q s α - 1 s - σ d q u = t α - 1 - t α - σ Γ q α β q 1 - σ , α .

Let g ( t ) = t α - 1 - t α - σ , t [ 0,1 ] .

Since g ( t ) = ( α - 1 ) t α - 2 - ( α - σ ) t α - σ - 1 , let g ( t ) = 0 ; we can get g ( t ) has a maximum at the point t 0 = A = ( ( α - 1 ) / ( α - σ ) ) 1 / ( 1 - σ ) .

Hence, (42) max 0 t 1 0 1 G ( t , q s ) s - σ d q s = A α - 1 - A α - σ Γ q ( α ) β q ( 1 - σ , α ) .

For the convenience, we denote max 0 t 1 0 1 G ( t , q s ) s - σ d q s by K .

Next, we denote the class of functions ϕ : [ 0 , ) [ 0 , ) by A satisfying

ϕ is nondecreasing;

ϕ ( x ) < x for any x > 0 ;

β ( x ) = ϕ ( x ) / x J , where J is the class of functions appearing in Theorem 11.

We give our main result as follows.

Theorem 14.

Let 0 < σ < 1 , 2 < α 3 , f : [ 0,1 ] × [ 0 , ) [ 0 , ) is continuous and lim t 0 + f ( t , · ) = , and t σ f ( t , y ) is a continuous function on [ 0,1 ] × [ 0 , ) . Assume that there exists 0 < λ 1 / K such that for x , y [ 0 , ) with y x and t [ 0,1 ] , (43) 0 t σ ( f ( t , y ) - f ( t , x ) ) λ ϕ ( y - x ) , where ϕ A . Then the BVP (3) has a unique positive solution (i.e., x ( t ) > 0 for t ( 0,1 ) ).

Proof.

We define the cone P by (44) P = { u C [ 0,1 ] : u ( t ) 0 } . It is clear that P is a complete metric space as P is a closed set of C [ 0,1 ] . It is also easy to check that P satisfies conditions (29) and (30) of Theorem 11.

We define the operator T by (45) T u t = 0 1 G t , q s f s , u s d q s T u t = 0 1 G ( t , q s ) s - σ s σ f ( s , u ( s ) ) d q s . In view of Lemma 12, T u C [ 0,1 ] . Moreover, it follows from the nonnegativeness of G ( t , q s ) and t σ f ( t , y ) that T u P for u P . Thus, T : P P .

Next, we will prove that assumptions in Theorem 11 are satisfied.

First, for u v , we have (46) T u t = 0 1 G t , q s f s , u s d q s T u t = 0 1 G t , q s s - σ s σ f s , u s d q s T u t 0 1 G t , q s s - σ s σ f s , v s d q s T u t = ( T v ) ( t ) . Hence, the operator T is nondecreasing. Besides, for u v and u v , (47) d T u , T v = max t 0,1 T u t - T v t = max t 0,1 T u t - T v t = max t 0,1 0 1 G t , q s f s , u s - f s , v s d q s = max t 0,1 0 1 G t , q s s - σ s σ f s , u s - f s , v s d q s max t [ 0,1 ] 0 1 G ( t , q s ) s - σ λ ϕ ( u ( s ) - v ( s ) ) d q s . Since ϕ is nondecreasing and u ( s ) - v ( s ) d ( u , v ) , (48) d T u , T v max t 0,1 0 1 G t , q s s - σ λ ϕ d u , v d q s d T u , T v = λ ϕ d u , v max t 0,1 0 1 G t , q s s - σ d q s d T u , T v = λ ϕ ( d ( u , v ) ) K . Moreover, when 0 < λ 1 / K , we get (49) d T u , T v ϕ d u , v d T u , T v = ϕ d u , v d u , v · d u , v d T u , T v = β ( d ( u , v ) ) · d ( u , v ) . Obviously, the last inequality is satisfied for u = v .

Taking into account that the zero function satisfies 0 T 0 , in view of Theorem 11, the operator T has a unique fixed point x ( t ) in P .

At last, we will prove x ( t ) is a positive solution. We assume that there exists 0 < t 1 < 1 such that x ( t 1 ) = 0 . Since x ( t ) of problem (3) is a fixed point of the operator T , we have (50) x ( t ) = 0 1 G ( t , q s ) f ( s , x ( s ) ) d q s , for 0 < t < 1 , x ( t 1 ) = 0 1 G ( t 1 , q s ) f ( s , x ( s ) ) d q s = 0 . For the nonnegative character of G ( t , q s ) and f ( s , x ) , the last relation gives (51) G t 1 , q s f s , x s = 0 a . e . ( s ) . f is continuous and lim t 0 + f ( t , · ) = ; then for M > 0 , we can find δ > 0 , and, for s [ 0,1 ] ( 0 , δ ) , we have f ( s , 0 ) > M . It is clear that [ 0,1 ] ( 0 , δ ) { s [ 0,1 ] : f ( s , x ( s ) ) > M } and μ ( [ 0,1 ] ( 0 , δ ) ) > 0 , where μ is the Lebesgue measure on [ 0,1 ] . That is to say, G ( t 1 , q s ) f ( s , x ( s ) ) = 0 a.e. ( s ) . This is a contradiction because G ( t 1 , q s ) is a rational function in s .

Therefore, x ( t ) > 0 for t ( 0,1 ) .

The proof is complete.

4. Example

Consider the following singular BVP: (52) D 1 / 2 5 / 2 u ( t ) + λ ( t 2 + 1 ) ln ( 1 + u ( t ) ) t 1 / 2 = 0 , 0 < t < 1 , λ > 0 , u ( 0 ) = u ( 1 ) = 0 , ( D 1 / 2 u ) ( 0 ) = 0 . Here, α = 2.5 , q = 1 / 2 , σ = 1 / 2 , and f ( t , u ) = λ ( t 2 + 1 ) ln ( 1 + u ( t ) ) / t 1 / 2 for ( t , u ) [ 0,1 ] × [ 0 , ) . Notice that f is continuous in [ 0,1 ] × [ 0 , ) and lim t 0 + f ( t , · ) = .

At first, we define ϕ by (53) ϕ : [ 0 , ) [ 0 , ) , ϕ ( x ) = ln ( 1 + x ) . It is clear that ϕ ( x ) = ln ( 1 + x ) is a nondecreasing function; for u v , we can get (54) ϕ ( u ) - ϕ ( v ) 0 .

Moreover, for u v , ϕ also satisfies (55) ϕ ( u ) - ϕ ( v ) ϕ ( u - v ) . In fact, when u v , (56) ϕ u - v - ϕ u - ϕ v = ln 1 + u - v ϕ u - v ϕ u - ϕ v    - ln 1 + u - ln 1 + v ϕ u - v ϕ u - ϕ v = ln 1 + u - v 1 + v 1 + u ϕ u - v ϕ u - ϕ v = ln ( 1 + ( u - v ) v 1 + u ) 0 , equivalently (57) ϕ ( u ) - ϕ ( v ) ϕ ( u - v ) . Above all, 0 ϕ ( u ) - ϕ ( v ) ϕ ( u - v ) for u v .

Second, for u v and t [ 0,1 ] , we have (58) 0 t 1 / 2 f t , u - f t , v 0 = λ t 2 + 1 ln 1 + u - ln 1 + v 0 λ t 2 + 1 ln 1 + u - v 0 2 λ ln ( 1 + u - v ) ; that is, f satisfies assumptions of Theorem 14.

Third, we should prove ϕ ( x ) belongs to A . By elemental calculus, it is easy to check that ϕ is nondecreasing and ϕ ( x ) < x , for x > 0 .

In order to prove β ( x ) = ϕ ( x ) / x J , we notice that if β ( t n ) 1 , then the sequence ( t n ) is a bounded sequence because in contrary case, that is, t n , we get (59) β ( t n ) = ln ( 1 + t n ) t n 0 . Now, we assume that t n 0 , and then we find ε > 0 such that for each n N there exists ρ n n with t ρ n ε .

Since ( t n ) is a bounded sequence, we can find a subsequence ( t k n ) of ( t ρ n ) with t k n a , for certain a [ 0,1 ) . When β ( t n ) 1 , it implies that (60) β ( t k n ) = ln ( 1 + t k n ) t k n 1 . and, as the unique solution of ln ( 1 + x ) = x is x 0 = 0 , we deduce that a = 0 . Therefore, t k n 0 and this contradicts the fact that t k n ε for every n N .

Thus, t n 0 and this proves that β J .

Finally, in view of Theorem 14, (61) 2 λ 1 K = 1 1 / 4 3 / 2 - 1 / 4 1 / 2 / Γ 1 / 2 3 / 2 · β 1 / 2 1 / 2 , 3 / 2 2 λ 10.96511985 ; that is, when λ 5.48256 , boundary value problem (52) has a unique positive solution.

Conflict of Interests

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

Acknowledgments

This paper is supported by the Natural Science Foundation of China (11201112), the Natural Science Foundation of Hebei Province (A2013208147), (A2014208152), and (A2015208114), and the Foundation of Hebei Education Department (Z2014062).

Bai Z. H. Positive solutions for boundary value problem of nonlinear fractional differential equation Journal of Mathematical Analysis and Applications 2005 311 2 495 505 10.1016/j.jmaa.2005.02.052 MR2168413 2-s2.0-25144460994 Campos L. M. B. C. On the solution of some simple fractional differential equations International Journal of Mathematics and Mathematical Sciences 1990 13 3 481 496 10.1155/S0161171290000709 MR1068011 Delbosco D. Rodino L. Existence and uniqueness for a nonlinear fractional differential equation Journal of Mathematical Analysis and Applications 1996 204 2 609 625 10.1006/jmaa.1996.0456 MR1421467 ZBL0881.34005 2-s2.0-0030528474 Kilbas A. A. Trujillo J. J. Differential equations of fractional order: methods, results and problems-I Applicable Analysis 2001 78 1-2 153 192 10.1080/00036810108840931 MR1887959 Li C. F. Luo X. N. Zhou Y. Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations Computers & Mathematics with Applications 2010 59 3 1363 1375 10.1016/j.camwa.2009.06.029 MR2579500 2-s2.0-74149083504 Ling Y. Ding S. S. A class of analytic functions defined by fractional derivation Journal of Mathematical Analysis and Applications 1994 186 2 504 513 10.1006/jmaa.1994.1313 MR1293007 2-s2.0-0000729947 Miller K. S. Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 New York, NY, USA John Wiley & Sons MR1219954 Qiu T. Bai Z. Existence of positive solutions for singular fractional differential equations Electronic Journal of Differential Equations 2008 146 1 9 MR2448901 Samko S. G. Kilbas A. A. Marichev O. I. Fractional Integral and Derivative. Theory and Applications 1993 Yverdon, Switzerland Gordon and Breach Science Publishers MR1347689 Zhang S. The existence of a positive solution for a nonlinear fractional differential equation Journal of Mathematical Analysis and Applications 2000 252 2 804 812 10.1006/jmaa.2000.7123 MR1800180 ZBL0972.34004 2-s2.0-0034670545 Zhou Y. Jiao F. Nonlocal Cauchy problem for fractional evolution equations Nonlinear Analysis: Real World Applications 2010 11 5 4465 4475 10.1016/j.nonrwa.2010.05.029 MR2683890 2-s2.0-77955515765 Kac V. Cheung P. Quantum Calculus 2002 New York, NY, USA Springer Jackson F. H. q -difference equations The American Journal of Mathematics 1910 32 4 305 314 10.2307/2370183 MR1506108 Carmichael R. D. The general theory of linear q -difference equations The American Journal of Mathematics 1912 34 2 147 168 10.2307/2369887 MR1506145 Mason T. E. On properties of the solutions of linear q -difference equations with entire function coefficients The American Journal of Mathematics 1915 37 4 439 444 10.2307/2370216 MR1506268 Adams C. R. On the linear ordinary q -difference equation Annals of Mathematics: Second Series 1928 30 1–4 195 205 10.2307/1968274 MR1502876 Page D. N. Information in black hole radiation Physical Review Letters 1993 71 23 3743 3746 10.1103/PhysRevLett.71.3743 MR1247149 2-s2.0-4243893296 Ahmad B. Ntouyas S. K. Boundary value problems for q -difference inclusions Abstract and Applied Analysis 2011 2011 15 292860 10.1155/2011/292860 MR2793778 Yu C. L. Wang J. F. Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives Advances in Difference Equations 2013 2013, article 124 10.1186/1687-1847-2013-124 MR3063222 2-s2.0-84886577631 Al-Salam W. A. Some fractional q -integrals and q -derivatives Proceedings of the Edinburgh Mathematical Society: Series II 1966 15 135 140 10.1017/S0013091500011469 MR0218848 Agarwal R. P. Certain fractional q-integrals and q-derivatives Proceedings of the Cambridge Philosophical Society 1969 66 365 370 MR0247389 Rajković P. M. Marinković S. D. Stanković M. S. Fractional integrals and derivatives in q -calculus Applicable Analysis and Discrete Mathematics 2007 1 1 311 323 MR2316607 Atici F. M. Eloe P. W. Fractional q -calculus on a time scale Journal of Nonlinear Mathematical Physics 2007 14 3 333 344 10.2991/jnmp.2007.14.3.4 MR2350094 Ferreira R. A. Nontrivial solutions for fractional q -difference boundary value problems Electronic Journal of Qualitative Theory of Differential Equations 2010 70 1 10 MR2740675 Ferreira R. A. C. Positive solutions for a class of boundary value problems with fractional q -differences Computers & Mathematics with Applications 2011 61 2 367 373 10.1016/j.camwa.2010.11.012 MR2754144 2-s2.0-78651236363 El-Shahed M. Al-Askar F. M. Positive solutions for boundary value problem of nonlinear fractional q -difference equation ISRN Mathematical Analysis 2011 2011 12 385459 MR2793792 10.5402/2011/385459 Graef J. R. Kong L. Positive solutions for a class of higher order boundary value problems with fractional q -derivatives Applied Mathematics and Computation 2012 218 19 9682 9689 10.1016/j.amc.2012.03.006 MR2916148 2-s2.0-84860430866 Ahmad B. Ntouyas S. K. Purnaras I. K. Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations Difference equations: new trends and applications in biology, medicine and biotechnology Advances in Difference Equations 2012 2012, article 140 10.1186/1687-1847-2012-140 2-s2.0-84873356580 Yu C. Wang J. Positive solutions of nonlocal boundary value problem for high-order nonlinear fractional q -difference equations Abstract and Applied Analysis 2013 2013 9 928147 MR3129338 10.1155/2013/928147 Caballero J. Harjani J. Sadarangani K. Positive solutions for a class of singular fractional boundary value problems Computers & Mathematics with Applications 2011 62 3 1325 1332 10.1016/j.camwa.2011.04.013 MR2824719 2-s2.0-79960973986 Amini-Harandi A. Emani H. A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations Nonlinear Analysis: Theory, Methods & Applications 2010 72 5 2238 2242 10.1016/j.na.2009.10.023 Nieto J. J. Rodríguez-López R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations Order 2005 22 3 223 239 10.1007/s11083-005-9018-5 MR2212687 2-s2.0-33644688928