**R E S E A R C H**

**Open Access**

## On nonlocal fractional sum-difference

## boundary value problems for Caputo

## fractional functional difference equations

## with delay

### Bunjong Kaewwisetkul

1_{and Thanin Sitthiwirattham}

2,3*
*_{Correspondence:}

thanin.s@sci.kmutnb.ac.th

2_{Department of Mathematics,}

Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand

3_{Mathematics Department, Faculty}

of Science and Technology, Suan Dusit University, Bangkok, Thailand Full list of author information is available at the end of the article

**Abstract**

In this article, we study the existence and uniqueness results for a nonlocal fractional sum-diﬀerence boundary value problem for a Caputo fractional functional diﬀerence equation with delay, by using the Schauder ﬁxed point theorem and the Banach contraction principle. Finally, we present some examples to display the importance of these results.

**MSC:** 39A05; 39A12

**Keywords:** Caputo fractional functional diﬀerence equations; boundary value
problem; delay; existence

**1 Introduction**

In this paper, we consider a nonlocal fractional sum boundary value problem for a Caputo fractional functional diﬀerence equation with delay of the form

*α _{C}u(t) =Ft*+

*α*– ,

*ut*+

*α*–,

*β*+

_{C}u(t*α*–

*β)*

, *t*∈N,*T*:={, , . . . ,*T*},

*γ _{C}u(α*–

*γ*– ) = ,

*u(T*+

*α) =ρ*–

*ωu(η*+

*ω),*

(.)

and *uα*– =*ψ*, where *ρ* = *η* (*ω*)((*α*–)[(*α*–)(–*γ*)+]+(*T*+*α*)[(*T*–*α*+)(–*γ*)–])
*s*=*α*–(*η*+*ω*–*σ*(*s*))*ω*–((*α*–)[(*α*–)(–*γ*)+]+(*η*+*ω*)[(*η*+*ω*–*α*+)(–*γ*)–])

, *α* ∈
(, ),*β*,*γ*,*ω*∈(, ),*η*∈N*α*–,*T*+*α*– are given constants,*F* ∈*C(*N*α*–,*T*+*α*×*Cr*×R,R)

and*ψ*is an element of the space

*C _{r}*+(α– ) :=

*ψ*∈

*Cr*:

*ψ*(α– ) = ,

*βCψ*(s–

*β*+ ) = ,s∈N

*α*–

*r*–,

*α*–

.

Forr∈N,*T*+we denote*Cr*is the Banach space of all continuous functions*ψ*:N*α*–*r*–,*α*–→
Rendowed with the norm

*ψCr* = max

*s*∈N*α*–*r*–,*α*–

*ψ(s)*.

If*u*:N*α*–*r*–,*α*–→R, then for any*t*∈N*α*–,*T*+*α*we denote by*ut* the element of*Cr*deﬁned

by

*ut*(θ) =*u(t*+*θ*) for*θ*∈N–*r*,.

Fractional diﬀerence calculus or discrete fractional calculus is a very new ﬁeld for math-ematicians. Basic deﬁnitions and properties of fractional diﬀerence calculus can be found in the book []. Some real-world phenomena are being studied with the assistance of frac-tional diﬀerence operators, one may refer to [, ] and the references therein. Good papers related to discrete fractional boundary value problems can be found in [–] and the ref-erences cited therein.

At present, the development of boundary value problems for fractional diﬀerence equa-tions which show an operation of the investigative function. The study may also have other functions related to the ones we are interested in. These creations are incorporating with nonlocal conditions which are both extensive and more complex. For example, Goodrich [] considered the discrete fractional boundary value problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–*ν _{y(t) =}_{λf}*

_{(t}

_{+}

_{ν}_{– ,}

_{y(t}_{+}

_{ν}_{– )),}

_{t}_{∈}

_{N},

*b*+,

*y(ν*– ) =*N _{i}*

_{=}

*F*

*i*(y(t*i*)),

*y(ν*+*b*+ ) =*M _{i}*

_{=}

*F*

*i*(y(t*i*)),

(.)

where*b*> is an integer, and*λ*> is a parameter,*ν*∈(, ] is a real number,*f* :N*ν*,*ν*+*b*×

R→Ris a continuous function,*F*

*i*,*Fi*:R→[,∞] are continuous functions for each*i*

and satisfy some growth conditions to be speciﬁed later, and{t* _{i}*}

*N*

_{i}_{=},{t

*}*

_{i}*M*

_{i}_{=}⊆N

*ν*–,

*ν*+

*b*. The

existence of positive solutions are obtained by Krasnosel’skii ﬁxed point theorem.
Reunsumrit*et al.*[] obtained suﬃcient conditions for the existence of positive
solu-tions for the three-point fractional sum boundary value problem for Caputo fractional
diﬀerence equations via an argument with a shift

*α _{C}u(t) +a(t*+

*α*– )f

*uθ*(t+

*α*– )= ,

*t*∈N,

*T*,

*u(α*– ) =*u(α*– ) = , (.)

*u(T*+*α) =λ*–*βu(η*+*β),*

where <*α*≤, <*β*≤,*η*∈N*α*–,*T*+*α*–,*αC*is the Caputo fractional diﬀerence operator

of order*α, andf* : [,∞)→[,∞) is a continuous function. The existence of at least one
positive solution is proved by using Krasnoselskii’s ﬁxed point theorem.

Recently, Sitthiwirattham [] investigated three-point fractional sum boundary value problems for sequential fractional diﬀerence equations of the forms

⎧ ⎨ ⎩

*α _{C}*[φ

*p*(

*β*

*Cx)](t) =f*(t+*α*+*β*– ,*x(t*+*α*+*β*– )),

*β _{C}x(α*– ) = ,

*x(α*+

*β*+

*T*) =

*ρ*–

*γ*

_{x(η}_{+}

_{γ}_{),}(.)

and uniqueness of solutions are obtained by the Banach ﬁxed point theorem and Schaefer’s ﬁxed point theorem.

The results mentioned above are the motivation for this research. The plan of this paper is as follows. In Section we recall some deﬁnitions and basic lemmas. Also, we derive a representation for the solution of (.) by converting the problem to an equivalent summa-tion equasumma-tion. In Secsumma-tion , we prove existence and uniqueness results of the problem (.) by using the Schauder ﬁxed point theorem and the Banach contraction principle. Some illustrative examples are presented in Section .

**2 Preliminaries**

In the following, there are notations, deﬁnitions, and lemma which are used in the main results. We brieﬂy recall the necessary concepts from the discrete fractional calculus; see [] for further information.

**Deﬁnition .** We deﬁne the generalized falling function by*tα*_{:=} (*t*+)

(*t*+–*α*), for any*t*and*α*
for which the right-hand side is deﬁned. If*t*+ –*α*is a pole of the Gamma function and
*t*+ is not a pole, then*tα*_{= .}

**Lemma .**([]) *Assume the following factorial functions are well deﬁned.If t*≤*r,then*

*tα*_{≤}_{r}α_{for any}_{α}_{> .}

**Deﬁnition .** For*α*> and*f*deﬁned onN*a*:={a,*a+ , . . .}, theαth-order fractional sum*

of*f* is deﬁned by

–*αf*(t) =–* _{a}αf*(t) :=

*(α)*

*t*–*α*

*s*=*a*

*t*–*σ(s)α*–*f*(s),

where*t*∈N*a*+*α*and*σ*(s) =*s*+ .

**Deﬁnition .** For*α*> and*f* deﬁned onN*a*, the*αth-order Caputo fractional diﬀerence*

of*f* is deﬁned by

* _{C}αf*(t) :=

*–(*

_{a}*N*–

*α*)

*Nf*(t) =

*(N*–

*α)*

*t*–(_{}*N*–*α*)

*s*=*a*

*t*–*σ*(s)*N*–*α*–*Nf*(s),

where*t*∈N*a*+*N*–*α*and*N*∈Nis chosen so that ≤*N*– <*α*<*N. Ifα*=*N, thenα _{C}f*(t) =

*N*

_{f}_{(t).}

**Lemma .**([]) *Assume thatα*> *and f deﬁned on*N*a*.*Then*

–_{a}_{+}*α _{N}*

_{–}

*+*

_{α}α_{C}y(t) =y(t) +C*C(t*–

*a)*+

*C(t*–

*a)*+· · ·+

*CN*–(t–

*a)N*–,

*for some Ci*∈R, ≤*i*≤*N*– *and*≤*N*– <*α*≤*N.*

**Lemma .** *Let* *ρ* = *η* (*ω*)((*α*–)[(*α*–)(–*γ*)+]+(*T*+*α*)[(*T*–*α*+)(–*γ*)–])

*s*=*α*–(*η*+*ω*–*σ*(*s*))*ω*–((*α*–)[(*α*–)(–*γ*)+]+(*η*+*ω*)[(*η*+*ω*–*α*+)(–*γ*)–]),

*α* ∈(, ),

*γ*,*ω*∈(, )*and f* ∈*C(*N*α*–,*T*+*α*–,R)*be given.Then the problem*

*α _{C}u(t) =f*(t+

*α*– ),

*t*∈N,

*T*, (.)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

*u(α*– ) = ,

*γ _{C}u(α*–

*γ*– ) = ,

*u(T*+*α) =ρ*–*ω _{u(η}*

_{+}

_{ω),}

_{η}_{∈}

_{N}

*α*–,*T*+*α*–,

(.)

*has the unique solution*

*u(t) =*

(α– )(α– )( –*γ*) + + t( –*α)( –γ*) – +*t*

×

*(α)*

*T*

*s*=

*T*+*α*–*σ(s)α*–*f*(s+*α*– ) – *ρ*
*(ω)(α)*

×
*η*–*α*

*s*=
*η*–*α*

*ξ*=*s*

*η*+*ω*–*α*–*σ*(ξ)*ω*–*ξ*+*α*–*σ(s)α*–*f*(s+*α*– )

+
*(α)*

*t*–*α*

*s*=

*t*–*σ*(s)*α*–*f*(s+*α*– ), (.)

*for t*∈N*α*–,*T*+*α*,*where*

= ( –*α)*(α– )( –*γ*) + + (T+*α)* – (T–*α*+ )( –*γ*)

+ *ρ*
*(ω)*

*η*

*s*=*α*–

*η*+*ω*–*σ*(s)*ω*–(α– )(α– )( –*γ*) + + (η+*ω)*

×(η+*ω*– α+ )( –*γ*) – . (.)

*Proof* Using the fractional sum of order*α*∈(, ) for (.) and from Lemma ., we obtain

*u(t) =C*+*C**t*+*C**t*+
*(α)*

*t*–*α*

*s*=

*t*–*σ*(s)*α*–*f*(s+*α*– ), (.)

for*t*∈N*α*–,*T*+*α*.

By substituting*t*=*α*– into (.) and employing the ﬁrst condition of (.), we obtain

*C*+*C(α*– )+*C(α*– )= . (.)

Using the Caputo fractional diﬀerence of order <*γ* < for (.), we obtain

*γ _{C}u(t) =*

*( –γ*)

*t*+_{}*γ*–

*s*=*α*–

*t*–*σ*(s)–*γC*+ Cs

+

*( –γ*)(α– )

*t*_{}+*γ*–

*s*=*α*–

*s*_{}–*α*+

*ξ*=

*t*–*σ(s)*–*γs*–*σ*(ξ)*α*–*f*(ξ+*α*– ), (.)

By substituting*t*=*α*–*γ*– into (.) and employing the second condition of (.)
im-plies

( –*γ*)C+ + (α– )( –*γ*)*C*= . (.)

Finally, taking the fractional sum of order <*ω*< for (.), we obtain

–*ωu(t) =*
condition of (.) implies

whereis deﬁned by (.) and the functional*Q*[f] is deﬁned as

*Q*[f] =
*(α)*

*T*

*s*=

*T*+*α*–*σ(s)α*–*f*(s+*α*– ) – *ρ*
*(ω)(α)*

×
*η*–*α*

*s*=
*η*–*α*

*ξ*=*s*

*η*+*ω*–*α*–*σ*(ξ)*ω*–*ξ*+*α*–*σ*(s)*α*–*f*(s+*α*– ).

Substituting the constants*C*,*C*and*C*into (.), we obtain (.).

**Corollary .** *Problem*(.)-(.)*has a unique solution of the form*

*u(t) =*

*T*

*s*=

*G(t,s)h(s*+*α*– ), (.)

*where*

*G(t,s) =*
*(α)*

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

*g(t,s),* *s*∈N,*t*–*α*∩N,*η*–*α*,

*g(t,s),* *s*∈N*t*–*α*+,*η*–*α*,

*g(t,s),* *s*∈N*η*–*α*+,*t*–*α*,

*g(t,s),* *s*∈N*t*–*α*+,*T*∩N*η*–*α*+,*T*,

(.)

*with gi*(t,*s), *≤*i*≤,*as*

*g*(t,*s) =:*

(α– )(α– )( –*γ*) + + ( –*α)( –γ*) – *t*+*t*

×

*T*+*α*–*σ(s)α*–– *ρ*
*(ω)*

*η*–*α*

*ξ*=*s*

*η*+*ω*–*α*–*σ*(ξ)*ω*–*ξ*+*α*–*σ*(s)*α*–

+*t*–*σ*(s)*α*–,

*g(t,s) =:*

(α– )(α– )( –*γ*) + + ( –*α)( –γ*) – *t*+*t*

×

*T*+*α*–*σ(s)α*–– *ρ*
*(ω)*

*η*–*α*

*ξ*=*s*

*η*+*ω*–*α*–*σ*(ξ)*ω*–*ξ*+*α*–*σ*(s)*α*–

,

*g*(t,*s) =:*

(α– )(α– )( –*γ*) + + ( –*α)( –γ*) – *t*+*t*

×*T*+*α*–*σ(s)α*–+*t*–*σ(s)α*–,

*g(t,s) =:*

(α– )(α– )( –*γ*) + + ( –*α)( –γ*) – *t*+*t*

×*T*+*α*–*σ*(s)*α*–.

**Lemma .**([]) *A bounded set in*R*n _{is relatively compact;}_{a closed bounded set in}*

_{R}

*n*

*is compact.*

**Lemma .**(Schauder ﬁxed point theorem []) *Assume that K is a convex compact set*
*in a Banach space X and that T* :*K*→*K is a continuous mapping.Then T has a ﬁxed*
*point.*

**3 Main results**

In this section, we wish to establish the existence results for the problem (.). To accom-plish this, we deﬁne the Banach space

*X*=*u*:*u*∈*C(*N*α*–*r*–,*T*+*α*,R),
*β*

*Cu*∈*C(*N*α*–*β*–*r*–,*T*+*α*–*β*+,R), <*β*<

with the norm deﬁned by

*u _{X}*=

*u*+

*β*, (.)

_{C}uwhere*u*=max_{t}_{∈N}_{α}_{–}_{r}_{–,}_{T}_{+}* _{α}*|

*u(t)*|and

*β*=max

_{C}u

_{t}_{∈N}

_{α}_{–}

_{r}_{–,}

_{T}_{+}

*|*

_{α}*ν*–

_{C}u(t*β*+ )|. For

*uα*–=

*ψ*, in view of the deﬁnitions of

*ut*and

*ψ*, we obtain

*uα*–=*uα*–(θ) =*u(θ*+*α*– ) =*ψ*(θ+*α*– ) for*θ*∈N–*r*,. (.)

Thus, we have

*u(t) =ψ*(t) for*t*∈N*α*–*r*–,*α*–. (.)

Since*F*∈*C(*N*α*–,*T*+*α*×*Cr*×R,R), set*F*[t,*ut*,

*β*

*Cu(t*–*β*+ )] :=*f*(t) in Lemma .. We

see by Lemma . that a function*u*is a solution of boundary value problem (.) if and
only if it satisﬁes

*u(t) =*

⎧ ⎨ ⎩

*T*

*s*=*G(t,s)F*[s+*α*– ,*us*+*α*–,
*β*

*Cu(s*+*α*–*β)],* *t*∈N*α*–,*T*+*α*,

*ψ(t),* *t*∈N*α*–*r*–,*α*–.

(.)

Deﬁne an operator*T* :*X*→*X* as follows:

(*Tu)(t) =*

⎧ ⎨ ⎩

*T*

*s*=*G(t,s)F*[s+*α*– ,*us*+*α*–,* _{C}βu(s*+

*α*–

*β*)],

*t*∈N

*α*–,

*T*+

*α*,

*ψ*(t), *t*∈N*α*–*r*–,*α*–,

(.)

and

= max

*t*∈N*α*–*r*–,*α*–

_{T}

*s*=

*G(t,s)φ(s*+*α*– )

, (.)

= max

*t*∈N*α*–*r*–,*α*–

_{T}

*s*=

* _{t}G(t*–

*β*+ ,

*s)φ(s*+

*α*– )

, (.)

*ϒ*=

||*( –β)*

(T* _{(α}*+

_{+ )}

*α)α*–

*ρ(T*+

*α*+

*ω)*

*(T)(α*+

*ω*+ )

(T+*α)( –β)*
×( –*γ*)T+*α*+ ( –*γ*)+ (T+*α) + *( –*α)( –γ*) –

+(T+*α)*
*α*+

*(α)*

+*α(T*+*α*–*β*+ )
–*β*

(T+ )( –*β*)

**Theorem .** *Assume the following properties:*
*Then boundary value problem*(.)*has at least one solution.*

*Proof* We shall use the Schauder ﬁxed point theorem to prove that the operator*T* deﬁned
by (.) has a ﬁxed point. We divide the proof into three steps.

*Step*I. Verify*T* maps bounded sets into bounded sets.
Suppose (A) holds, choose

≤

For the second cases, if (A) holds, choose

and by the same argument as above, we obtain

* _{T}u(t)_{X}*≤+

*(T*+

*α*–

*β*+ ) –

*β*

*( –β)* +

*λ*|u*s*+*α*–|*τ*+*λ**βCu(s*+*α*–*β)*

*τ*

*ϒ*

≤*L*
+

*L*
+

*L*

=*L,* (.)

which implies that*T* :*P*→*P*.

*Step*II. The continuity of the operator*T* follows from the continuity of*F*and*G.*
*Step*III. By Lemma . and Lemma .,*P* is compact.

Hence, by the Schauder ﬁxed point theorem, we can conclude that problem (.) has at least one solution. The proof is completed.

The second result is the existence and uniqueness of a solution to problem (.), by using the Banach contraction principle.

**Theorem .** *Assume the following properties:*

(A) *There exists a constantκ*> *such that*

* _{F}*[t,

*u,u*] –

*F*[t,

*v,v]*≤

*κ*|u–

*v|*+|u–

*v|*,

*for eachu,v*∈*Cr*,*u,v*∈R.
(A) *κ(*+*) < *,*where*

**=(T+*α)*
*α*

*(α*+ ) +
||

(T* _{(α}*+

_{+ )}

*α)α*–

*ρ(T*+

*α*+

*ω)*

*(T*)(α+

*ω*+ )

×(T+*α)*( –*γ*)T+*α*+ ( –*γ*), (.)

=

(T+*α*–*β*+ )–*β*
*( –β*)

(T* _{(α}*+

_{+ )}

*α)α*–

*ρ(T*+

*α*+

*ω)*

*(T)(α*+

*ω*+ )

×

||

*T*+*α*+ ( –*α)( –γ*) – +(T+*α)*
*α*–

*(α)*

. (.)

*Then the problem*(.)*has a unique solution.*

*Proof* Consider the operator*T* :*X*→*X* deﬁned by (.). Clearly, the ﬁxed point of the
operator*T* is the solution of boundary value problem (.). We will use the Banach
con-traction principle to prove that*T* has a ﬁxed point. We ﬁrst show that*T* is a contraction.
For each*t*∈N*α*–,*T*+*α*, we have

(*Tu)(t) – (Tv)(t)*

=

*T*

*s*=

*G(t,s)Fs*+*α*– ,*us*+*α*–,
*β*

*Cu(s*+*α*–*β)*

–*Fs*+*α*– ,*vs*+*α*–,
*β*

*Cv(s*+*α*–*β)*

≤*κ*u–*vX*

×_{}

By (A) implies*T* is a contraction. Hence, by the Banach contraction principle, we see
that*T* has a ﬁxed point which is a unique solution of the problem (.).

**4 Examples**

**Example .** Consider the following fractional diﬀerence boundary value problem:
satisﬁed. Therefore, by Theorem ., boundary value problem (.) has at least one
solu-tion.

**Example .** Consider the following fractional diﬀerence boundary value problem:

and for*t*∈N_{–}

,, we have

_{F}

*t,ut*,*βCu*

–*Ft,vt*,*βCv*≤

,

|u*t*–*vt*|+

*Cu*

*t*+

–

*Cv*

*t*+

,

so (A) holds with*κ*=

,. Also, we can show that

|| ≈., ≈., ≈,.,

and

*κ*(+)≈. < .

Therefore (A) holds, by Theorem ., boundary value problem (.) has a unique solu-tion.

**Acknowledgements**

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GEN-59-21.

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

The authors declare that they carried out all the work in this manuscript and read and approved the ﬁnal manuscript.

**Author details**

1_{Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin, Nakhon}

Pathom, Thailand. 2_{Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology}

North Bangkok, Bangkok, Thailand.3_{Mathematics Department, Faculty of Science and Technology, Suan Dusit University,}

Bangkok, Thailand.

**Publisher’s Note**

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 6 April 2017 Accepted: 18 July 2017
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