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Multi-strip fractional q-integral boundary value problems for nonlinear fractional q-difference equations

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Abstract

In this article, we study the existence and uniqueness of solutions for multi-strip fractional q-integral boundary value problems of nonlinear fractional q-difference equations. By using the Banach contraction principle, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory some interesting results are obtained. Some examples are presented to illustrate the results.

MSC:34A08, 34B18, 39A13.

1 Introduction

In this article, we investigate the following nonlinear fractional q-difference equation for multi-strip fractional q-integral boundary condition:

{ D q α u ( t ) = f ( t , u ( t ) ) , t ( 0 , T ) , u ( 0 ) = 0 , u ( T ) = i = 1 m γ i ( I q i β i u ) | η i ξ i = i = 1 m γ i ( I q i β i u ( ξ i ) I q i β i u ( η i ) ) ,
(1.1)

where 1<α2, 0<q, q i <1, β i >0, 0 η i < ξ i T, γ i R for all i=1,2,,m are given constants, D q α is the fractional q-derivative of Riemann-Liouville type of order α, I q i β i is the fractional q i -integral of order β i and f:[0,T]×RR is a continuous function.

q-Difference calculus or quantum calculus was initiated by Jackson [1]. Basic definitions and properties of quantum calculus can be found in the book [2]. The fractional q-difference calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. For some recent work on the subject, we refer to [512] and the references cited therein.

Strip conditions appear in the mathematical modeling of certain real world problems. For motivation, discussion on multi-strip boundary conditions, examples and a consistent bibliography on these problems, we refer to the papers [1320] and the references therein. As it is pointed out in [20], the boundary condition in (1.1) can be interpreted in the sense that a controller at the right-end of the considered interval is influenced by a discrete distribution of finite many nonintersecting strips of arbitrary length expressed in terms of fractional integral boundary conditions.

The significance of investigating problem (1.1) is that the multi-strip fractional q-integral boundary condition is very general and includes many conditions as special cases. In particular, if β i =1 for i=1,2,,m, then the condition of (1.1) is reduced to the multi-strip q-integral condition as follows:

u(0)=0,u(T)= γ 1 η 1 ξ 1 u(s) d q 1 s+ γ 2 η 2 ξ 2 u(s) d q 2 s++ γ m η m ξ m u(s) d q m s.

Moreover, we emphasize that we have different quantum numbers and as far as we know this is new in the literature.

The rest of the paper is organized as follows. In Section 2 we briefly give some basic notations, definitions and lemmas. In Section 3 we collect some auxiliary results needed in the proofs of our main results. Section 4 contains the main results concerning existence and uniqueness results for problem (1.1), which are shown by applying the Banach contraction principle, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory. Some examples are presented in Section 5 to illustrate the results.

2 Preliminaries

To make this paper self-contained, below we recall some known facts on fractional q-calculus. The presentation here can be found in, for example, [21, 22].

For q(0,1), define

[ a ] q = 1 q a 1 q ,aR.
(2.1)

The q-analogue of the power function ( a b ) k with k N 0 :={0,1,2,} is

( a b ) ( 0 ) =1, ( a b ) ( k ) = i = 0 k 1 ( a b q i ) ,kN,a,bR.
(2.2)

More generally, if γR, then

( a b ) ( γ ) = a γ i = 0 1 ( b / a ) q i 1 ( b / a ) q γ + i ,a0.
(2.3)

Note if b=0, then a ( γ ) = a γ . We also use the notation 0 ( γ ) =0 for γ>0. The q-gamma function is defined by

Γ q (x)= ( 1 q ) ( x 1 ) ( 1 q ) x 1 ,xR{0,1,2,}.
(2.4)

Obviously, Γ q (x+1)= [ x ] q Γ q (x).

The q-derivative of a function h is defined by

( D q h)(x)= h ( x ) h ( q x ) ( 1 q ) x for x0and( D q h)(0)= lim x 0 ( D q h)(x),
(2.5)

and q-derivatives of higher order are given by

( D q 0 h ) (x)=h(x)and ( D q k h ) (x)= D q ( D q k 1 h ) (x),kN.
(2.6)

The q-integral of a function h defined on the interval [0,b] is given by

( I q h)(x)= 0 x h(s) d q s=x(1q) i = 0 h ( x q i ) q i ,x[0,b].
(2.7)

If a[0,b] and h is defined in the interval [0,b], then its integral from a to b is defined by

a b h(s) d q s= 0 b h(s) d q s 0 a h(s) d q s.
(2.8)

Similar to derivatives, an operator I q k is given by

( I q 0 h ) (x)=h(x)and ( I q k h ) (x)= I q ( I q k 1 h ) (x),kN.
(2.9)

The fundamental theorem of calculus applies to these operators D q and I q , i.e.,

( D q I q h)(x)=h(x),
(2.10)

and if h is continuous at x=0, then

( I q D q h)(x)=h(x)h(0).
(2.11)

Definition 2.1 Let ν0 and h be a function defined on [0,T]. The fractional q-integral of Riemann-Liouville type is given by ( I q 0 h)(x)=h(x) and

( I q ν h ) (x)= 1 Γ q ( ν ) 0 x ( x q s ) ( ν 1 ) h(s) d q s,ν>0,x[0,T].
(2.12)

Definition 2.2 The fractional q-derivative of Riemann-Liouville type of order ν0 is defined by ( D q 0 h)(x)=h(x) and

( D q ν h ) (x)= ( D q l I q l ν h ) (x),ν>0,
(2.13)

where l is the smallest integer greater than or equal to ν.

Definition 2.3 For any x,s>0,

B q (x,s)= 0 1 u ( x 1 ) ( 1 q u ) ( s 1 ) d q u
(2.14)

is called the q-beta function.

From [2], the expression of q-beta function in terms of the q-gamma function can be written as

B q (x,s)= Γ q ( x ) Γ q ( s ) Γ q ( x + s ) .

Lemma 2.4 [4]

Let α,β0 and f be a function defined in [0,T]. Then the following formulas hold:

  1. (1)

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

  2. (2)

    ( D q α I q α f)(x)=f(x).

Lemma 2.5 [22]

Let α>0 and ν be a positive integer. Then the following equality holds:

( I q α D q ν f ) (x)= ( D q ν I q α f ) (x) k = 0 ν 1 x α ν + k Γ q ( α + k ν + 1 ) ( D q k f ) (0).
(2.15)

3 Some auxiliary lemmas

Lemma 3.1 Let α,β>0 and 0<q<1. Then we have

0 η ( η q s ) ( α 1 ) s ( β ) d q s= η α + β B q (α,β+1).
(3.1)

Proof Using the definitions of q-analogue of power function and q-beta function, we have

0 η ( η q s ) ( α 1 ) s ( β ) d q s = ( 1 q ) η n = 0 q n ( η q η q n ) ( α 1 ) ( η q n ) β = ( 1 q ) η n = 0 q n η α 1 ( 1 q q n ) ( α 1 ) η β q n β = ( 1 q ) η α + β n = 0 q n ( 1 q q n ) ( α 1 ) q n β = η α + β 0 1 ( 1 q s ) ( α 1 ) s ( β ) d q s = η α + β B q ( α , β + 1 ) .

The proof is complete. □

Lemma 3.2 Let α,β>0 and 0<p,q<1. Then we have

0 η 0 x ( η p x ) ( α 1 ) ( x q y ) ( β 1 ) d q y d p x= η α + β [ β ] q Γ p ( α ) Γ p ( β + 1 ) Γ p ( α + β + 1 ) .
(3.2)

Proof Taking into account Lemma 3.1, we have

0 η 0 x ( η p x ) ( α 1 ) ( x q y ) ( β 1 ) d q y d p x = 0 η ( η p x ) ( α 1 ) 0 x ( x q y ) ( β 1 ) d q y d p x = 1 [ β ] q 0 η ( η p x ) ( α 1 ) x ( β ) d p x = 1 [ β ] q η α + β B p ( α , β + 1 ) = η α + β [ β ] q Γ p ( α ) Γ p ( β + 1 ) Γ p ( α + β + 1 ) .

This completes the proof. □

For convenience, we set a nonzero constant

Λ= T α 1 i = 1 m γ i Γ q i ( α ) Γ q i ( α + β i ) ( ξ i α + β i 1 η i α + β i 1 ) .
(3.3)

Lemma 3.3 Let β i >0, 0<q, q i <1, γ i R, η i , ξ i (0,T) and η i < ξ i for all i=1,2,,m. Then, for a given yC([0,1],R), the unique solution of the linear q-difference equation

D q α u(t)=y(t),t(0,T),1<α2,
(3.4)

subject to the multi-strip fractional q-integral condition

u(0)=0,u(T)= i = 1 m γ i ( I q i β i u ) | η i ξ i = i = 1 m γ i ( I q i β i u ( ξ i ) I q i β i u ( η i ) ) ,
(3.5)

is given by

u ( t ) = t α 1 Λ { 0 T ( T q s ) ( α 1 ) Γ q ( α ) y ( s ) d q s i = 1 m γ i Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) y ( x ) d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) y ( x ) d q x d q i s ) } + 0 t ( t q s ) ( α 1 ) Γ q ( α ) y ( s ) d q s ,
(3.6)

where Λ is defined by (3.3).

Proof Since 1<α2, we take n=2. In view of Definition 2.2 and Lemma 2.4, the linear q-difference equation (3.4) can be written as

( I q α D q 2 I q 2 α u ) (t)= ( I q α y ) (t).

Using Lemma 2.5, we obtain

u(t)= c 1 t α 1 + c 2 t α 2 + 0 t ( t q s ) ( α 1 ) Γ q ( α ) y(s) d q s
(3.7)

for some constants c 1 , c 2 R. Since u(0)=0, we get c 2 =0.

Applying the Riemann-Liouville fractional q i -integral of order β i >0 with c 2 =0 for (3.7) and taking into account Lemma 3.1, we have

I q i β i u ( ξ i ) = 0 ξ i ( ξ i q i s ) ( β i 1 ) Γ q i ( β i ) ( c 1 s α 1 + 0 s ( s q x ) ( α 1 ) Γ q ( α ) y ( x ) d q x ) d q i s = 1 Γ q i ( β i ) Γ q ( α ) 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) y ( x ) d q x d q i s + c 1 Γ q i ( β i ) 0 ξ i ( ξ i q i s ) ( β i 1 ) s α 1 d q i s = 1 Γ q i ( β i ) Γ q ( α ) 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) y ( x ) d q x d q i s + c 1 Γ q i ( α ) ξ i α + β i 1 Γ q i ( α + β i ) .
(3.8)

Repeating the above process with t= η i and using the second condition of (3.5), we get a constant c 1 as follows:

c 1 = 1 Λ { i = 1 m γ i Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) y ( x ) d q x d q i s 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) y ( x ) d q x d q i s ) 0 T ( T q s ) ( α 1 ) Γ q ( α ) y ( s ) d q s } .
(3.9)

Substituting the values of constants c 1 and c 2 in the linear solution (3.7), the desired result in (3.6) is obtained. □

4 Main results

Let C=C([0,T],R) denote the Banach space of all continuous functions from [0,T] to endowed with the supremum norm defined by u= sup t [ 0 , T ] |u(t)|. In view of Lemma 3.3, we define an operator A:CC by

( A u ) ( t ) = t α 1 Λ { 0 T ( T q s ) ( α 1 ) Γ q ( α ) f ( s , u ( s ) ) d q s i = 1 m γ i Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) f ( x , u ( x ) ) d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) f ( x , u ( x ) ) d q x d q i s ) } + 0 t ( t q s ) ( α 1 ) Γ q ( α ) f ( s , u ( s ) ) d q s ,
(4.1)

with Λ0. It should be noticed that problem (1.1) has solutions if and only if the operator A has fixed points.

For the sake of convenience, we put

Φ = T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + T α Γ q ( α + 1 ) .
(4.2)

The first existence and uniqueness result is based on the Banach contraction mapping principle.

Theorem 4.1 Let f:[0,T]×RR be a continuous function satisfying the assumption

(H1) there exists a constant L>0 such that |f(t,u)f(t,v)|L|uv| for each t[0,T] and u,vR.

If

LΦ<1,
(4.3)

where a constant Φ is given by (4.2), then the multi-strip boundary value problem (1.1) has a unique solution on [0,T].

Proof We transform problem (1.1) into a fixed point problem, u=Au, where the operator A is defined by (4.1). Applying the Banach contraction mapping principle, we will show that the operator A has a fixed point which is a unique solution of problem (1.1).

Setting sup t [ 0 , T ] |f(t,0)|= M 0 < and choosing

r M 0 Φ 1 L Φ ,

with L Φ satisfying (4.3), we will show that A B r B r , where the set B r ={uC:ur}. For any u B r , and taking into account Lemma 3.2, we have

A u sup t [ 0 , T ] { t α 1 | Λ | { 0 T ( T q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s + i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s ) } + 0 t ( t q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s } T α 1 | Λ | { 0 T ( T q s ) ( α 1 ) Γ q ( α ) ( | f ( s , u ( s ) ) f ( s , 0 ) | + | f ( s , 0 ) | ) d q s + i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) × ( | f ( x , u ( x ) ) f ( x , 0 ) | + | f ( x , 0 ) | ) d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) × ( | f ( x , u ( x ) ) f ( x , 0 ) | + | f ( x , 0 ) | ) d q x d q i s ) } + 0 T ( T q s ) ( α 1 ) Γ q ( α ) ( | f ( s , u ( s ) ) f ( s , 0 ) | + | f ( s , 0 ) | ) d q s ( L r + M 0 ) { T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + T α Γ q ( α + 1 ) } = ( L r + M 0 ) Φ r .

It follows that A B r B r .

For u,vC and for each t[0,T], we have

| A u ( t ) A v ( t ) | T α 1 | Λ | { 0 T ( T q s ) ( α 1 ) Γ q ( α ) ( | f ( s , u ( s ) ) f ( s , v ( s ) ) | ) d q s + i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) × ( | f ( s , u ( s ) ) f ( s , v ( s ) ) | ) d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) ( | f ( s , u ( s ) ) f ( s , v ( s ) ) | ) d q x d q i s ) } + 0 T ( T q s ) ( α 1 ) Γ q ( α ) ( | f ( s , u ( s ) ) f ( s , v ( s ) ) | ) d q s L u v { T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + T α Γ q ( α + 1 ) } = L Φ u v .

The above result leads to AuAvLΦuv. As LΦ<1, by (4.3), therefore A is a contraction. Hence, by the Banach contraction mapping principle, we deduce that A has a fixed point which is the unique solution of problem (1.1). □

Next, we prove the existence of at least one solution by using Krasnoselskii’s fixed point theorem.

Lemma 4.2 (Krasnoselskii’s fixed point theorem [23])

Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (a) Ax+ByM whenever x,yM; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists zM such that z=Az+Bz.

Theorem 4.3 Assume that f:[0,T]×RR is a continuous function satisfying assumption (H1). In addition, we suppose that

(H2) |f(t,u)|ψ(t), (t,u)[0,T]×R and ψC([0,T], R + ).

If the following condition holds

L Γ q ( α + 1 ) ( T 2 α 1 | Λ | + T α ) <1,
(4.4)

then the multi-strip boundary value problem (1.1) has at least one solution on [0,T].

Proof We define sup t [ 0 , T ] |ψ(t)|=ψ and choose a suitable constant R such that

RψΦ,

where Φ is defined by (4.2). Furthermore, we define the operators A 1 and A 2 on B R ={uC:uR} by

( A 1 u ) ( t ) = t α 1 Λ i = 1 m γ i Γ q i ( β i ) Γ q ( α ) 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) f ( x , u ( x ) ) d q x d q i s t α 1 Λ i = 1 m γ i Γ q i ( β i ) Γ q ( α ) 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) f ( x , u ( x ) ) d q x d q i s ,

and

( A 2 u)(t)= t α 1 Λ 0 T ( T q s ) ( α 1 ) Γ q ( α ) f ( s , u ( s ) ) d q s+ 0 t ( t q s ) ( α 1 ) Γ q ( α ) f ( s , u ( s ) ) d q s.

It should be noticed that A= A 1 + A 2 .

For any u,v B R , we have

A 1 u + A 2 v ψ { T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α B q i ( β i , α + 1 ) Γ q i ( β i ) + i = 1 m | γ i | η i β i + α B q i ( β i , α + 1 ) Γ q i ( β i ) ) + T α Γ q ( α + 1 ) } = ψ Φ R .

Therefore ( A 1 u)+( A 2 v) B R . Obviously, condition (4.4) implies that A 2 is a contraction mapping.

Finally, we will show that A 1 is compact and continuous. The continuity of f coupled with assumption (H2) implies that the operator A 1 is continuous and uniformly bounded on B R . We define sup ( t , u ) [ 0 , T ] × B R |f(t,u)|= M <. For t 1 , t 2 [0,T], t 2 < t 1 and u B R , we have

| ( A 1 u ) ( t 1 ) ( A 1 u ) ( t 2 ) | | t 1 α 1 t 2 α 1 | | Λ | i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) × 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s + | t 1 α 1 t 2 α 1 | | Λ | i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) × 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s M | t 1 α 1 t 2 α 1 | | Λ | Γ q ( α + 1 ) { i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) } .

Actually, as | t 1 t 2 |0 the right-hand side of the above inequality tends to zero independently of u. So A 1 is relatively compact on B R . Therefore, by the Arzelá-Ascoli theorem, A 1 is compact on B R . Thus all the assumptions of Lemma 4.2 are satisfied. Thus, the boundary value problem (1.1) has at least one solution on [0,T]. The proof is complete. □

Remark 4.4 In the above theorem we can interchange the roles of the operators A 1 and A 2 to obtain the second result replacing (4.4) by the following condition:

L T α 1 | Λ | Γ q ( α + 1 ) ( i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) <1.

Now, our third existence result is based on Leray-Schauder’s nonlinear alternative.

Lemma 4.5 (Nonlinear alternative for single-valued maps [24])

Let E be a Banach space, C be a closed, convex subset of E, U be an open subset of C and 0U. Suppose that F: U ¯ C is a continuous, compact (that is, F( U ¯ ) is a relatively compact subset of C) map. Then either

  1. (i)

    F has a fixed point in U ¯ , or

  2. (ii)

    there is uU (the boundary of U in C) and λ(0,1) with u=λF(u).

Theorem 4.6 Assume that f:[0,T]×RR is a continuous function. In addition we suppose that:

(H3) there exist a continuous nondecreasing function ϕ:[0,)(0,) and a function pC([0,T], R + ) such that

| f ( t , u ) | p(t)ϕ ( | u | ) for each (t,u)[0,T]×R;

(H4) there exists a constant N>0 such that

N p ϕ ( N ) Φ >1,

where Φ is defined by (4.2).

Then the multi-strip boundary value problem (1.1) has at least one solution on [0,T].

Proof Firstly, we will show that the operator A defined by (4.1) maps bounded sets (balls) into bounded sets in C. For a positive number ρ, let B ρ ={uC:uρ} be a bounded ball in C. Then, for t[0,T], we have

| A u ( t ) | T α 1 | Λ | { 0 T ( T q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s + i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s ) } + 0 T ( T q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s T α 1 | Λ | Γ q ( α + 1 ) ( p ϕ ( u ) T α + p ϕ ( u ) i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + p ϕ ( u ) i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + p ϕ ( u ) T α Γ q ( α + 1 ) p ϕ ( ρ ) T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + p ϕ ( ρ ) T α Γ q ( α + 1 ) : = K .

Therefore, we deduce that AuK.

Secondly, we will show that A maps bounded sets into equicontinuous sets of C. Let sup ( t , u ) [ 0 , T ] × B ρ |f(t,u)|= K <, τ 1 , τ 2 [0,T] with τ 2 < τ 1 and u B ρ . Then we have

| ( A u ) ( τ 1 ) ( A u ) ( τ 2 ) | | τ 1 α 1 τ 2 α 1 | | Λ | { 0 T ( T q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s + i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s ) } + | 0 τ 1 ( τ 1 q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s 0 τ 2 ( τ 2 q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s | | τ 1 α 1 τ 2 α 1 | K | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + | τ 1 α 1 τ 2 α 1 | K Γ q i ( α + 1 ) .

Obviously, the right-hand side of the above inequality tends to zero independently of x B ρ as τ 2 τ 1 . Therefore it follows by the Arzelá-Ascoli theorem that A:CC is completely continuous.

Let u be a solution of problem (1.1). Then, for t[0,T], and following similar computations as in the first step with (H3), we have

u T α 1 | Λ | Γ q ( α + 1 ) ( p ϕ ( u ) T α + p ϕ ( u ) i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + p ϕ ( u ) i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) + p ϕ ( u ) T α Γ q ( α + 1 ) = p ϕ ( u ) Φ .

Consequently, we have

u p ϕ ( u ) Φ 1.

In view of (H4), there exists a constant N>0 such that uN. Let us set

U= { x C : u < N } .

Note that the operator A: U ¯ C is continuous and completely continuous. From the choice of U, there is no uU such that u=λAu for some λ(0,1). Consequently, by nonlinear alternative of Leray-Schauder type (Lemma 4.5), we deduce that A has a fixed point in U ¯ , which is a solution of the boundary value problem (1.1). This completes the proof. □

As the forth result, we prove the existence of solutions of (1.1) by using Leray-Schauder degree theory.

Theorem 4.7 Let f:[0,T]×RR be a continuous function. Assume that

(H5) there exist constants 0ω< Φ 1 , where Φ are given by (4.2), and Ψ>0 such that

| f ( t , u ) | ω|u|+Ψfor each (t,u)[0,T]×R.

Then the multi-strip boundary value problem (1.1) has at least one solution on [0,T].

Proof Let A be the operator defined by (4.1). We will prove that there exists at least one solution uC of the operator equation u=Au.

Setting a ball B ρ C, where a constant radius ρ >0, by

B ρ = { u C : sup t [ 0 , T ] | u ( t ) | < ρ } ,

it is sufficient to show that A: B ¯ ρ C satisfies

uθAu,u B ρ ,θ[0,1].
(4.5)

Now, we set

H(θ,u)=θAu,uC,θ[0,1].

As shown in Theorem 4.6, we have that the operator A is continuous, uniformly bounded and equicontinuous. Then, by the Arzelá-Ascoli theorem, a continuous map h θ (u)=uH(θ,u)=uθAu is completely continuous. If (4.5) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that

deg ( h θ , B ρ , 0 ) = deg ( I θ A , B ρ , 0 ) = deg ( h 1 , B ρ , 0 ) = deg ( h 0 , B ρ , 0 ) = deg ( I , B ρ , 0 ) = 1 0 , 0 B ρ ,

where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, we have h 1 (u)=uAu=0 for at least one u B ρ . Let us assume that u=θAu for some θ[0,1]. Then, for all t[0,T], we have

| u ( t ) | = | θ ( A u ) ( t ) | T α 1 | Λ | { 0 T ( T q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s + i = 1 m | γ i | Γ q i ( β i ) Γ q ( α ) ( 0 ξ i 0 s ( ξ i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s + 0 η i 0 s ( η i q i s ) ( β i 1 ) ( s q x ) ( α 1 ) | f ( x , u ( x ) ) | d q x d q i s ) } + 0 T ( T q s ) ( α 1 ) Γ q ( α ) | f ( s , u ( s ) ) | d q s ( ω | u | + Ψ ) { T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α B q i ( β i , α + 1 ) Γ q i ( β i ) + i = 1 m | γ i | η i β i + α B q i ( β i , α + 1 ) Γ q i ( β i ) ) + T α Γ q ( α + 1 ) } = ( ω | u | + Ψ ) Φ .

Taking norm sup t [ 0 , T ] |u(t)|=u and solving for u, we get

u Ψ Φ 1 ω Φ .

Choosing ρ = Ψ Φ 1 ω Φ +1, then we deduce that (4.5) holds. This completes the proof. □

5 Examples

In this section, we present some examples to illustrate our results.

Example 5.1 Consider the following multi-strip fractional q-integral boundary value problem:

{ D 1 2 7 4 u ( t ) = 4 | u ( t ) | ( 5 + t ) 2 ( 1 + | u ( t ) | ) , t ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = 2 ( I 1 4 2 3 u ) | 1 6 1 5 + 3 4 ( I 1 2 4 5 u ) | 2 5 2 3 + 10 ( I 1 5 7 6 u ) | 1 2 1 .
(5.1)

Here α=7/4, q=1/2, T=1, m=3, γ 1 =2, γ 2 =3/4, γ 3 =10, β 1 =2/3, β 2 =4/5, β 3 =7/6, q 1 =1/4, q 2 =1/2, q 3 =1/5, ξ 1 =1/5, ξ 2 =2/3, ξ 3 =1, η 1 =1/6, η 2 =2/5, η 3 =1/2 and f(t,u)=(4|u(t)|)/( ( 5 + t ) 2 (1+|u(t)|)). Since

| f ( t , u ) f ( t , v ) | 4 25 |uv|,

then (H1) is satisfied with L=4/25. Using the Maple program, we find that

Λ = T α 1 i = 1 m γ i Γ q i ( α ) Γ q i ( α + β i ) ( ξ i α + β i 1 η i α + β i 1 ) Λ 5.259895840 , Φ = T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) Φ = + T α Γ q ( α + 1 ) Φ 2.200354723 .

Therefore, we get

LΦ= 4 25 (2.200354723)0.352056756<1.

Hence, by Theorem 4.1, the boundary value problem (5.1) has a unique solution on [0,1].

Example 5.2 Consider the following multi-strip fractional q-integral boundary value problem:

{ D 5 8 4 3 u ( t ) = 1 u 2 + 3 π 2 sin ( π u 4 ) + 1 20 π ( 5 + sin ( π t ) ) , t ( 0 , 2 ) , u ( 0 ) = 0 , u ( 2 ) = ( I 1 3 1 2 u ) | 1 5 1 4 + 2 3 ( I 1 2 6 5 u ) | 1 2 3 ( I 1 8 3 2 u ) | 1 3 1 + 4 5 ( I 1 4 3 4 u ) | 2 5 1 2 .
(5.2)

Here α=4/3, q=5/8, T=2, m=4, γ 1 =1, γ 2 =2/3, γ 3 =3, γ 4 =4/5, β 1 =1/2, β 2 =6/5, β 3 =3/2, β 4 =3/4, q 1 =1/3, q 2 =1/2, q 3 =1/8, q 4 =1/4, ξ 1 =1/4, ξ 2 =2, ξ 3 =1, ξ 4 =1/2, η 1 =1/5, η 2 =1, η 3 =1/3, η 4 =2/5 and f(t,u)=((sin(πu/4))/( u 2 +3 π 2 ))+((5+sin(πt))/(20π)). By using the Maple program, we find that

Λ = T α 1 i = 1 m γ i Γ q i ( α ) Γ q i ( α + β i ) ( ξ i α + β i 1 η i α + β i 1 ) Λ 2.448357686 , Φ = T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) Φ = + T α Γ q ( α + 1 ) Φ 5.843174987 .

Clearly,

| f ( t , u ) | =| 1 u 2 + 3 π 2 sin ( π u 4 ) + 5 + sin ( π t ) 20 π | ( 5 + sin ( π t ) ) ( 5 | u | + 3 60 π ) .

Choosing p(t)=5+sin(πt) and ψ(|u|)=(5|u|+3)/(60π), we can show that

N ( 6 ) ( 5 N + 3 60 π ) ( 3.493621065 ) >1,

which implies that N>7.967778981. Hence, by Theorem 4.6, the boundary value problem (5.2) has at least one solution on [0,2].

Example 5.3 Consider the following multi-strip fractional q-integral boundary value problem:

{ D 3 8 7 5 u ( t ) = 1 12 π tan 1 ( 3 u ) + 2 | u ( t ) | 1 + | u ( t ) | , t ( 0 , 3 ) , u ( 0 ) = 0 , u ( 3 ) = 2 ( I 1 2 6 5 u ) | 3 2 5 3 + 11 ( I 1 4 7 3 u ) | 1 5 7 4 + ( 9 5 ) ( I 2 3 6 7 u ) | 5 4 5 2 u ( 3 ) = + 5 ( I 2 5 3 4 u ) | 3 7 1 2 + ( 4 7 ) ( I 3 5 2 3 u ) | 1 3 9 5 .
(5.3)

Here α=7/5, q=3/8, T=3, m=5, γ 1 =2, γ 2 =11, γ 3 =9/5, γ 4 =5, γ 5 =4/7, β 1 =6/5, β 2 =7/3, β 3 =6/7, β 4 =3/4, β 5 =2/3, q 1 =1/2, q 2 =1/4, q 3 =2/3, q 4 =2/5, q 5 =3/5, ξ 1 =5/3, ξ 2 =7/4, ξ 3 =5/2, ξ 4 =1/2, ξ 5 =9/5, η 1 =3/2, η 2 =1/5, η 3 =5/4, η 4 =3/7, η 5 =1/3 and f(t,u)=(( tan 1 (3u))/(12π))+((2|u(t)|)/(1+|u(t)|)). By using the Maple program, we find that

Λ = T α 1 i = 1 m γ i Γ q i ( α ) Γ q i ( α + β i ) ( ξ i α + β i 1 η i α + β i 1 ) Λ 33.26381181 , Φ = T α 1 | Λ | Γ q ( α + 1 ) ( T α + i = 1 m | γ i | ξ i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) + i = 1 m | γ i | η i β i + α Γ q i ( α + 1 ) Γ q i ( α + β i + 1 ) ) Φ = + T α Γ q ( α + 1 ) Φ 7.22159847 .

We observe that

| f ( t , u ) | =| 1 12 π tan 1 (3u)+ 2 | u ( t ) | 1 + | u ( t ) | | | u | 4 π +2.

Therefore, we have Ψ=2 and

ω=1/4π< Φ 1 =0.13847350.

Hence, by Theorem 4.7, the boundary value problem (5.3) has at least one solution on [0,3].

Authors’ information

The fourth author is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

References

  1. 1.

    Jackson FH: q -Difference equations. Am. J. Math. 1970, 32: 305-314.

  2. 2.

    Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.

  3. 3.

    Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 15(2):135-140. 10.1017/S0013091500011469

  4. 4.

    Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060

  5. 5.

    Ernst, T: The history of q-calculus and a new method. UUDM Report 2000:16, Department of Mathematics, Uppsala University (2000)

  6. 6.

    Ferreira R: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70: 1-10.

  7. 7.

    Goodrich CS: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61: 191-202. 10.1016/j.camwa.2010.10.041

  8. 8.

    Ma J, Yang J: Existence of solutions for multi-point boundary value problem of fractional q -difference equation. Electron. J. Qual. Theory Differ. Equ. 2011, 92: 1-10.

  9. 9.

    Graef JR, Kong L: Positive solutions for a class of higher order boundary value problems with fractional q -derivatives. Appl. Math. Comput. 2012, 218: 9682-9689. 10.1016/j.amc.2012.03.006

  10. 10.

    Ahmad B, Ntouyas SK, Purnaras IK: Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 140

  11. 11.

    Ahmad B, Ntouyas SK: Existence of solutions for nonlinear fractional q -difference inclusions with nonlocal Robin (separated) conditions. Mediterr. J. Math. 2013, 10: 1333-1351. 10.1007/s00009-013-0258-0

  12. 12.

    Li X, Han Z, Sun S: Existence of positive solutions of nonlinear fractional q -difference equation with parameter. Adv. Differ. Equ. 2013., 2013: Article ID 260

  13. 13.

    Ahmad B, Ntouyas SK: Existence of solutions for fractional differential inclusions with nonlocal strip conditions. Arab J. Math. Sci. 2012, 18: 121-134. 10.1016/j.ajmsc.2012.01.005

  14. 14.

    Ahmad B, Ntouyas SK: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with strip conditions. Bound. Value Probl. 2012., 2012: Article ID 55

  15. 15.

    Ahmad B, Ntouyas SK: Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions. Electron. J. Differ. Equ. 2012, 98: 1-22.

  16. 16.

    Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415

  17. 17.

    Ahmad B, Ntouyas SK, Alsaedi A, Al-Hutami H: Nonlinear q -fractional differential equations with nonlocal and sub-strip type boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2014, 26: 1-12.

  18. 18.

    Ahmad B, Nieto JJ, Alsaedi A, Al-Hutami H: Existence of solutions for nonlinear fractional q -difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 2014, 351: 2890-2909. 10.1016/j.jfranklin.2014.01.020

  19. 19.

    Alsaedi A, Ahmad B, Al-Hutami H: A study of nonlinear fractional q -difference equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 410505

  20. 20.

    Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013, 20: 1-19.

  21. 21.

    Annaby MH, Mansour ZS Lecture Notes in Mathematics 2056. In q-Fractional Calculus and Equations. Springer, Berlin; 2012.

  22. 22.

    Ferreira RAC: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70: 1-10.

  23. 23.

    Krasnoselskii MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123-127.

  24. 24.

    Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.

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Acknowledgements

The research of J. Tariboon and S. Asawasamrit is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

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Correspondence to Jessada Tariboon.

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The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Keywords

  • fractional q-calculus
  • boundary value problems
  • existence
  • uniqueness