Skip to main content

Theory and Modern Applications

Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces

Abstract

This paper is concerned with the existence of a unique solution to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence and uniqueness result is established by using the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main result.

MSC:26A33, 39A05, 39A10, 39A12.

1 Introduction

For a,b∈R, such that b−a is a nonnegative integer, we define N a ={a,a+1,a+2,…} and N a b ={a,a+1,…,b} throughout this paper. It is also worth noting that, in what follows, for any Banach-valued function u defined on N a , we appeal to the convention ∑ s = k 1 k 2 u(s)=θ, where k 1 , k 2 ∈ N a with k 1 > k 2 and θ is the zero element of a given Banach space.

In this paper, we will consider the existence of a unique solution to the following discrete fractional mixed type sum-difference equation boundary value problem in the Banach space E:

{ Δ α u ( t ) + f ( t + α − 1 , u ( t + α − 1 ) , ( T u ) ( t ) , ( S u ) ( t ) ) = θ , t ∈ N 0 , u ( α − n ) = Δ u ( α − n ) = Δ 2 u ( α − n ) = ⋯ = Δ n − 2 u ( α − n ) = θ , Δ α − 1 u ( ∞ ) = u ∞ ,
(1.1)

where n−1<α≤n, n∈ N 2 , Δ α denotes the discrete Riemann-Liouville fractional difference of order α, f: N α − 1 ×E×E×E→E is continuous, θ represents the zero element of E, Δ α − 1 u(∞)= lim t → + ∞ Δ α − 1 u(t)= u ∞ ∈E and

(Tu)(t)= ∑ s = 0 t k(t,s)u(s+α−1),(Su)(t)= ∑ s = 0 ∞ h(t,s)u(s+α−1),

where k:D→R, D={(t,s)∈ N 0 × N 0 :s≤t}, h: N 0 × N 0 →R.

Discrete fractional calculus is a generalization of ordinary difference and summation on arbitrary order that can be non-integer, and it has gained considerable popularity due mainly to its demonstrated applications in describing some real-world phenomena [1, 2]. Among all the topics, the branch of discrete fractional boundary value problems is currently undergoing active investigation; see, for example, [3–17] and the references therein.

Boundary value problems for differential equations in Banach spaces have been studied by many authors [18–36]. Especially for the study of nonlinear mixed type integro-differential equations which arise from many nonlinear problems in science [36], a series of excellent results have been obtained in recent years [21, 23, 30–36].

On the other hand, it is well known that discrete analogues of differential equations can be very useful in applications [37, 38], in particular for using computer to simulate the behavior of solutions for certain dynamic equations. However, compared to continuous case, significantly less is known about discrete difference calculus in Banach spaces [39–45]. Furthermore, as far as we know, the theory of discrete fractional mixed type sum-difference equations boundary value problems in Banach spaces is still a new research area. So, in this paper, we focus on this gap and provide some sufficient conditions for the existence and uniqueness of solutions to problem (1.1).

The remainder of this paper is organized as follows. Section 2 preliminarily presents some necessary basic knowledge for the theory of discrete fractional calculus in Banach spaces. In Section 3, the existence and uniqueness result for the solution to problem (1.1) will be established with the help of the contraction mapping principle. Finally, in Section 4, two concrete examples are provided to illustrate the possible applications of the established analytical result.

2 Preliminaries

In this section, we firstly present the definitions for the discrete Riemann-Liouville fractional difference and the discrete fractional sum for Banach-valued functions similar to the corresponding definitions for real-valued functions [46–49].

Definition 2.1 ([46])

For any t and ν, the falling factorial function is defined as

t ν ̲ = Γ ( t + 1 ) Γ ( t + 1 − ν )

provided that the right-hand side is well defined. We appeal to the convention that if t+1−ν is a pole of the gamma function and t+1 is not a pole, then t ν ̲ =0.

Definition 2.2 The ν th discrete fractional sum of a function f: N a →E, for ν>0, is defined by

Δ a − ν f(t)= 1 Γ ( ν ) ∑ s = a t − ν ( t − s − 1 ) ν − 1 ̲ f(s),t∈ N a + ν .

Also, we define the trivial sum Δ a − 0 f(t)=f(t), t∈ N a .

Definition 2.3 The ν th discrete Riemann-Liouville fractional difference of a function f: N a →E, for ν>0, is defined by

Δ a ν f(t)= Δ n Δ a − ( n − ν ) f(t),t∈ N a + n − ν ,

where n is the smallest integer greater than or equal to ν and Δ n is the n th order forward difference operator. If ν=n∈ N 1 , then Δ a n f(t)= Δ n f(t).

Remark 2.1 From Definitions 2.2 and 2.3, it is easy to see that Δ a − ν maps functions defined on N a to functions defined on N a + ν and Δ a ν maps functions defined on N a to functions defined on N a + n − ν , where n is the smallest integer greater than or equal to ν. Also, it is worth reminding the reader that the t in Δ a ν f(t) (or Δ a − ν f(t)) represents an input for the function Δ a ν f (or Δ a − ν f) and not for the function f. For ease of notation, throughout this paper we omit the subscript a in Δ a ν f(t) and Δ a − ν f(t) when it does not lead to domain confusion and general ambiguity.

Now, we present the following two results, which are analogues to the ordinary case for the real-valued function.

Lemma 2.1 Let f: N a →E and ν,μ>0. Then

Δ a + μ − ν Δ a − μ f(t)= Δ a − ν − μ f(t)= Δ a + ν − μ Δ a − ν f(t),t∈ N a + μ + ν .

Lemma 2.2 Let f: N a →E, ν>0 and p be a positive integer. Then

Δ a − ν Δ p f(t)= Δ p Δ a − ν f(t)− ∑ i = 1 p ( t − a ) v − i ̲ Γ ( ν − i + 1 ) Δ p − i f(a).

Remark 2.2 Lemma 2.1 and Lemma 2.2 are natural analogues of Theorem 2.2 in [46] and Theorem 2.2 in [47] for real-valued functions. Their proofs are similar to the ordinary case. So, here we omit them. Additionally, by using Lemma 2.1, we can easily obtain the equality Δ ν Δ − ν f(t)=f(t), ν>0 holds, for any Banach-valued function f.

At last, we need to state the following lemmas, which will be important in the sequel.

Lemma 2.3 Let ν>0 and f: N a →E. Then

Δ a + n − ν − ν Δ a ν f ( t ) = f ( t ) + c 1 ( t − a − n + ν ) ν − 1 ̲ + c 2 ( t − a − n + ν ) ν − 2 ̲ + ⋯ + c n ( t − a − n + ν ) ν − n ̲ ,
(2.1)

where c i ∈E, i=1,2,…,n, and n is the smallest integer greater than or equal to ν.

Proof By Definition 2.3, Lemma 2.1, Lemma 2.2 and Remark 2.2, we have

Δ a + n − ν − ν Δ a ν f ( t ) = ( Δ a + n − ν − ν Δ n Δ a − ( n − ν ) f ) ( t ) = ( Δ n Δ a + n − ν − ν Δ a − ( n − ν ) f ) ( t ) − ∑ i = 1 n ( t − a − n + ν ) v − i ̲ Γ ( ν − i + 1 ) [ ( Δ n − i Δ a − ( n − ν ) f ) ( a + n − ν ) ] = f ( t ) − ∑ i = 1 n ( t − a − n + ν ) v − i ̲ Γ ( ν − i + 1 ) [ ( Δ n − i Δ a − ( n − ν ) f ) ( a + n − ν ) ] .

Setting c i =− ( Δ n − i Δ a − ( n − ν ) f ) ( a + n − ν ) Γ ( ν − i + 1 ) , i=1,2,…,n; then we get (2.1). So the proof is complete. □

Lemma 2.4 ([48])

Let a∈R and μ>0 be given. Then

Δ ( t − a ) μ ̲ =μ ( t − a ) μ − 1 ̲

for any t for which both sides are well defined. Furthermore, for n−1<ν≤n, n∈ N 1 and μ∈R∖(− N 1 ),

Δ a + μ − ν ( t − a ) μ ̲ = μ − ν ̲ ( t − a ) μ + ν ̲ ,t∈ N a + μ + ν ,

and

Δ a + μ ν ( t − a ) μ ̲ = μ ν ̲ ( t − a ) μ − ν ̲ ,t∈ N a + μ + n − ν .

3 Main results

In this section, we establish the existence of a unique solution to problem (1.1). To accomplish this, we firstly list here the following conditions.

(C1) There exist constants k ∗ and h ∗ such that

k ∗ = sup t ∈ N 0 ∑ s = 0 t | k ( t , s ) | < + ∞ , h ∗ = sup t ∈ N 0 1 1 + ( t + α − 1 ) α − 1 ̲ ∑ s = 0 ∞ | h ( t , s ) | [ 1 + ( s + α − 1 ) α − 1 ̲ ] < + ∞ .

(C2) f ∗ = ∑ t = α − 1 ∞ ∥f(t,θ,θ,θ)∥<+∞, and there exist nonnegative numbers a, b, c and a function p: N α − 1 →[0,∞) with p ∗ = ∑ t = α − 1 ∞ p(t)(1+ t α − 1 ̲ )<+∞ such that

∥ f ( t , u , v , w ) − f ( t , u ¯ , v ¯ , w ¯ ) ∥ ≤p(t) ( a ∥ u − u ¯ ∥ + b ∥ v − v ¯ ∥ + c ∥ w − w ¯ ∥ )

for t∈ N α − 1 , u,v,w, u ¯ , v ¯ , w ¯ ∈E.

Next, we define

X= { u : N α − n → E | sup t ∈ N α − n ∥ u ( t ) ∥ 1 + t α − 1 ̲ < + ∞ }

equipped with the norm

∥ u ∥ X = sup t ∈ N α − n ∥ u ( t ) ∥ 1 + t α − 1 ̲ .

Furthermore, by means of the linear functional analysis theory, we can easily prove that (X, ∥ ⋅ ∥ X ) is a Banach space.

Next, we state and prove the following lemmas, which will be used to establish the existence result of solutions to problem (1.1).

Lemma 3.1 If (C1) and (C2) hold, then, for any u∈X,

∑ t = 0 ∞ ∥ f ( t + α − 1 , u ( t + α − 1 ) , ( T u ) ( t ) , ( S u ) ( t ) ) ∥ ≤ p ∗ ( a + b k ∗ + c h ∗ ) ∥ u ∥ X + f ∗ .
(3.1)

Proof Setting u ¯ = v ¯ = w ¯ =θ in (C2), we have

∥ f ( t , u , v , w ) ∥ ≤ p ( t ) ( a ∥ u ∥ + b ∥ v ∥ + c ∥ w ∥ ) + ∥ f ( t , θ , θ , θ ) ∥ , ( t , u , v , w ) ∈ N α − 1 × E × E × E .

So, for any u∈X, t∈ N 0 , using (C2) again produces

∥ f ( t + α − 1 , u ( t + α − 1 ) , ( T u ) ( t ) , ( S u ) ( t ) ) ∥ ≤ p ( t + α − 1 ) ( a ∥ u ( t + α − 1 ) ∥ + b ∥ ( T u ) ( t ) ∥ + c ∥ ( S u ) ( t ) ∥ ) + ∥ f ( t + α − 1 , θ , θ , θ ) ∥ = p ( t + α − 1 ) [ 1 + ( t + α − 1 ) α − 1 ̲ ] ( a ∥ u ( t + α − 1 ) ∥ 1 + ( t + α − 1 ) α − 1 ̲ + b ∥ ( T u ) ( t ) ∥ 1 + ( t + α − 1 ) α − 1 ̲ + c ∥ ( S u ) ( t ) ∥ 1 + ( t + α − 1 ) α − 1 ̲ ) + ∥ f ( t + α − 1 , θ , θ , θ ) ∥ ≤ p ( t + α − 1 ) [ 1 + ( t + α − 1 ) α − 1 ̲ ] ( a ∥ u ∥ X + b k ∗ ∥ u ∥ X + c h ∗ ∥ u ∥ X ) + ∥ f ( t + α − 1 , θ , θ , θ ) ∥ .
(3.2)

Summating both sides of (3.2), we can get (3.1). The proof is completed. □

Lemma 3.2 Let h: N 0 →E be given and n−1<α≤n, n∈ N 2 . The unique solution of

{ Δ α u ( t ) + h ( t ) = θ , t ∈ N 0 , u ( α − n ) = Δ u ( α − n ) = Δ 2 u ( α − n ) = ⋯ = Δ n − 2 u ( α − n ) = θ , Δ α − 1 u ( ∞ ) = u ∞ ,
(3.3)

is

u(t)= ∑ s = 0 ∞ G(t,s)h(s)+ u ∞ Γ ( α ) t α − 1 ̲ ,t∈ N α − n ,

where

G(t,s)= 1 Γ ( α ) { t α − 1 ̲ − ( t − s − 1 ) α − 1 ̲ , s ∈ N 0 t − α , t α − 1 ̲ , s ∈ N t − α + 1 .
(3.4)

Proof Suppose that u: N α − n →E satisfies the equation of problem (3.3), then Lemma 2.3 implies that

u(t)=− 1 Γ ( α ) ∑ s = 0 t − α ( t − s − 1 ) α − 1 ̲ h(s)+ c 1 t α − 1 ̲ + c 2 t α − 2 ̲ +⋯+ c n t α − n ̲

for some c i ∈E, i=1,2,…,n, t∈ N α − n . By u(α−n)=θ, we get c n =θ.

Furthermore, in view of Lemma 2.4, we have

Δ u ( t ) = − 1 Γ ( α − 1 ) ∑ s = 0 t − ( α − 1 ) ( t − s − 1 ) α − 2 ̲ h ( s ) + c 1 ( α − 1 ) t α − 2 ̲ + c 2 ( α − 2 ) t α − 3 ̲ + ⋯ + c n − 1 ( α − n + 1 ) t α − n ̲ .
(3.5)

Substituting Δu(α−n)=θ in (3.5) gives c n − 1 =θ.

Repeating the above steps with Δ 2 u(α−n)=⋯= Δ n − 2 u(α−n)=θ, we can get

c n − 2 = c n − 3 =⋯= c 2 =θ.

Therefore,

u(t)=− 1 Γ ( α ) ∑ s = 0 t − α ( t − s − 1 ) α − 1 ̲ h(s)+ c 1 t α − 1 ̲ ,t∈ N α − n .
(3.6)

By virtue of Lemma 2.4 again, we have

Δ α − 1 u(t)=− ∑ s = 0 t − 1 h(s)+ c 1 Γ(α),t∈ N 0 .
(3.7)

Using the condition Δ α − 1 u(∞)= u ∞ in (3.7), we obtain

c 1 = 1 Γ ( α ) ( ∑ s = 0 ∞ h ( s ) + u ∞ ) .

Now, substitution of c 1 into (3.6) gives

u ( t ) = − 1 Γ ( α ) ∑ s = 0 t − α ( t − s − 1 ) α − 1 ̲ h ( s ) + 1 Γ ( α ) ∑ s = 0 ∞ t α − 1 ̲ h ( s ) + u ∞ Γ ( α ) t α − 1 ̲ = ∑ s = 0 ∞ G ( t , s ) h ( s ) + u ∞ Γ ( α ) t α − 1 ̲ , t ∈ N α − n ,

where G(t,s) is defined by (3.4). The proof is complete. □

Remark 3.1 From the expression of G(t,s), we can easily find that G(t,s)≥0 and G ( t , s ) 1 + t α − 1 ̲ < 1 Γ ( α ) for (t,s)∈ N n − α × N 0 .

With the above auxiliary results in hand, we now establish the main result as follows.

Theorem 3.1 If (C1), (C2) hold and

σ= p ∗ ( a + b k ∗ + c h ∗ ) Γ ( α ) <1,
(3.8)

then problem (1.1) has a unique solution u in X.

Proof Define an operator F:X→X by

(Fu)(t)= ∑ s = 0 ∞ G(t,s)f ( s + α − 1 , u ( s + α − 1 ) , ( T u ) ( s ) , ( S u ) ( s ) ) + u ∞ Γ ( α ) t α − 1 ̲ ,

where t∈ N α − n , and due to Lemma 3.1, we have

∥ ( F u ) ( t ) ∥ 1 + t α − 1 ̲ ≤ ∑ s = 0 ∞ G ( t , s ) 1 + t α − 1 ̲ ∥ f ( s + α − 1 , u ( s + α − 1 ) , ( T u ) ( s ) , ( S u ) ( s ) ) ∥ + ∥ u ∞ ∥ t α − 1 ̲ Γ ( α ) ( 1 + t α − 1 ̲ ) ≤ 1 Γ ( α ) { p ∗ ( a + b k ∗ + c h ∗ ) ∥ u ∥ X + f ∗ + ∥ u ∞ ∥ } = σ ∥ u ∥ X + ϱ , t ∈ N α − n .

Therefore,

∥ F u ∥ X ≤σ ∥ u ∥ X +ϱ,u∈X,

here ϱ=( f ∗ +∥ u ∞ ∥)/Γ(α) and σ is defined by (3.8). So, the operator ℱ is well defined. Furthermore, from Lemma 3.2, we can transform problem (1.1) as an operator equation u=Fu, and it is clear to see that u is a solution of problem (1.1) is equivalent to a fixed point of ℱ.

Next, for any u,v∈X, we denote

A ( u , v ) ( t ) = f ( t + α − 1 , u ( t + α − 1 ) , ( T u ) ( t ) , ( S u ) ( t ) ) − f ( t + α − 1 , v ( t + α − 1 ) , ( T v ) ( t ) , ( S v ) ( t ) ) , t ∈ N 0 .

In view of (C2), we have

∥ ( F u ) ( t ) − ( F v ) ( t ) ∥ 1 + t α − 1 ̲ ≤ ∑ s = 0 ∞ G ( t , s ) 1 + t α − 1 ̲ ∥ A ( u , v ) ( s ) ∥ ≤ 1 Γ ( α ) ∑ s = 0 ∞ [ p ( s + α − 1 ) ( a ∥ u ( s + α − 1 ) − v ( s + α − 1 ) ∥ + b ∥ ( T u ) ( s ) − ( T v ) ( s ) ∥ + c ∥ ( S u ) ( s ) − ( S v ) ( s ) ∥ ) ] ≤ 1 Γ ( α ) ∑ s = 0 ∞ [ p ( s + α − 1 ) [ 1 + ( s + α − 1 ) α − 1 ̲ ] × ( a ∥ u − v ∥ X + b k ∗ ∥ u − v ∥ X + c h ∗ ∥ u − v ∥ X ) ] ≤ 1 Γ ( α ) p ∗ ( a + b k ∗ + c h ∗ ) ∥ u − v ∥ X = σ ∥ u − v ∥ X .

So we get

∥ F u − F v ∥ X ≤σ ∥ u − v ∥ X ,

which, together with the assumption that σ<1, implies that ℱ is a contraction mapping. By means of the Banach contraction mapping principle, we get that ℱ has a unique fixed point in E; that is problem (1.1) has a unique solution. This completes the proof. □

4 Examples

In this section, we illustrate the possible application of the above established analytical result with the following two concrete examples.

Example 4.1 Consider the following problem:

{ Δ 7 / 2 u n ( t ) + 3 − ( t + 1 ) n [ 1 + ( t + 5 / 2 ) 5 / 2 ̲ ] 2 sin [ n 2 ( t + 5 / 2 ) + u n ( t + 5 / 2 ) ] + 2 − ( t + 1 ) ( n + 2 ) 2 [ 1 + ( t + 5 / 2 ) 5 / 2 ̲ ] 3 ln { 1 + [ ∑ s = 0 t 1 ( t + s + 2 ) 2 ̲ u 3 n ( s + 5 / 2 ) ] 2 } + e − ( t + 1 ) n [ 3 + sin ( t + 5 / 2 ) + ( t + 5 / 2 ) 5 / 2 ̲ ] × { ∑ s = 0 ∞ cos ( t 2 s ) ( s + 2 ) 2 ̲ [ 1 + ( s + 5 / 2 ) 5 / 2 ̲ ] u n + 1 ( s + 5 / 2 ) } = 0 , t ∈ N 0 , u n ( − 1 / 2 ) = Δ u n ( − 1 / 2 ) = Δ 2 u n ( − 1 / 2 ) = 0 , Δ 5 / 2 u n ( ∞ ) = 1 n ! , n = 1 , 2 , 3 , … .
(4.1)

Conclusion Problem (4.1) has a unique solution { u n (t)} such that u n (t)→0 as n→∞ for t∈ N − 1 / 2 .

Proof Let E= c 0 ={u=( u 1 , u 2 ,…, u n ,…): u n →0}. Evidently, (E,∥⋅∥) is a Banach space with the norm ∥u∥= sup n | u n | for any u∈E. Then the infinite discrete fractional difference system (4.1) can be regarded as a boundary value problem of the form (1.1) in the Banach space E. In this situation, α=7/2, θ=(0,0,…,0,…)∈E, u ∞ =(1,1/2!,…,1/n!,…)∈E,

k(t,s)= 1 ( t + s + 2 ) 2 ̲ ,h(t,s)= cos ( t 2 s ) ( s + 2 ) 2 ̲ [ 1 + ( s + 5 / 2 ) 5 / 2 ̲ ] ,

and f=( f 1 , f 2 ,…, f n ,…), in which

f n ( t , u , v , w ) = 3 − ( t − 3 / 2 ) n [ 1 + t 5 / 2 ̲ ] 2 sin ( n 2 t + u n ) + 2 − ( t − 3 / 2 ) ( n + 2 ) 2 [ 1 + t 5 / 2 ̲ ] 3 ln ( 1 + v 3 n 2 ) + e − ( t − 3 / 2 ) n [ 3 + sin t + t 5 / 2 ̲ ] w n + 1 ,

where t∈ N 5 / 2 and u=( u 1 , u 2 ,…, u n ,…), v=( v 1 , v 2 ,…, v n ,…), w=( w 1 , w 2 ,…, w n ,…)∈E. From the expression of f n , it is easy to see that f: N 5 / 2 ×E×E×E→E is continuous. Furthermore, for any t∈ N 5 / 2 , u,v,w, u ¯ , v ¯ , w ¯ ∈E, we have

| f n ( t , u , v , w ) − f n ( t , u ¯ , v ¯ , w ¯ ) | ≤ 3 − ( t − 3 / 2 ) n [ 1 + t 5 / 2 ̲ ] 2 | sin ( n 2 t + u n ) − sin ( n 2 t + u ¯ n ) | + 2 − ( t − 3 / 2 ) ( n + 2 ) 2 [ 1 + t 5 / 2 ̲ ] 3 | ln ( 1 + v 3 n 2 ) − ln ( 1 + v ¯ 3 n 2 ) | + e − ( t − 3 / 2 ) n [ 3 + sin t + t 5 / 2 ̲ ] | w n + 1 − w ¯ n + 1 | ≤ 3 − ( t − 3 / 2 ) n [ 1 + t 5 / 2 ̲ ] 2 | u n − u ¯ n | + 2 − ( t − 3 / 2 ) ( n + 2 ) [ 1 + t 5 / 2 ̲ ] 3 | v 3 n − v ¯ 3 n | + e − ( t − 3 / 2 ) n [ 3 + sin t + t 5 / 2 ̲ ] | w n + 1 − w ¯ n + 1 | ≤ 2 − ( t − 3 / 2 ) 1 + t 5 / 2 ̲ [ | u n − u ¯ n | + 1 / 3 | v 3 n − v ¯ 3 n | + | w n + 1 − w ¯ n + 1 | ] ,

and therefore,

∥ f ( t , u , v , w ) − f ( t , u ¯ , v ¯ , w ¯ ) ∥ ≤ 2 − ( t − 3 / 2 ) 1 + t 5 / 2 ̲ [ ∥ u − u ¯ ∥ + 1 / 3 ∥ v − v ¯ ∥ + ∥ w − w ¯ ∥ ] ,

where a=c=1, b= 1 3 , p(t)= 2 − ( t − 3 / 2 ) 1 + t 5 / 2 ̲ , which imply that (C2) holds together with the following facts:

p ∗ = ∑ t = 5 / 2 ∞ p(t) ( 1 + t 5 / 2 ̲ ) = ∑ t = 5 / 2 ∞ 2 − ( t − 3 / 2 ) =1<∞

and

f ∗ = ∑ s = 5 / 2 ∞ ∥ f ( t , θ , θ , θ ) ∥ ≤ ∑ s = 5 / 2 ∞ 3 − ( t − 3 / 2 ) =1/2<∞.

On the other hand, we can verify that

k ∗ = sup t ∈ N 0 ∑ s = 0 t 1 ( t + s + 2 ) 2 ̲ = sup t ∈ N 0 1 2 ( t + 1 ) = 1 2 < ∞ , h ∗ = sup t ∈ N 0 1 1 + ( t + 5 / 2 ) 5 / 2 ̲ ∑ s = 0 ∞ | cos ( t 2 s ) | [ 1 + ( s + 5 / 2 ) 5 / 2 ̲ ] [ 1 + ( s + 5 / 2 ) 5 / 2 ̲ ] ( s + 2 ) 2 ̲ ≤ sup t ∈ N 0 1 1 + ( t + 5 / 2 ) 5 / 2 ̲ ∑ s = 0 ∞ 1 ( s + 2 ) 2 ̲ ≤ 1 1 + Γ ( 7 / 2 ) < 1 4 < ∞ .

So (C1) is also satisfied. Finally, by a simple calculation, we can obtain

σ= p ∗ ( a + b k ∗ + c h ∗ ) Γ ( α ) ≤ ( 1 + 1 / 6 + 1 / 4 ) Γ ( 7 / 2 ) <0.43<1.

Thus, all the conditions of Theorem 3.1 are satisfied and our conclusion follows from Theorem 3.1. □

Example 4.2 Consider the following problem:

{ Δ 7 / 3 ω ( t , x ) + 2 − ( t + 1 ) 4 [ 1 + ( t + 4 / 3 ) 4 / 3 ̲ ] cos [ ω ( t + 4 / 3 , x ) ] + 3 − ( t + 1 ) e 2 [ 1 + ( t + 4 / 3 ) 4 / 3 ̲ ] [ ∑ s = 0 t 1 ( t + s + 2 ) 2 ̲ ω ( s + 4 / 3 , x ) ] + e − ( t + 1 ) e 3 [ 2 + cos ( t + 4 / 3 ) + ( t + 4 / 3 ) 4 / 3 ̲ ] { ∑ s = 0 ∞ sin ( t + e s ) ( s + 2 ) 2 ̲ [ 1 + ( s + 4 / 3 ) 4 / 3 ̲ ] ω ( s + 4 / 3 , x ) } = 0 , t ∈ N 0 , x ∈ [ 0 , 1 ] , ω ( − 2 / 3 , x ) = Δ t ω ( − 2 / 3 , x ) = 0 , Δ 4 / 3 ω ( ∞ , x ) = x 2 .
(4.2)

Here, Δ 7 / 3 ω(t,x) represents the discrete Riemann-Liouville fractional difference of order 7/3 for the function ω(t,x) with respect to its first variable t.

Conclusion Problem (4.2) has a unique solution ω: N − 2 / 3 ×[0,1]→R such that for each given t∈ N − 2 / 3 , ω(t,x) is continuous for x∈[0,1].

Proof Let E=C[0,1]={g:[0,1]→R is continuous}; then (E,∥⋅∥) is a Banach space equipped with the norm ∥g∥= sup x ∈ [ 0 , 1 ] |g(x)|, g∈E. Define u: N − 2 / 3 →E by u(t)=ω(t,â‹…)∈E; then the discrete fractional partial difference system (4.2) can be transformed into the form of problem (1.1), where θ=0, u ∞ = x 2 ,

k(t,s)= 1 ( t + s + 2 ) 2 ̲ ,h(t,s)= sin ( t + e s ) ( s + 2 ) 2 ̲ [ 1 + ( s + 4 / 3 ) 4 / 3 ̲ ] ,

and

f(t,u,v,w)= 2 − ( t − 1 / 3 ) 4 [ 1 + t 4 / 3 ̲ ] cosu+ 3 − ( t − 1 / 3 ) e 2 [ 1 + t 4 / 3 ̲ ] v+ e − ( t − 1 / 3 ) e 3 [ 2 + cos t + t 4 / 3 ̲ ] w

for (t,u,v,w)∈ N 4 / 3 ×E×E×E. It is obvious that f is continuous.

Choosing a= 1 4 , b= 1 e 2 , c= 1 e 3 and p(t)= 2 − ( t − 1 / 3 ) 1 + t 4 / 3 ̲ , t∈ N 4 / 3 ; then we can verify that p ∗ =1, f ∗ <1/4, k ∗ =1/2, h ∗ <0.4566, σ<0.3631 and

∥ f ( t , u , v , w ) − f ( t , u ¯ , v ¯ , w ¯ ) ∥ ≤p(t) ( a ∥ u − u ¯ ∥ + b ∥ v − v ¯ ∥ + c ∥ w − w ¯ ∥ )

holds for any t∈ N 4 / 3 , u,v,w, u ¯ , v ¯ , w ¯ ∈E.

Clearly, all the conditions of Theorem 3.1 are fulfilled. Therefore, we can conclude that problem (4.2) has a unique solution. □

References

  1. Wu G, Baleanu D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 2014, 75: 283-287. 10.1007/s11071-013-1065-7

    Article  MathSciNet  MATH  Google Scholar 

  2. Atıcı F, Şengül S: Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369: 1-9. 10.1016/j.jmaa.2010.02.009

    Article  MathSciNet  MATH  Google Scholar 

  3. Atıcı F, Eloe P: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17: 445-456. 10.1080/10236190903029241

    Article  MathSciNet  MATH  Google Scholar 

  4. Goodrich C: Solutions to a discrete right-focal fractional boundary value problem. Int. J. Differ. Equ. 2010, 5: 195-216.

    MathSciNet  Google Scholar 

  5. Goodrich C: Continuity of solutions to discrete fractional initial value problems. Comput. Math. Appl. 2010, 59: 3489-3499. 10.1016/j.camwa.2010.03.040

    Article  MathSciNet  MATH  Google Scholar 

  6. Goodrich C: 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

    Article  MathSciNet  MATH  Google Scholar 

  7. Goodrich C: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011, 217: 4740-4753. 10.1016/j.amc.2010.11.029

    Article  MathSciNet  MATH  Google Scholar 

  8. Goodrich C: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 2012, 18: 397-415. 10.1080/10236198.2010.503240

    Article  MathSciNet  MATH  Google Scholar 

  9. Goodrich C: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385: 111-124. 10.1016/j.jmaa.2011.06.022

    Article  MathSciNet  MATH  Google Scholar 

  10. Goodrich C: On semipositone discrete fractional boundary value problems with non-local boundary conditions. J. Differ. Equ. Appl. 2013, 19: 1758-1780. 10.1080/10236198.2013.775259

    Article  MathSciNet  MATH  Google Scholar 

  11. Dahal R, Duncan D, Goodrich C: Systems of semipositone discrete fractional boundary value problems. J. Differ. Equ. Appl. 2014, 20: 473-491. 10.1080/10236198.2013.856073

    Article  MathSciNet  MATH  Google Scholar 

  12. Holm M:Solutions to a discrete, nonlinear, (N−1,1) fractional boundary value problem. Int. J. Dyn. Syst. Differ. Equ. 2011, 3: 267-287.

    MathSciNet  MATH  Google Scholar 

  13. Pan Y, Han Z, Sun S, Hou C: The existence of solutions to a class of boundary value problems with fractional difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 275

    Google Scholar 

  14. Ferreira R: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 2013, 19: 712-718. 10.1080/10236198.2012.682577

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen F, Zhou Y: Existence and Ulam stability of solutions for discrete fractional boundary value problem. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 459161

    Google Scholar 

  16. Lv W: Existence of solutions for discrete fractional boundary value problems with a p -Laplacian operator. Adv. Differ. Equ. 2012., 2012: Article ID 163

    Google Scholar 

  17. Lv W: Solvability for discrete fractional boundary value problems with a p -Laplacian operator. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 679290

    Google Scholar 

  18. Liu Y: Boundary value problems for second order differential equations on unbounded domains in a Banach space. Appl. Math. Comput. 2003, 135: 569-583. 10.1016/S0096-3003(02)00070-X

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang X, Feng M, Ge W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 2008, 69: 3310-3321. 10.1016/j.na.2007.09.020

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang X, Feng M, Ge W: Existence and nonexistence of positive solutions for a class of n th-order three-point boundary value problems in Banach spaces. Nonlinear Anal. 2009, 70: 584-597. 10.1016/j.na.2007.12.028

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang X, Feng M, Ge W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 2010, 233: 1915-1926. 10.1016/j.cam.2009.07.060

    Article  MathSciNet  MATH  Google Scholar 

  22. Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 2008, 222: 351-363. 10.1016/j.cam.2007.11.003

    Article  MathSciNet  MATH  Google Scholar 

  23. Feng M, Pang H: A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 2009, 70: 64-82. 10.1016/j.na.2007.11.033

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen H, Li P: Existence of solutions of three-point boundary value problems in Banach spaces. Math. Comput. Model. 2009, 49: 780-788. 10.1016/j.mcm.2008.05.003

    Article  MathSciNet  MATH  Google Scholar 

  25. Chen H, Zhao Y: Triple positive solutions for nonlinear boundary value problems in Banach space. Comput. Math. Appl. 2009, 58: 1780-1787. 10.1016/j.camwa.2009.07.061

    Article  MathSciNet  MATH  Google Scholar 

  26. Jiang W, Wang B: Positive solutions for second-order multi-point boundary value problems in Banach spaces. Electron. J. Differ. Equ. 2011., 2011: Article ID 22

    Google Scholar 

  27. Hao X, Liu L, Wu Y, Xu N: Multiple positive solutions for singular n th-order nonlocal boundary value problems in Banach spaces. Comput. Math. Appl. 2011, 61: 1880-1890. 10.1016/j.camwa.2011.02.017

    Article  MathSciNet  MATH  Google Scholar 

  28. Su X: Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. Nonlinear Anal. 2011, 74: 2844-2852. 10.1016/j.na.2011.01.006

    Article  MathSciNet  MATH  Google Scholar 

  29. Guo D, Lakshmikantham V: Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces. J. Math. Anal. Appl. 1988, 129: 211-222. 10.1016/0022-247X(88)90243-0

    Article  MathSciNet  MATH  Google Scholar 

  30. Guo D: A boundary value problem for n th-order integro-differential equations in a Banach space. Appl. Math. Comput. 2003, 136: 571-592. 10.1016/S0096-3003(02)00082-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Guo D: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in a Banach space. Nonlinear Anal. 2004, 56: 985-1006. 10.1016/j.na.2003.10.023

    Article  MathSciNet  MATH  Google Scholar 

  32. Guo D: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 2005, 63: 618-641. 10.1016/j.na.2005.05.023

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu Y, Zhang H: Multiple positive solutions of a boundary value problem for a class of 2 n th-order m -point singular integro-differential equations in Banach spaces. Appl. Math. Comput. 2009, 214: 607-617. 10.1016/j.amc.2009.04.021

    Article  MathSciNet  MATH  Google Scholar 

  34. Xu Y, Zhang H: Multiple positive solutions of a m -point boundary value problem for 2 n th-order singular integro-differential equations in Banach spaces. Nonlinear Anal. 2009, 70: 3243-3253. 10.1016/j.na.2008.04.026

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang L, Ahmad B, Wang G, Agarwal R: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.

    Article  MathSciNet  MATH  Google Scholar 

  36. Lakshmikantham V: Some problems in integro-differential equations of Volterra type. J. Integral Equ. 1985, 10: 137-146.

    MathSciNet  MATH  Google Scholar 

  37. Hilscher R, Zeidan V: Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey. J. Differ. Equ. Appl. 2005, 11: 857-875. 10.1080/10236190500137454

    Article  MathSciNet  MATH  Google Scholar 

  38. Kelley W, Peterson A: Difference Equations: An Introduction with Applications. Academic Press, New York; 1991.

    MATH  Google Scholar 

  39. Agarwal R, O’Regan D: Difference equations in Banach spaces. J. Aust. Math. Soc. A 1998, 64: 277-284. 10.1017/S1446788700001762

    Article  MATH  Google Scholar 

  40. Agarwal R, O’Regan D: A fixed-point approach for nonlinear discrete boundary value problems. Comput. Math. Appl. 1998, 36: 115-121.

    Article  MathSciNet  MATH  Google Scholar 

  41. González C, Jiménez-Melado A: An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions of difference equations in Banach spaces. J. Math. Anal. Appl. 2000, 247: 290-299. 10.1006/jmaa.2000.6877

    Article  MathSciNet  MATH  Google Scholar 

  42. Tabor J: Oscillation of linear difference equations in Banach spaces. J. Differ. Equ. 2003, 192: 170-187. 10.1016/S0022-0396(03)00040-8

    Article  MathSciNet  MATH  Google Scholar 

  43. Bay N, Phat V: Stability analysis of nonlinear retarded difference equations in Banach spaces. Comput. Math. Appl. 2003, 45: 951-960. 10.1016/S0898-1221(03)00068-3

    Article  MathSciNet  MATH  Google Scholar 

  44. González C, Jiménez-Melado A: Set-contractive mappings and difference equations in Banach spaces. Comput. Math. Appl. 2003, 45: 1235-1243. 10.1016/S0898-1221(03)00094-4

    Article  MathSciNet  MATH  Google Scholar 

  45. Agarwal R, Thompson H, Tisdell C: Difference equations in Banach spaces. Comput. Math. Appl. 2003, 45: 1437-1444. 10.1016/S0898-1221(03)00100-7

    Article  MathSciNet  MATH  Google Scholar 

  46. Atıcı F, Eloe P: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2: 165-176.

    MathSciNet  Google Scholar 

  47. Atıcı F, Eloe P: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 2009, 137: 981-989.

    MathSciNet  MATH  Google Scholar 

  48. Holm M: Sum and difference compositions in discrete fractional calculus. CUBO 2011, 13: 153-184. 10.4067/S0719-06462011000300009

    Article  MathSciNet  MATH  Google Scholar 

  49. Miller K, Ross B: Fractional difference calculus. In Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications. Nihon University, Koriyama; 1989:139-152.

    Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The research is supported by the National Natural Science Foundation of China 11261032, the Longdong University Grant XYZK-1207 and XYZK-1402.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Weidong Lv.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lv, W., Feng, J. Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces. Adv Differ Equ 2014, 184 (2014). https://doi.org/10.1186/1687-1847-2014-184

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2014-184

Keywords