Existence of solutions of discrete fractional problem coupled to mixed fractional boundary conditions

In this paper, we introduce a two-point nonlinear boundary value problem for a finite fractional difference equation. An associated Green’s function is constructed as a series of functions and some of its properties are obtained. Some existence results are deduced from fixed point theory and lower and upper solutions.


Introduction
In this paper, we consider the following discrete fractional problem coupled to mixed fractional boundary conditions, − α y(t) + a(t + α − 1)y(t + α − 1) = f (t + α − 1, y(t + α − 1)), t ∈ N T +1 0 , y(α − 2) = β y(α + T + 1 − β) = 0, (1) where α is the standard Riemman-Liouville type discrete fractional difference operator, t ∈ N T +1 0 , 1 < α ≤ 2, 0 ≤ β ≤ 1, T is a positive integer and function f : N α+T α−1 × R −→ R is continuous. Here we denote N K +r r = {r , . . . , K + r } for any r ∈ R and K a positive integer. So, we will look for solutions y : N T +α+1 The continuous fractional calculus has a long history, but the discrete fractional calculus has been investigated recently. The development of this theory starts with the paper of Díaz and Osler [5] where is defined a fractional difference as an infinite series and a generalization of the binomial formula for the n-th order difference n f operator. This study continues with the work of Miller and Ross [10] which deals with the linear υ-th order fractional differential equation as an analogue of the linear n-th order ordinary differential equation. Since them, a great progress has been made in the study of boundary value problems for fractional difference equations (see [1,2,[6][7][8] and references therein).
We recall some classical definitions from discrete fractional calculus theory and preliminary results.

Definition 1
We define t (ν) = (t + 1)/ (t + 1 − ν), for any t and ν for which the right hand side is well defined. We also 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
The ν-th fractional sum of a function f, for ν > 0 and t ∈ N r +ν , is defined as We define the ν-th fractional difference for ν > 0, by The paper is scheduled as follows: In next section we deduce some properties of the Green's function related to the linear problem for the particular case of a(t + α − 1) = 0 for all t ∈ N T +1 0 . The case 0 < α − β < 1 has been treated in [9], we will continue this study, by improving some of the obtained results in [9] and by considering the case 1 ≤ α − β ≤ 2. Section 3 is devoted to deduce the expression of the Green's function related to the problem (2) for a nontrivial function a(t), with small enough bounded absolute value. Moreover, some a priori bounds of the Green's function are obtained. The arguments are in the line of the ones given in the paper [3]. In last section we show the applicability of the given results by obtaining some existence and uniqueness results of the nonlinear problem (1).

Properties of function G 0
In this section we will extend previous results related to the Green's function G 0 of problem (2) with a identically zero. Some of these results have been given in [9] for the case 0 < α−β < 1, we improve some of them and study also the case 1 ≤ α − β ≤ 2.
Lemma 4 [9] The unique solution of the linear fractional mixed boundary value problem In the following Lemma, we will show the properties of the Green's function, which extends the one given in [9, Lemma 3.3].
(ii) The case 0 < α−β < 1 has been proved in [9,Lemma 3 Thus, to show that (α) t G 0 (t, s) < 0, we must prove that So, by using that As a direct consequence, we deduce that As a result, we get that and the proof is concluded.
In the following, we obtain a bound from above of the Green's function.

Theorem 6
Let G 0 (t, s) be the Green's function defined by (3). Then, for any (t, s) , the following inequalities are fulfilled: Proof From previous result we know that So, the bounds come from the ones of the previous inequality. Since, we have that Moreover, it is clear that (6) vanishes if and only if It is immediate to verify that s 0 (α, β) ∈ {0, . . . , T + 1} if and only if α − β > 1.
In this situation we have that Thus (ii) Assume now that 1 < α − β. Now the maximum is attained at s 0 (α, β), in consequence: and the proof is concluded.

Green's function of problem (2)
In this section we deduce the existence of the Green's function G related to Problem (2). The expression of the function G is given as a functional series and we prove its convergence for suitable values of the function a(t). The arguments are in the line to the ones given in [3].
where G 0 (t, s) is given by (3) and In order to express the Green's function associated with the linear problem (2) we shall use the spectral theory in Banach space given by the following Lemma: Let be the Banach space with norm Consider the following assumption: (H ) There exists a > 0 such that where C 1 (T , α, β) and C 2 (T , α, β) are introduced in Theorem 6.
In next result we prove the existence and uniqueness of solution of problem (2) by means of the construction of its related Green's function.

Moreover, G(t, s) is the Green's function related to problem (2).
Proof The solution y of (2) satisfies, for all t ∈ N T +α+1 α−2 , the following equality: where G 0 is defined by (3).
This expression can be reformulated as Now, denote operators A and B by Then, Eq. (9) becomes First, let us show that B < 1. For any y ∈ X with y = 1 and t ∈ N T +α+1 α−2 , by (11), we have (i) If 0 < α − β ≤ 1, then, using Theorem 6 (i) and condition (H ), we have that for all t ∈ N T +α+1 α−2 the following inequalities are fulfilled: (ii) If α − β > 1, then using Theorem 6 (ii) and condition (H ), we get Therefore, by Lemma 7, we deduce that By using analogous arguments to [3], we show that for n ∈ N 0 = {0, 1, 2, . . .} the following identity holds: In the sequel we obtain the following a priori bounds for function G n for all n ∈ N 0 : (ii) If α − β > 1, then where a is defined in (H ).
By induction, (14) if fulfilled for any n ∈ N 0 . Arguing as at the first case, we conclude that (15) is true for all n ∈ N 0 . Finally, by (H ) and (7), for all (t, s) ∈ N T +α+1 , we deduce the following inequalities:
As a direct consequence of Theorem 8 and Eqs. (7) and (8), we deduce the following result In next result, we obtain an upper bound for function |G(t, s)| for any s fixed.
, where G is given by (7) .

Existence and uniqueness of solutions
In this section, we discuss the existence and uniqueness of solutions to problem (1). Define the operator S : X −→ X by The proof of that S is completely continuous is analogous to the one done in [3]. It follows that the fixed points of the operator S are the solutions of the boundary value problem (1).
Arguing as in [3], one can prove the following result

Theorem 11 Assume that condition (H) holds and f satisfies the following condition:
(H * ) There exists a constant K ∈ (0, T +1 s=0 G(s) −1 ) (G given in Lemma 10) such that Then Problem (1) has a unique solution.

Example 12
Let a satisfy condition (H) and K ∈ (0, with g : N α+T α−1 → R an arbitrary nontrivial given function. It is not difficult to verify that for any t ∈ N α+T From Theorem 11 the considered problem has a unique solution. In the sequel we deduce an existence result for the nonlinear Problem (1).

Theorem 13 Assume that condition (H) holds and f satisfies the following condition:
where h : [0, +∞) → [0, +∞) is a continuous function such that Then Problem (1) has at least one solution.
Proof We know that the solutions of problem (1) are given as the fixed points of operator S defined on (19). So, to deduce the existence of a fixed point we consider, for any λ ∈ (0, 1), a function y ∈ X , such that y = λ S y. As a consequence, for all t ∈ N α+T α we have: Thus, we have that So, from (21) we deduce that there is a constant K > 0, independent of λ, such that In consequence, we deduce from Schaefer's fixed point Theorem [12], that operator S has a fixed point and it is a solution of Problem (1) Example 14 Let a satisfying condition (H ). Let us consider Problem (1) with function which satisfies inequality (20) and property (21) for functions h(s) = √ s + 1, s ≥ 0, and g(t) = t 2−α , t ∈ N α+T α−1 . So, from Theorem 13, the considered problem has at least one solution.
Remark 15 Notice that in the two previous theorems we cannot ensure that the obtained solution is not trivial. This property can be deduced when f (t, 0) ≡ 0 on N α+T α . In particular, the obtained solutions in the two previous examples are non trivial. Now we will develop the monotone iterative technique for problem (1). To this end we must assume the existence of a pair of well ordered lower and upper solutions, which are defined as follows.