R E S E A R C H
Open Access
Existence of positive solutions for Caputo
fractional difference equation
Huiqin Chen
1, Zhen Jin
2and Shugui Kang
1**Correspondence:
dtkangshugui@126.com 1School of Mathematics and
Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we studied the following Caputo fractional difference boundary value problem (FBVP):νCy(t) = –f(t+
ν
– 1,y(t+ν
– 1)),y(ν
– 3) =y(b+ν
) =2y(ν
– 3) = 0, where 2 <ν
≤3 is a real number,νCy(t) is the standard Caputo difference. By means of cone theoretic fixed point theorems, some results on the existence of one or more positive solutions for the above Caputo fractional boundary value problems are obtained.MSC: 26A33; 39A05; 39A12
Keywords: Caputo fractional difference; boundary value problem; Green’s function; existence of positive solution
1 Introduction
In this paper, we discuss the following Caputo fractional difference boundary value prob-lem (FBVP):
ν
Cy(t) = –f(t+ν– ,y(t+ν– )),
y(ν– ) =y(b+ν) =y(ν– ) = , ()
wheret∈[,b+ ]N,b≥ is an integer.f : [ν– ,ν– , . . . ,b+ν]Nν–×[, +∞)→[, +∞)
is continuous andf is not identically zero, <ν≤, andν
Cy(t) is the standard Caputo
difference.
Fractional difference equations appear in useful biological models []. Influenced by the results of [–], the existence of one or two positive solutions for fractional difference equations has been studied (see [–]). Among them, in [, ], the authors introduced the fractional sum and difference operators, studied their behavior and developed a complete theory governing their compositions. In [], Atici and Eloe studied the following two-point boundary value problem for fractional difference equation:
–νy(t) =f(t+ν– ,y(t+ν– )),
y(ν– ) = , y(b+ν+ ) = , ()
where <ν≤. They obtained the existence of positive solutions by means of the Kras-nosel’ski˘ı fixed point theorem. Goodrich gave some new existence results for () in [].
Goodrich [] also considered the following fractional boundary value problem with non-local conditions:
–νy(t) =f(t+ν– ,y(t+ν– )),
y(ν– ) =g(y), y(b+ν+ ) = , ()
where <ν≤. He showed a uniqueness theorem by means of the contraction and an existence theorem by means of the Brouwer theorem for (). He also presented the exis-tence of at least one positive solution to () by using the Krasnosel’ski˘ı theorem. In [– ], the authors deduced the existence of one or more positive solutions for fractional difference equations. Recently, Wu and Baleanu introduced some applications of the Ca-puto fractional difference to discrete chaotic maps in [, ]. However, to the best of our knowledge, few papers can be found in the literature dealing with the Caputo fractional difference equation boundary value problems. Motivated by [], we make some attempt to fill this gap in the existing literature in this paper. We consider the existence of one or more positive solutions for the boundary value problem of the Caputo fractional difference equation ().
The article is structured as follows. In Section , we deduce the Green’s function of the FBVP (). Then we prove that Green’s function satisfies the useful properties. In Section , we present and prove our main results. In Section , we give some examples to illustrate our results.
2 The Green’s function
We first review some basic results about fractional sums and differences. For anyt∈[, b+ ]Nandν> , we define
tν= (t+ )
(t+ –ν),
for which the right-hand side is defined. We appeal to the convention that ift+ –νis a pole of the Gamma function andt+ is not a pole, thentν= .
Theνth fractional sum of a functionf is defined by
–ν
f(t) =
(ν)
t–ν
s=a
(t–s– )ν–f(s),
forν> andt∈ {a+ν,a+ν+ , . . .}=Na+ν. We also define theνth Caputo fractional
difference forν> by
ν
Cf(t) =–(n–ν)nf(t) =
(n–ν)
t–(n–ν) s=a
(t–s– )n–ν–n af(s),
wheren– <ν≤n.
Lemma .[] Assume thatν> and f is defined on domainsNa,then
–ν
a+(n–ν)
ν
Cf(t) =f(t) – n–
k=
ck(t–a)k,
where ck∈R,k= , , . . .n– ,and n– <ν≤n.
Lemma .[] Let f :Na+ν×Na→Rbe given.Then
t–ν
s=a f(t,s)
=
t–ν
s=a
tf(t,s) +f(t+ ,t+ –ν), for t∈Na+ν.
In order to get our main results, we first give an important lemmas. This lemma will give a representation for the solution of (), provided that the solution exists.
Lemma . Let <ν≤and g: [ν– ,ν– , . . . ,b+ν]Nν–→Rbe given.Then the solution
of the FBVP
ν
Cy(t) = –g(t+ν– ),
y(ν– ) =y(b+ν) =y(ν– ) = ()
is given by
y(t) =
b+
s=
G(t,s)g(s+ν– ),
where the Green’s function G: [ν– ,ν– , . . . ,b+ν]Nν–×[,b+ ]N→Ris defined by
G(t,s) =
(ν)
⎧ ⎪ ⎨ ⎪ ⎩
(ν– )(t–ν+ )(b+ν–s– )ν–– (t–s– )ν–,
≤s<t–ν+ ≤b+ ,
(ν– )(t–ν+ )(b+ν–s– )ν–, ≤t–ν+ ≤s≤b+ .
Proof We apply Lemma . to reduce () to an equivalent summation equation
y(t) = –
(ν)
t–ν
s=
(t–s– )ν–g(s+ν– ) +c+ct+ct
for someci∈R,i= , , . By Lemma ., we have
y(t) = –
(ν)
t+–ν
s=
(ν– )(t–s– )ν–g(s+ν– ) +c+ ct,
y(t) = –
(ν)
t+–ν
s=
(ν– )(ν– )(t–s– )ν–g(s+ν– ) + c.
From (), we get
c= –
ν–
(ν– )
b+
s=
c=
Therefore, the solution of the FBVP () is
y(t) = –
Lemma . The Green’s function G has the following properties:
Now, we want to get
So () holds, which completes the proof.
Lemma . If f : [ν– ,ν– , . . . ,b+ν]Nν–×[, +∞)→[,∞)is a continuous function.
Then the solutions y of the FBVP()satisfy
min
Proof Taking into account Lemma ., we have
min
The following fixed point theorem will play a major role in our main results.
Lemma .[] LetBa Banach space and letK⊆Bbe a cone.Assume thatand
For the sake of convenience, we denote
f∞=lim inf
We use Lemma . to establish the existence of positive solutions to the FBVP (). To this end, one or several of the following conditions will be needed.
(H) f : [ν– ,ν– , . . . ,b+ν]Nν–×[, +∞)→[, +∞)is continuous.
Now consider the operatorTdefined by
(Ty)(t) = point of T. We shall obtain sufficient conditions for the existence of fixed points ofT. First, we notice thatTis a summation operator on a discrete finite set. Hence,Tis trivially completely continuous. From (), then
Theorem . Assume that there exist two different positive numbers r and R such that f satisfies condition(H)at r and condition(H)at R.Then the FBVP()has at least one positive solution y∈Ksatisfyingmin{r,R} ≤ y ≤max{r,R}.
>B·
Therefore, by Lemma ., we complete the proof.
By the proof of Theorem ., we obtain the following corollary.
Letr∈(,r), fory∈∂r, we get
(Ty)(t) =
b+
s=
G(t,s)fs+ν– ,y(s+ν– )
≤ b+
s=
G(b+ν,s)(A–ε)r
<Ar b+
s=
G(b+ν,s)
=r,
from which we see thatTy<yfory∈∂r.
On the other hand, sincef∞<A, there exist <σ<AandR> such that
f(t,y)≤σy, y≥R,t∈[ν– ,b+ν]Nν–.
Let M=max(t,y)∈[ν–,b+ν]×[,R]f(t,y), then ≤f(t,y)≤σy+M, ≤y< +∞. LetR >
max{p,A–Mσ}, fory∈∂R, we have
Ty ≤ b+
s=
G(b+ν,s)fs+ν– ,y(s+ν– )
≤σy+M b+
s=
G(b+ν,s)
= (σR+M)·
A <R.
Therefore, we haveTy ≤ yfory∈∂R.
Finally, for anyy∈∂p, sincep≤y(t)≤pforb+ν≤t≤(b+ν), we have
(Ty)
b–ν
+ν
=
b+
s= G
b–ν
+ν,s
fs+ν– ,y(s+ν– )
>B· p
[(b+ν)–ν+] s=[b+ν–ν+]
G
b–ν
+ν,s
=p=y,
from which we see thatTy>yfory∈K∩∂p.
By Lemma ., the proof is complete.
From the proof of Theorem ., we get the following corollary.
Corollary . Assume that(H)and(H)hold, (H)is replaced by(H∗).Then the FBVP
From the proof of Theorem . and ., we get some theorems and corollaries.
Theorem . Suppose that condition(H)and(H)are satisfied.Then the FBVP()has at least one positive solution.
Corollary . Suppose that(H)holds.Also assume that f= +∞,f∞= .Then the FBVP
()has at least one positive solution.
Theorem . Suppose that(H)and(H)hold.Then the FBVP()has at least one positive solution.
Corollary . Suppose that(H)holds.Also assume that f= ,f
∞= +∞.Then the FBVP
()has at least one positive solution.
4 Some examples
In this section, we present some examples to validate our main conclusions. In the fol-lowing examples, we take ν= andb= . A computation shows thatA≈., B≈..
Example . Consider the following Caputo fractional difference boundary value prob-lem:
⎧ ⎨ ⎩
C y(t) = –et–( y
(t+
) + y
(t+
)),
y(–) =y( ) =y(–) = ,
()
wheref(t,y) =et– (
y
+
y
), thusf=f∞= +∞. Takingp= , we get
f(t,y)≤
·
+
·
= . <Ap
for ≤y≤pandν– ≤t≤b+ν. All conditions in Corollary . are satisfied. Applying Corollary ., the FBVP () has at least two positive solutionsyandysuch that <y<
<y.
Example . Consider the following boundary value problem:
⎧ ⎨ ⎩
C y(t) = –( +cos(t+))y(t+ )e–y(t+
),
y(–) =y( ) =y(–) = .
()
Letf(t,y) = ( +cost)ye–y, thenf=f∞= . We choosep= , whenp≤y≤pand
b+ν
≤t≤
(b+ν) , we get
f(t,y)≥( – )pe–p≈.p≥Bp.
Example . Consider the following boundary value problem:
⎧ ⎨ ⎩
C y(t) = – y(t+ )( +
+y(t+
)
),
y(–) =y( ) =y(– ) = ,
()
wheref(t,y) = y( +
+y), it is easy to compute thatf∞= . <Aandf= . > B, which yields the condition (H). By Theorem ., the FBVP () has at least one positive
solution.
Example . Consider the following boundary value problem:
⎧ ⎨ ⎩
C y(t) = –f(t+ν– ,y(t+ν– )), y(–) =y( ) =y(–
) = ,
()
wheref(t,y) = +t ·
y
e–y+e–y+, it is easy to compute thatf≈. <Aandf∞≈ . >B. Thus, the conditions (H) and (H) hold. Applying Theorem ., we can find
that the FBVP () has at least one positive solution.
Remark Iff(t,y) =h(t)g(y), our conclusions hold also.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HQ conceived of the study and participated in its design and coordination. SK drafted the manuscript. ZJ participated in the design of the study and the sequence correction. All authors read and approved the final manuscript.
Author details
1School of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, P.R. China. 2Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, P.R. China.
Acknowledgements
The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper. Project supported by the National Natural Science Foundation of China (Grant No. 11271235) and Shanxi Datong University Institute (2009-Y-15, 2010-B-01, 2013K5) and the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111020).
Received: 7 November 2014 Accepted: 7 January 2015 References
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