Parameter dependence for existence , nonexistence and multiplicity of nontrivial solutions for an Atıcı – Eloe fractional difference Lidstone BVP

Dependence on a parameter λ are established for existence, nonexistence and multiplicity results for nontrivial solutions to a nonlinear Atıcı–Eloe fractional difference equation ∆y(t− 2)− β∆ν−2y(t− 1) = λ f (t + ν− 1, y(t + ν− 1)), with 3 < ν ≤ 4 a real number, under Lidstone boundary conditions. In particular, the uniqueness of solutions and the continuous dependence of the unique solution on the parameter λ are also studied.


Introduction
Currently, there is increasing interest in Atıcı-Eloe fractional difference equations, with pioneering papers by Atıcı and Eloe [2][3][4] and Goodrich [6,7] driving much of this interest.It is natural to investigate questions for Atıcı-Eloe fractional difference equations devoted to the important results, such as those obtained in [1,5,9,10,12].That is the goal of this paper for fractional difference equations involving Lidstone boundary conditions.
In a recent paper [10], under the same boundary conditions, Graef et al. studied a nonlinear discrete fourth-order equation with dependence on two parameters: for t ∈ {a + 1, a + 2, . . ., b − 1}.Two sequences were constructed so that they converged uniformly to its unique solution.
Motivated by the above works, in this paper, for b ∈ N and b ≥ 3, we are concerned with the parameter dependence for existence, nonexistence and multiplicity of nontrivial solutions, as well as the uniqueness of solutions, for the νth order Atıcı-Eloe fractional difference equation, for t ∈ {1, 2, . . ., b}, satisfying the discrete Lidstone boundary conditions where ∆ ν is the νth Atıcı-Eloe fractional difference with 3 < ν ≤ 4 a real number, β > 0 and λ > 0 are parameters, and The rest of this paper is organized as follows.In Section 2, we give some preliminary definitions and theorems from the theory of cones in Banach spaces that are employed to establish the main results.In Section 3, we give main results.We first construct some Green's functions, evaluate bounds for the Green's functions and define a suitable cone in a Banach space.Then, we derive existence, nonexistence and multiplicity results for nontrivial solutions to the BVP (1.1)-(1.2) in terms of different values of λ, as well as the unique solution for the BVP, which depends continuously on the parameter λ.

Preliminaries
We shall state some definitions from fractional difference equations along with some definitions and theorems from cone theory on which the paper's main results depend.Definition 2.1 ([2, 8]).Let n − 1 < ν ≤ n be a real number and t ∈ {a + ν, a + ν + 1, . ..}.The νth Atıcı-Eloe fractional sum of the function u is defined by where t (ν) = Γ(t + 1)/Γ(t + 1 − ν) is the falling function.If t + 1 − ν is a pole of the Gamma function and t + 1 is not a pole, then t (ν) = 0. Also, the νth Atıcı-Eloe fractional difference of the function u is defined by where ∆ is the forward difference defined as ∆u(t) = u(t + 1) − u(t), and ∆ i u(t) = ∆(∆ i−1 u(t)), i = 2, 3, . . .Remark 2.2.We note that for u defined on {a, a + 1, . ..}, then ∆ −ν a u is defined on {a + ν, a + ν + 1, . ..}.We shall suppress the dependence on a in ∆ −ν a u(t) since domains will be clear by context.Remark 2.3.From the definition of νth Atıcı-Eloe fractional difference, we have We also require the following operational properties of fractional sum operator.
Let (B, • ) be a real Banach space.P ⊂ B is a cone provided (i) αu + βv ∈ P, for all α, β ≥ 0 and for all u, v ∈ P, and (ii) P ∩ (−P ) = {0}.A cone P in a real Banach space B induces a partial order on B; namely, for u, v ∈ B, u v with respect to P, if v − u ∈ P.
For our existence results, we will employ the theorem below which is due to Krasnosel'skiȋ [11].
Theorem 2.6.Let B be a Banach space, P ⊂ B be a cone, and suppose that Ω 1 , Ω 2 are bounded open balls of B centered at the origin, with Ω 1 ⊂ Ω 2 .Suppose further that A : P ∩ (Ω 2 \ Ω 1 ) → P is a completely continuous operator such that either holds.Then A has a fixed point in P ∩ (Ω 2 \ Ω 1 ).

Main results
First, let us consider the following boundary value problems respectively.Anderson and Minhós [1] derived the expression for the Green's function G 1 (t, s) for the BVP (3.1), where Also, by direct computation, we can get the Green's function G 2 (t, s) for the BVP (3.2), where Next, we consider the Banach space (B, • ) of real-valued functions on {ν − 4, ν − 3, . . ., ν + b − 2} with the norm From the following result we can see that the Green's function of the νth order boundary value problem is a convolution of (3.3) and (3.4).
has the solution ) with [[r]] denoting the smallest integer larger than or equal to r.