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  • Open Access

On nonlocal boundary value problems of nonlinear q-difference equations

Advances in Difference Equations20122012:81

https://doi.org/10.1186/1687-1847-2012-81

  • Received: 6 March 2012
  • Accepted: 13 June 2012
  • Published:

Abstract

This paper studies a nonlocal boundary value problem of nonlinear third-order q-difference equations. Our results are based on Leray-Schauder degree theory and some standard fixed point theorems.

MSC 2000: 39A05; 39A13.

Keywords

  • q-difference equations
  • nonlocal boundary conditions
  • Leray-Schauder degree theory
  • fixed point theorems

1 Introduction

In this paper, we study a nonlocal nonlinear boundary value problem (BVP) of third-order q-difference equations given by
D q 3 u ( t ) = f ( t , u ( t ) ) , t I q , u ( 0 ) = 0 , D q u ( 0 ) = 0 , u ( 1 ) = α u ( η ) ,
(1.1)

where f C(I q × ,), I q = {q n : n } {0,1}, q (0,1) is a fixed constant, η {q n : n } and α1/η2 is a real number.

The subject of q-difference equations has evolved into a multidisciplinary subject in the last few decades. In fact, it is a truly operational subject and its operational formulas were often used with great success in the theory of classical orthogonal polynomials and Bessel functions [1, 2]. For some pioneer work on q-difference equations, we refer the reader to [1, 35], whereas the recent development of the subject can be found in [617] and references therein. However, the theory of boundary value problems for nonlinear q-difference equations is still in the initial stages and many aspects of this theory need to be explored. In particular, the study of nonlocal boundary value problems for nonlinear q-difference equations is yet to be initiated.

The aim of our paper is to present some existence results for the problem (1.1). The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we apply Banach's contraction principle to prove the uniqueness of the solution of the problem, while the third result is based on Krasnoselskii's fixed point theorem. The methods used are standard; however, their exposition in the framework of problem (1.1) is new. In Sect. 2, we present some basic material that we need in the sequel and Sect. 3 contains main results of the paper. Some illustrative examples are also discussed.

2 Preliminaries

Let us recall some basic concepts of q-calculus [8, 9].

For 0 < q < 1, we define the q-derivative of a real-valued function f as
D q f ( t ) = f ( t ) - f ( q t ) ( 1 - q ) t , t I q - { 0 } , D q f ( 0 ) = lim t 0 D q f ( t ) .
Note that
lim q 1 - D q f ( t ) = f ( t ) .
The higher order q-derivatives are defined inductively as
D q 0 f ( t ) = f ( t ) , D q n f ( t ) = D q D q n - 1 f ( t ) , n .

For example, D q ( t k ) = [ k ] q t k - 1 , where k is a positive integer and the q-bracket [k] q = (q k - 1)/(q - 1). In particular, D q (t2) = (1 + q)t.

For y ≥ 0, let us set J y = { y q n : n { 0 } } { 0 } and define the definite q-integral of a function f : J y by
I q f ( y ) = 0 y f ( s ) d q s = n = 0 y ( 1 - q ) q n f ( y q n )
provided that the series converges. For b 1 , b 2 J y ( b 1 = y q n 1 , b 2 = y q n 2 for some n 1 , n 2 ) , we define
b 1 b 2 f ( s ) d q s = I q f ( b 2 ) - I q f ( b 1 ) = ( 1 - q ) n = 0 q n [ b 2 f ( b 2 q n ) - b 1 f ( b 1 q n ) ] .
Similarly, we have
I q 0 f ( t ) = f ( t ) , I q n f ( t ) = I q I q n - 1 f ( t ) , n .
Observe that
D q I q f ( x ) = f ( x ) ,
(2.1)
and if f is continuous at x = 0, then
I q D q f ( x ) = f ( x ) - f ( 0 ) .

This implies that if D q f(t) = σ(t), then f(t) = I q σ(t) + c, where c is an arbitrary constant.

In q-calculus, the product rule and integration by parts formula are
D q ( g h ) ( t ) = D q g ( t ) h ( t ) + g ( q t ) D q h ( t ) ,
(2.2)
0 x f ( t ) D q g ( t ) d q t = f ( t ) g ( t ) 0 x - 0 x D q f ( t ) g ( q t ) d q t .
(2.3)

In the limit q → 1-, the above results correspond to their counterparts in standard calculus.

For f , g : J y , it is possible to introduce an inner product
f , g = 0 1 f ( t ) g ( t ) d q t

and the resulting Hilbert space is denoted by L q 2 ( 0 , 1 ) .

As argued in [16], we can write the solution of the third-order q-difference equation D q 3 u ( t ) = v ( t ) in the following form:
u ( t ) = 0 t α 1 ( q ) t 2 + α 2 ( q ) t s + α 3 ( q ) s 2 v ( s ) d q s + a 0 + a 1 t + a 2 t 2 ,
(2.4)

where a0, a1, a2 are arbitrary constants and α1(q), α2(q), α3(q) can be fixed appropriately.

Choosing α1(q) = 1/(1 + q), α2(q) = -q, α3(q) = q3/(1 + q) and using (2.1) and (2.2), we find that
D q u ( t ) = 0 t t v ( s ) d q s - 0 t q s v ( s ) d q s , D q 2 u ( t ) = 0 t v ( s ) d q s , D q 3 u ( t ) = v ( t ) .
Thus, the solution (2.4) of D q 3 u ( t ) = v ( t ) takes the form
u ( t ) = 0 t t 2 + q 3 s 2 1 + q - q t s v ( s ) d q s + a 0 + a 1 t + a 2 t 2 .
(2.5)
Lemma 2.1 The BVP (1.1) is equivalent to the integral equation
u ( t ) = 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + t 2 1 - α η 2 × α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s .
(2.6)
Proof. In view of (2.5), the solution of D q 3 u = f ( t , u ) can be written as
u ( t ) = 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + a 0 + a 1 t + a 2 t 2 ,
(2.7)
where a1, a2, a2 are arbitrary constants. Using the boundary conditions of (1.1) in (2.7), we find that a0 = 0, a1 = 0 and
a 2 = 1 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s .

Substituting the values of a0, a1 and a2 in (2.7), we obtain (2.6). This completes the proof.

We define
G 1 = max t I q 0 t ( t - s ) 2 2 d q s + t 2 1 - α η 2 α 0 η ( η - s ) 2 2 d q s - 0 1 ( 1 - s ) 2 2 d q s = max γ ( 1 + q ) q 2 ( 1 + q + q 2 ) 4 , α η 2 ( 1 - η ) 1 - α η 2 ( 1 + q ) ( 1 + q + q 2 ) ,
(2.8)
where
γ = η + 1 - η 1 - α η 2 .
Remark 2.1 For q → 1-, equation (2.6) takes the form
u ( t ) = 0 t ( t - s ) 2 2 f ( s , u ( s ) ) d s + t 2 1 - α η 2 α 0 η ( η - s ) 2 2 f ( s , u ( s ) ) d s - 0 1 ( 1 - s ) 2 2 f ( s , u ( s ) ) d s .
which is equivalent to the solution of a classical third-order nonlocal boundary value problem
u ( t ) = f ( t , u ( t ) ) , u ( 0 ) = 0 , u ( 0 ) = 0 , u ( 1 ) = α u ( η ) , 0 t 1 , 0 < η < 1 .
(2.9)

3 Existence results

Let C q = C ( I q , ) denote the Banach space of all continuous functions from I q endowed with the norm defined by x = sup{|x(t)| : t I q }.

Theorem 3.1 Assume that there exist constants M1 ≥ 0 and M2 > 0 such that M1G1 < 1 and |f(t, u)| ≤ M1|u| + M2 for all t I q , u , where G1 is given by (2.8). Then the problem (1.1) has at least one solution.

Proof. Let B R C q be a suitable ball with radius R > 0. Define an operator Ϝ : B ̄ R - C q as
[ Ϝ u ] ( t ) = 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η 2 f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s .
In view of Lemma 2.1, we just need to prove the existence of at least one solution u C q such that u = Ϝ u . Thus, it is sufficient to show that the operator Ϝ satisfies
u λ Ϝ u , u B R and λ [ 0 , 1 ] .
(3.1)
Let us define
H ( λ , y ) = λ Ϝ u , u C q , λ [ 0 , 1 ] .
Then, by Arzela-Ascoli theorem, h λ ( u ) = u - H ( λ , u ) = u - λ Ϝ u is completely continuous. If (3.1) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that
deg ( h λ , B R , 0 ) = deg ( I - λ Ϝ , B R , 0 ) = deg ( h 1 , B R , 0 ) = deg ( h 0 , B R , 0 ) = deg ( I , B R , 0 ) = 1 0 , 0 B R ,
where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, h 1 ( t ) = u - λ Ϝ u = 0 for at least one u B R . Let us set
B R = u C q : u < R ,
where R will be fixed later. In order to prove (3.1), we assume that u = λ Ϝ u for some λ [0,1] and for all t I q so that
u ( t ) = λ [ Ϝ u ] ( t ) 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s 0 t t 2 + q 3 s 2 1 + q - q t s M 1 u ( s ) + M 2 d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s M 1 u ( s ) + M 2 d q s - 0 1 1 + q 3 s 2 1 + q - q s M 1 u ( s ) + M 2 d q s M 1 u + M 2 max t I q 0 t t 2 + q 3 s 2 1 + q - q t s d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s d q s - 0 1 1 + q 3 s 2 1 + q - q s d q s M 1 u + M 2 G 1 ,
which implies that
u M 2 G 1 1 - M 1 G 1 .

Letting R = M 2 G 1 1 - M 1 G 1 + 1 , (3.1) holds. This completes the proof.

Theorem 3.2 Let f : I q × be a jointly continuous function satisfying the Lipschitz condition
f ( t , u ) - f ( t , v ) L u - v , t I q , u , v ,

where L is a Lipschitz constant. Then the boundary value problem (1.1) has a unique solution provided L < 1/G1, where G1 is given by (2.8).

Proof. Let us define an operator Ϝ : C q C q by
[ Ϝ u ] ( t ) = 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s .
Let us set max t I q f ( t , 0 ) = M and choose
r M G 1 1 - L G 1
(3.2)
Then we show that Ϝ B r B r , where B r = { u C q : u r } . For u B r , we have
Ϝ u = max t I q 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s = max t I q 0 t t 2 + q 3 s 2 1 + q - q t s [ ( f ( s , u ( s ) ) - f ( s , 0 ) ) + f ( s , 0 ) ] d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s [ ( f ( s , u ( s ) ) - f ( s , 0 ) ) + f ( s , 0 ) ] d q s - 0 1 1 + q 3 s 2 1 + q - q s [ ( f ( s , u ( s ) ) - f ( s , 0 ) ) + f ( s , 0 ) ] d q s L u + M max t I q 0 t t 2 + q 3 s 2 1 + q - q t s d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s d q s - 0 1 1 + q 3 s 2 1 + q - q s d q s G 1 ( L r + M ) r .

where we have used (3.2).

Now, for u, v , we obtain
Ϝ u - Ϝ v = max t I q [ Ϝ u ] ( t ) - [ Ϝ v ] ( t ) max t I q 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s L max t I q 0 t t 2 + q 3 s 2 1 + q - q t s d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s d q s - 0 1 1 + q 3 s 2 1 + q - q s d q s u - v L G 1 u - v .

As L < 1/G1, therefore Ϝ is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof.

To prove the next existence result, we need the following known fixed point theorem due to Krasnoselskii [18].

Theorem 3.3 Let M be a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i) A x + B y M whenever x , y M ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists z M such that z = Az + Bz.

Theorem 3.4 Assume that f : I q × is a continuous function such that
f ( t , u ) - f ( t , v ) L u - v , t I q , u , v .
(3.3)
Furthermore, |f(t, u)| ≤ μ(t), (t, u) I q × , with μ C(I q , +). Then the boundary value problem (1.1) has at least one solution on I q if
1 - α η 3 1 - α η 2 ( 1 + q ) ( 1 + q + q 2 ) < 1 .
(3.4)
Proof. Letting sup t I q μ ( t ) = μ , we fix r ̄ μ G 1 (G1 is given by (2.8) and consider B r ̄ = { u : u r ̄ } . We define the operators P 1 and P 2 on B r ̄ as
[ P 1 u ] ( t ) = 0 t t 2 + q 3 s 2 1 + q - q t s f ( s , u ( s ) ) d q s , [ P 2 u ] ( t ) = t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s .
For u , v B r ̄ , we find that
P 1 u + P 2 v μ max t I 0 t t 2 + q 3 s 2 1 + q + q t s d q s + t 2 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s d q s - 0 1 1 + q 3 s 2 1 + q - q s d q s = μ G 1 r ̄ .
Thus, P 1 u + P 2 v B r ̄ . It follows from (3.3) and (3.4) that P 2 is a contraction mapping. Continuity of f implies that the operator P 1 is continuous. Also, P 1 is uniformly bounded on B r ̄ as
P 1 u 1 ( 1 + q ) ( 1 + q + q 2 ) .

Now we prove the compactness of the operator P 1 .

In view of (H1), we define sup ( t , u ) I q × B r f ( t , u ) = f ̄ , and consequently we have
[ P 1 u ] ( t 1 ) - [ P 1 u ] ( t 2 ) = 0 t 1 t 1 2 + q 3 s 2 1 + q - q t 1 s f ( s , u ( s ) ) d q s - 0 t 2 t 2 2 + q 3 s 2 1 + q - q t 2 s f ( s , u ( s ) ) d q s + ( t 1 2 + t 2 2 ) 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η 2 f ( s , u ( s ) ) d q s - 0 1 1 + q 3 s 2 1 + q - q s f ( s , u ( s ) ) d q s f ̄ 0 t 1 ( t 1 - t 2 ) [ t 1 + t 2 - q ( 1 + q ) s ] 1 + q d q s - t 1 t 2 t 2 2 + q 3 s 2 1 + q - q t 2 s d q s + ( t 1 2 - t 2 2 ) 1 - α η 2 α 0 η η 2 + q 3 s 2 1 + q - q η s d q s - 0 1 1 + q 3 s 2 1 + q - q s d q s ,

which is independent of u and tends to zero as t2t1. So P 1 is relatively compact on B r ̄ . Hence, by the Arzelá-Ascoli Theorem, P 1 is compact on B r ̄ . Thus all the assump tions of Theorem 3.3 are satisfied. So the conclusion of Theorem 3.3 implies that (1.1) has at least one solution on I q . This completes the proof.

Remark 3.1 In the limit q → 1-, our results reduce to the ones for a classical third-order nonlocal nonlinear boundary value problem (2.9).

Example 3.1. Consider the following problem
D 1 2 3 u ( t ) = M 1 ( 2 π ) sin ( 2 π u ) + u 1 + u + t 2 , t [ 0 , 1 ] 1 / 2 , u ( 0 ) = 0 , D 1 2 u ( 0 ) = 0 , u ( 1 ) = 2 u ( 1 / 2 ) .
(3.5)
Here q = 1/2 and M1 will be fixed later. Observe that
f ( t , u ) = M 1 ( 2 π ) sin ( 2 π u ) + u 1 + u + t 2 M 1 u + 2 ,
and
G 1 = max γ ( 1 + q ) q 2 ( 1 + q + q 2 ) 4 , α η 2 ( 1 - η ) 1 - α η 2 ( 1 + q ) ( 1 + q + q 2 ) = α η 2 ( 1 - η ) 1 - α η 2 ( 1 + q ) ( 1 + q + q 2 ) = 4 / 21 .

Clearly M2 = 2 and we can choose M 1 < 1 G 1 = 21 / 4 . Thus, Theorem 3.1 applies to the problem (3.5).

Example 3.2. Consider the following problem with unbounded nonlinearity
D 1 2 3 u ( t ) = 5 u + c o s u + ( u 2 / ( 1 + u 2 ) ) , t [ 0 , 1 ] 1 / 2 , u ( 0 ) = 0 , D 1 2 u ( 0 ) = 0 , u ( 1 ) = 2 u ( 1 / 2 ) .
(3.6)
Clearly
f ( t , u ) = 5 u + c o s u + ( u 2 / ( 1 + u 2 ) ) 5 u + 2 ,

with M1 = 5 < 1/G1 = 21/4 (G1 is given in Example 3.1) and M2 = 2. Thus, by the conclusion of Theorem 3.1, the problem (3.6) has a solution.

Example 3.3. Consider
D 3 4 3 u ( t ) = L cos t + tan - 1 u , t [ 0 , 1 ] 3 / 4 , u ( 0 ) = 0 , D 3 4 u ( 0 ) = 0 , u ( 1 ) = u ( 1 / 4 ) .
(3.7)
With f(t, u) = L (cos t + tan-1 u), we find that
f ( t , u ) - f ( t , v ) L tan - 1 u - tan - 1 v L u - v
and
G 1 = max γ ( 1 + q ) q 2 ( 1 + q + q 2 ) 4 , α η 2 ( 1 - η ) 1 - α η 2 ( 1 + q ) ( 1 + q + q 2 ) = γ ( 1 + q ) q 2 ( 1 + q + q 2 ) 4 = 86704128 2398926080 .

Fixing L < 1 G 1 27 . 668 , it follows by Theorem 3.2 that the problem (3.7) has a unique solution.

Declarations

Acknowledgements

The research of Bashir Ahmad was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the reviewers for their useful comments.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
(2)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago, Spain

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Copyright

© Ahmad and Nieto; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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