Skip to main content

Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations

Abstract

In this paper, we establish the existence of positive solutions of a boundary value problem for nonlinear singular third-order q-difference equations D q 3 u(t)+λa(t)f(u(t))=0, t I q , u(0)=0, D q u(0)=0, α D q u(1)+β D q 2 u(1)=0, by using Krasnoselskii’s fixed-point theorem on a cone, where λ is a positive parameter. Finally, we give an example to demonstrate the use of the main result of this paper. The conclusions in this paper essentially extend and improve known results.

1 Introduction

The q-difference equations initiated in the beginning of the 20th century [14], is a very interesting field in difference equations. In the last few decades, it has evolved into a multidisciplinary subject and plays an important role in several fields of physics, such as cosmic strings and black holes [5], conformal quantum mechanics [6], and nuclear and high-energy physics [7]. For some recent work on q-difference equations, we refer the reader to [812]. However, the theory of boundary value problems (BVPs) for nonlinear q-difference equations is still in an early stage and many aspects of this theory need to be explored. To the best of our knowledge, for the BVPs of nonlinear third-order q-difference equations, a few works were done, see [13, 14] and the references therein.

Recently, in [15], El-Shahed has studied the existence of positive solutions for the following nonlinear singular third-order BVP:

{ u ( t ) + λ a ( t ) f ( u ( t ) ) = 0 , 0 t 1 , u ( 0 ) = u ( 0 ) = 0 , α u ( 1 ) + β u ( 1 ) = 0 ,

by Krasnoselskii’s fixed-point theorem on a cone.

More recently, in [13] Ahmad has studied the existence of positive solutions for the following nonlinear BVP of third-order q-difference equations:

{ D q 3 u ( t ) = f ( t , u ( t ) ) , 0 t 1 , u ( 0 ) = 0 , D q u ( 0 ) = 0 , u ( 1 ) = 0 ,

by Leray-Schauder degree theory and some standard fixed-point theorems.

Motivated by the work above, in this paper, we will study the following BVP of nonlinear singular third-order q-difference equations:

{ D q 3 u ( t ) + λ a ( t ) f ( u ( t ) ) = 0 , t I q , u ( 0 ) = 0 , D q u ( 0 ) = 0 , α D q u ( 1 ) + β D q 2 u ( 1 ) = 0 ,
(1.1)

where λ>0 is a positive parameter, a:(0,1)[0,) is continuous and 0< 0 1 a(t) d q t<, f is a continuous function, I q ={ q n :nN}{0,1}, q(0,1) is a fixed constant, and α,β0, α+β>0.

Obviously, when q 1 , BVP (1.1) reduces to the standard BVP in [15].

Throughout this paper, we always suppose the following conditions to hold:

(C1) fC([0,1],[0,+));

(C2) α,β0, α+β>0 and α β α + β q.

2 Preliminary results

In this section, firstly, let us recall some basic concepts of q-calculus [16, 17].

Definition 2.1 For 0<q<1, we define the q-derivative of a real-value 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).

Definition 2.2 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),nN.

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)/(q1). In particular, D q ( t 2 )=(1+q)t.

Definition 2.3 The q-integral of a function f defined in the interval [a,b] is given by

a x f(t) d q t:= n = 0 x(1q) q n f ( x q n ) af ( a q n ) ,x[a,b],

and for a=0, we denote

I q f(x)= 0 x f(t) d q t= n = 0 x(1q) q n f ( x q n ) ,

then

a b f(t) d q t= 0 b f(t) d q t 0 a f(t) d q t.

Similarly, we have

I q 0 f(t)=f(t), I q n f(t)= I q I q n 1 f(t),nN.

Observe that

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).

In q-calculus, the product rule and integration by parts formula are

D q (gh)(t)= D q g(t)h(t)+g(qt) D q h(t),
(2.1)
0 x f(t) D q g(t) d q t= [ f ( t ) g ( t ) ] 0 x 0 x D q f(t)g(qt) d q t.
(2.2)

Remark 2.1 In the limit q 1 , the above results correspond to their counterparts in standard calculus.

Definition 2.4 Let E be a real Banach space. A nonempty closed convex set PE is called a cone if it satisfies the following two conditions:

  1. (i)

    xP, λ0 implies λxP;

  2. (ii)

    xP, xP implies x=0.

Theorem 2.1 (Krasnoselskii) [18]

Let E be a Banach space and let KE be a cone in E. Assume that Ω 1 and Ω 2 are open subsets of E with 0 Ω 1 and Ω ¯ 1 Ω 2 . Let T:K( Ω ¯ 2 Ω 1 )K be a completely continuous operator. In addition, suppose either

(H1) Tuu, uK Ω 1 and Tuu, uK Ω 2 or

(H2) Tuu, uK Ω 2 and Tuu, uK Ω 1

holds. Then T has a fixed point in K( Ω ¯ 2 Ω 1 ).

Lemma 2.1 Let yC[0,1], then the BVP

{ D q 3 u ( t ) + y ( t ) = 0 , t I q , u ( 0 ) = 0 , D q u ( 0 ) = 0 , α D q u ( 1 ) + β D q 2 u ( 1 ) = 0 ,
(2.3)

has a unique solution

u(t)= 0 1 G(t,s;q)y(s) d q s,

where

G(t,s;q)= 1 ( 1 + q ) ( α + β ) { α t 2 ( 1 q s ) + β t 2 ( t q s ) ( t q 2 s ) ( α + β ) , 0 s t 1 , α t 2 ( 1 q s ) + β t 2 , 0 t s 1 .

Proof Integrate the q-difference equation from 0 to t, we get

D q 2 u(t)= 0 t y(s) d q s+ a 2 .
(2.4)

Integrate (2.4) from 0 to t, and change the order of integration, we have

D q u(t)= 0 t (tqs)y(s) d q s+ a 2 t+ a 1 .
(2.5)

Integrating (2.5) from 0 to t, and changing the order of integration, we obtain

u(t)= 0 t ( t 2 + q 3 s 2 1 + q q t s ) y(s) d q s+ a 2 1 + q t 2 + a 1 t+ a 0 ,
(2.6)

where a 2 , a 1 , a 0 are arbitrary constants. Using the boundary conditions u(0)=0, D q u(0)=0, α D q u(1)+β D q 2 u(1)=0 in (2.6), we find that a 0 = a 1 =0, and

a 2 = 1 α + β ( α 0 1 ( 1 q s ) y ( s ) d q s + β 0 1 y ( s ) d q s ) .

Substituting the values of a 2 , a 1 , and a 0 in (2.6), we obtain

u ( t ) = 0 t ( t 2 + q 3 s 2 1 + q q t s ) y ( s ) d q s + t 2 ( 1 + q ) ( α + β ) ( α 0 1 ( 1 q s ) y ( s ) d q s + β 0 1 y ( s ) d q s ) = 0 1 G ( t , s ; q ) y ( s ) d q s ,

where

G(t,s;q)= 1 ( 1 + q ) ( α + β ) { α t 2 ( 1 q s ) + β t 2 ( t q s ) ( t q 2 s ) ( α + β ) , 0 s t 1 , α t 2 ( 1 q s ) + β t 2 , 0 t s 1 .

This completes the proof. □

Remark 2.2 For q1, equation (2.6) takes the form

u(t)= 1 2 0 t ( t s ) 2 y(s) d q s+ a 2 2 t 2 + a 1 t+ a 0 ,

which is the solution of a classical third-order ordinary differential equation u (t)+y(t)=0 and the associated form of Green’s function for the classical case is

G(t,s)= 1 2 ( α + β ) { α t 2 ( 1 s ) + β t 2 ( t s ) 2 ( α + β ) , 0 s t 1 , α t 2 ( 1 s ) + β t 2 , 0 t s 1 .

It is obvious that, when (C2) holds, G(t,s;q)0, and G(t,s;q)G(1,s;q), 0t,s1.

Lemma 2.2 Let (C2) hold, then G(t,s;q)g(t)G(1,s;q) for 0t,s1, where g(t)= 4 β 5 ( α + β ) t 2 .

Proof If ts, then

G ( t , s ; q ) G ( 1 , s ; q ) = α t 2 ( 1 q s ) ( 1 + q ) ( α + β ) + β t 2 ( 1 + q ) ( α + β ) α ( 1 q s ) ( 1 + q ) ( α + β ) + β ( 1 + q ) ( α + β ) = t 2 α q s α + β t 2 1 α q s α + β t 2 α α + β t 2 = β α + β t 2 4 β 5 ( α + β ) t 2 .

If ts, then

G ( t , s ; q ) G ( 1 , s ; q ) = α t 2 ( 1 q s ) ( 1 + q ) ( α + β ) + β t 2 ( 1 + q ) ( α + β ) t 2 + q 3 s 2 1 + q + q t s α ( 1 q s ) ( 1 + q ) ( α + β ) + β ( 1 + q ) ( α + β ) 1 + q 3 s 2 1 + q + q s = α q s α + β t 2 q 3 s 2 + ( 1 + q ) q t s α q s α + β q 3 s 2 + ( 1 + q ) q s ( 1 + q ) q t 2 α q s α + β t 2 q 3 s 2 ( 1 + q ) q s q 3 s 2 = ( 1 + q ) t 2 α s α + β t 2 q 2 s 2 ( 1 + q ) s q 2 s 2 t 2 α α + β t 2 1 + q q 2 4 β 5 ( α + β ) t 2 .

The proof is complete. □

We consider the Banach space C q =C( I q ,R) equipped with standard norm u=sup{|u(t)|,t I q }, u C q . Define a cone P by

P= { u C q | u ( t ) 0 , u ( t ) g ( t ) u , t I q } .

It is easy to see that if uP, then u=u(1).

Define an integral operator T:P C q by

Tu(t)=λ 0 1 G(t,s;q)a(s)f ( u ( s ) ) d q s,t I q ,uP.
(2.7)

Obviously, T is well defined and uP is a solution of BVP (1.1) if and only if u is a fixed point of T.

Remark 2.3 By Lemma 2.2, we obtain, for uP, Tu(t)0 on I q and

T u ( t ) = λ 0 1 G ( t , s ; q ) a ( s ) f ( u ( s ) ) d q s λ g ( t ) 0 1 G ( 1 , s ; q ) a ( s ) f ( u ( s ) ) d q s λ g ( t ) sup t I q 0 1 G ( t , s ; q ) a ( s ) f ( u ( s ) ) d q s = g ( t ) T u .

Thus T(P)P.

We adopt the following assumption:

(C3) a(t)C((0,1), R + ) may be singular at t=0,1, 0< 0 1 a(t) d q t<+, and 0< 0 1 G(1,s;q)a(t) d q t<+.

Lemma 2.3 Assume (C1), (C2), and (C3) hold, then T:PP is completely continuous.

Proof Define the functions a n (t) for n2 by

a n (t)= { inf { a ( t ) , a ( 1 n ) } , 0 t 1 n , a ( t ) , 1 n t 1 1 n , inf { a ( t ) , a ( 1 1 n ) } , 1 1 n t 1 .

Next, for n2, we define the operator T n :PP by

T n u(t)=λ 0 1 G(t,s;q) a n (s)f ( u ( s ) ) d q s,t I q ,uP.

Obviously, T n is completely continuous on P for any n2 by an application of the Ascoli-Arzelá theorem. Denote B K ={uP:uK}. Then T n converges uniformly to T as n. In fact, for any t I q , for each fixed K>0 and u B K , from (C1), we obtain

| T n u ( t ) T u ( t ) | = | λ 0 1 G ( t , s ; q ) [ a ( s ) a n ( s ) ] f ( u ( s ) ) d q s | λ 0 1 n G ( 1 , s ; q ) | a ( s ) a n ( s ) | f ( u ( s ) ) d q s + λ 1 n 1 1 n G ( 1 , s ; q ) | a ( s ) a n ( s ) | f ( u ( s ) ) d q s + λ 1 1 n 1 G ( 1 , s ; q ) | a ( s ) a n ( s ) | f ( u ( s ) ) d q s 0 ( n ) ,

where we have used the fact that G(t,s;q)0, and G(t,s;q)G(1,s;q), 0t,s1. Hence, T n converges uniformly to T as n, and therefore T is completely continuous also. This completes the proof. □

3 Main results

In this section, we will apply Krasnoselskii’s fixed-point theorem to the eigenvalue problem (1.1). First, we define some important constants:

A q = 0 1 G ( 1 , s ; q ) a ( s ) g ( s ) d q s , B q = 0 1 G ( 1 , s ; q ) a ( s ) d q s , F 0 = lim u 0 + sup f ( u ) u , f 0 = lim u 0 + inf f ( u ) u , F = lim u + sup f ( u ) u , f = lim u + inf f ( u ) u .

Here we assume that 1 A q f =0 if f = and 1 B q F 0 = if F 0 =0 and 1 A q f 0 =0 if f 0 = and 1 B q F = if F =0.

The main result of this paper is the following.

Theorem 3.1 Suppose that (C1), (C2) and (C3) hold and A q f > B q F 0 . Then for each λ( 1 A q f , 1 B q F 0 ), BVP (1.1) has at least one positive solution.

Proof By the definition of F 0 , we see that there exists an l 1 >0, such that f(u)( F 0 +ε)u for 0u l 1 . If uP with u= l 1 , we have

Tu=Tu(1)=λ 0 1 G(1,s;q)a(s)f ( u ( s ) ) d q sλ( F 0 +ε)u B q .

Choose ε>0 sufficiently small such that λ( F 0 +ε) B q 1. Then we obtain Tuu. Thus if we let Ω 1 ={u C q |u< l 1 }, then Tuu for uP Ω 1 .

From the definition of f , we see that there exist an l 3 >0 and l 3 > l 1 , such that f(u)( f ε)u for u> l 2 . Let l 2 > l 3 , if uP with u= l 2 we have

T u = T u ( 1 ) = λ 0 1 G ( 1 , s ; q ) a ( s ) f ( u ( s ) ) d q s λ 0 1 G ( 1 , s ; q ) a ( s ) g ( s ) f ( u ( s ) ) d q s λ ( f ε ) u A q .

Choose ε>0 sufficiently small such that λ( f ε) A q 1. Then we have Tuu. Let Ω 2 ={u C q |u< l 2 }, then Ω 1 Ω ¯ 2 and Tuu for uP Ω 2 .

Condition (H1) of Krasnoselskii’s fixed-point theorem is satisfied. Hence, by Theorem 2.1, the result of Theorem 3.1 holds. This completes the proof of Theorem 3.1. □

Theorem 3.2 Suppose that (C1), (C2) and (C3) hold and A q f 0 > B q F . Then for each λ( 1 A q f 0 , 1 B q F ), BVP (1.1) has at least one positive solution.

Proof It is similar to the proof of Theorem 3.1. □

Theorem 3.3 Suppose that (C1), (C2) and (C3) hold and λ B q f(u)<u for u(0,+). Then BVP (1.1) has no positive solution.

Proof Assume to the contrary that u is a positive solution of BVP (1.1). Then

u ( 1 ) = λ 0 1 G ( 1 , s ; q ) a ( s ) f ( u ( s ) ) d q s < 1 B q 0 1 G ( 1 , s ; q ) a ( s ) u ( s ) d q s u ( 1 ) B q 0 1 G ( 1 , s ; q ) a ( s ) d q s = u ( 1 ) .

This is a contradiction and completes the proof. □

Theorem 3.4 Suppose that (C1), (C2) and (C3) hold and λ A q f(u)>u for u(0,+). Then BVP (1.1) has no positive solution.

Proof It is similar to the proof of Theorem 3.3. □

4 Example

Consider the following BVP:

{ D 1 2 3 u ( t ) + λ t 1 2 10 u 2 + u u + 1 ( 5 + sin u ) = 0 , t I q , u ( 0 ) = 0 , D 1 2 u ( 0 ) = 0 , D 1 2 u ( 1 ) + 3 D 1 2 2 u ( 1 ) = 0 .
(4.1)

Then F 0 =6, f 0 =4, F =60, f =40, and 4uf(u)60u. By direct calculations, we obtain A q =0.110963 and B q =0.271661. From Theorem 3.1 we see that if λ(0.225299,0.613510) then the problem (4.1) has a positive solution. From Theorem 3.3 we see that if λ<0.061351 then the problem (4.1) has no positive solution. By Theorem 3.4 we see that if λ>2.252986 then the problem (4.1) has no positive solution.

References

  1. 1.

    Jackson FH: On q -difference equations. Am. J. Math. 1910, 32: 305-314. 10.2307/2370183

    Article  Google Scholar 

  2. 2.

    Carmichael RD: The general theory of linear q -difference equations. Am. J. Math. 1912, 34: 147-168. 10.2307/2369887

    Article  Google Scholar 

  3. 3.

    Mason TE: On properties of the solutions of linear q -difference equations with entire function coefficients. Am. J. Math. 1915, 37: 439-444.

    Google Scholar 

  4. 4.

    Adams CR: On the linear ordinary q -difference equation. Ann. Math. 1928, 30: 195-205. 10.2307/1968274

    Article  Google Scholar 

  5. 5.

    Strominger A: Information in black hole radiation. Phys. Rev. Lett. 1993, 71: 3743-3746. 10.1103/PhysRevLett.71.3743

    MathSciNet  Article  Google Scholar 

  6. 6.

    Youm D: q -Deformed conformal quantum mechanics. Phys. Rev. D 2000., 62: Article ID 095009

    Google Scholar 

  7. 7.

    Lavagno A, Swamy PN: q -Deformed structures and nonextensive statistics: a comparative study. Physica A 2002, 305(1-2):310-315. Non extensive thermodynamics and physical applications (Villasimius, 2001) 10.1016/S0378-4371(01)00680-X

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ahmad B, Ntouyas SK: Boundary value problems for q -difference inclusions. Abstr. Appl. Anal. 2011., 2011: Article ID 292860

    Google Scholar 

  9. 9.

    Dobrogowska A, Odzijewicz A: Second order q -difference equations solvable by factorization method. J. Comput. Appl. Math. 2006, 193: 319-346. 10.1016/j.cam.2005.06.009

    MathSciNet  Article  Google Scholar 

  10. 10.

    Ahmad B: A study of second-order q -difference equations with boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 35 10.1186/1687-1847-2012-35

    Google Scholar 

  11. 11.

    El-Shahed M, Hassan HA: Positive solutions of q -difference equation. Proc. Am. Math. Soc. 2010, 138: 1733-1738.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Yu CL, Wang JF: Existence of solutions for nonlinear second-order q -difference equations with first-order q -derivatives. Adv. Differ. Equ. 2013., 2013: Article ID 124

    Google Scholar 

  13. 13.

    Ahmad B: Boundary value problems for nonlinear third-order q -difference equations. Electron. J. Differ. Equ. 2011., 2011: Article ID 94

    Google Scholar 

  14. 14.

    Ahmad B, Nieto J: On nonlocal boundary value problem of nonlinear q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 81 10.1186/1687-1847-2012-81

    Google Scholar 

  15. 15.

    El-Shahed M: Positive solutions for nonlinear singular third order boundary value problem. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 424-429. 10.1016/j.cnsns.2007.10.008

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gasper G, Rahman M: Basic Hypergeometric Series. Cambridge University Press, Cambridge; 1990.

    Google Scholar 

  17. 17.

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

    Book  Google Scholar 

  18. 18.

    Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of China (10901045), (11201112) and (61304106), the Natural Science Foundation of Hebei Province (A2013208147) and (A2011208012).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Changlong Yu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, CY and JW contributed to each part of this work equally and read and approved the final version of the manuscript.

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Yu, C., Wang, J. Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations. Adv Differ Equ 2014, 21 (2014). https://doi.org/10.1186/1687-1847-2014-21

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2014-21

Keywords

  • q-difference equations
  • positive solutions
  • singular boundary value problem
  • Krasnoselskii’s fixed-point theorem