SOLVABILITY OF FRACTIONAL FUNCTIONAL BOUNDARY-VALUE PROBLEMS WITH P-LAPLACIAN OPERATOR ON A HALF-LINE AT RESONANCE

This paper aims to consider the existence of solutions for p-Laplacian functional boundary-value problems at resonance on a half-line with two dimensional kernel. By employing some operators which satisfies suitable conditions and the Re and Gen extension of coincidence degree theory, a new result on the existence of solutions is acquired.


Introduction
Boundary value problems on the half-line arise in various applications such as in the study of the unsteady flow of gas through a semi-infinite porous medium, in analyzing the heat transfer in radial flow between circle disks, in the study of plasma physics,in an analysis of the mass transfer on a rotating disk in a non-Newtonian fluid, etc. [1]. Su and Zhang [25] studied the following fractional differential equations on the half-line, using Schauder's fixed point theorem, It's well-known that Leibenson [20] firstly introduced the p-Laplacian equation which is 19, 21, [26][27][28]) and the references therein. In [9], using the Ge and Ren extension of Mawhin coincidence theory [4], the author investigated the following p-Laplacian third order integral and m-point boundary value problem at resonance      (ϕ p (u ′′ (t))) ′ = ω(t, u(t), u ′ (t), u ′′ (t)), t ∈ (0, 1); where the function ω : α i ξ i = 1 and η ∈ (0, 1). Fractional differential equations appear naturally in various fields of science and engineering. This is due to the fact that the differential equations of arbitrary order provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials. For recent publication on fractional calculus and fractional differential, we refer the reader to see [17,22,23]. In [26], the author discussed the existence of solutions for the following multipoint boundary value problem of fractional p-Laplacian equation where 1 < α ≤ 2, D α 0 + is the standard Riemann-Liouville fractional derivative, 0 < ξ 1 < ξ 2 < . . . < ξ n < +∞, α i > 0, Σ n i=1 α i = 1. Motivated by the above results, in this paper, we study boundary value problem                        2) in infinite interval, where 0 < β ≤ 1, n − 1 < α ≤ n, n ≥ 3, φ p (s) = |s| p−2 s, p > 1, Γ 1 , Γ 2 are continuous linear functionals with the resonance condition: Γ 1 (t α−1 ) = Γ 2 (t α−n+1 ) = Γ 1 (t α−n+1 ) = Γ 2 (t α−1 ) = 0. Boundary value problem (1.2) is to be at resonance if D β 0 + (φ p (D α 0 + u))(t) = 0 subject to boundary conditions has a non-trivial solution.
To the best of our knowledge, this is the first paper to study p-Laplacian functional boundary-value problems at resonance on a half-line with two dimensional kernel, which contains two-point boundary conditions, multi-point boundary conditions, differential boundary conditions, integral boundary conditions and integral differential boundary conditions that are commonly studied, and has a high degree of generality. And the main difficulties are that we have to construct suitable Banach spaces for the problem and establish operators P and Q. Of course, when n − 1 < α ≤ n, we need to be more careful in dealing with some details. It is worth noting that the operator Q is not a projector.
In this paper, we will always suppose that the following conditions hold.

Preliminaries
Definition 2.1 (see [4,15]). Let X and Z be two Banach spaces with norms ∥ · ∥ X , ∥ · ∥ Z , respectively. A continuous operator M : where dom M denotes the domain of the operator M .
In this paper, an operator T : Let X 1 = Ker M and X 2 be the complement space of X 1 in X. Then X = X 1 ⊕ X 2 . Let P : X → X 1 be projector and Ω ⊂ X be an open and bounded set with the origin θ ∈ Ω. Definition 2.2 (see [4]). Suppose that N λ : Ω → Z, λ ∈ [0, 1] is a continuous and bounded operator.
By Lemam 2.7 and (1) in Lemma 2.8, it is easy to deduce that By Definition 2.5, we have Thus, we obtain Consequently, we conclude that {X, ∥ · ∥ X } is a Banach space, which completes the proof of Lemma 2.11.

Main result
In order to obtain the main result, we will introduce the following assumptions: shown in the following lemma.
Proof. It is easy to get that where c and d are two arbitrary constants.
By the continuity of T , we get that Im M ⊂ Z is closed. So, M is quasi-linear.
Proof of Lemma 3.1. Take a projector P : X → X 1 and an operator Q : Z → Z 1 as follows: , By Lemma 3.1 and condition (C 2 ), we carefully check that Obviously, QZ := Z 1 , Q(I − Q) = 0, dimZ 1 = dimX 1 and KerQ = ImM . It follows from Lemma 3.1 and g(t) ∈ Y that Q : Z → Z 1 is continuous and bounded.
For u ∈ X, set u = u − P u + P u. It is easy to check that and it is also elementary to confirm the identity ImP = KerM and ImP KerP = Define an operator R as Proof. Firstly, we prove that R : Ω × [0, 1] → X 2 and P u + R(u, λ) ∈ domM, u ∈ Ω, λ ∈ [0, 1]. Considering the continuity of Q and f , we can easily check that By the continuity of f , |T i y| ≤ ∥y∥ ∞ and sup t∈[0,+∞) |I β 0 + µ| < +∞, we get that for any r > 0, there exists a constant M r > 0 such that if Therefore, R(u, λ) ∈ X. It is clear that Meanwhile, note that (I − Q)N λ u ∈ KerQ = ImM , we obtain that Secondly, we show that R is continuous.
Since Ω is bounded, there exists a constant r > 0 such that ∥u∥ X ≤ r, u ∈ Ω. By (H 1 ), (H 2 ) and boundedness of Q, there exist two constants M r and C r such that By the uniform continuity of φ q (x) in [−C r , max{1, C r }], we obtain that for any such that if u, v ∈ Ω, ∥u − v∥ X < δ η , then These, together with It is easy to get that Since t n−α e (q+n−1)t , t α and t are uniformly continuous on [0, T ], we get that Take It follows from the uniform continuity of t, Considering the uniform continuity of 1 e (q+i−1)t , t i and t, i = 1, 2, . . . , n − 1 on For any ε > 0, there exists a constant T 1 > 0 such that for any t > T 1 , Obviously, there exists a constant T > T 1 such that for any t > T For any t 2 > t 1 > T , we have By Theorem 2.4, we get that {R(u, λ)|u ∈ Ω, λ ∈ [0, 1]} is relatively compact. The proof is completed. Now, we will show that N λ is M -quasi-compact in Ω, where Ω ⊂ X is an open and bounded set with θ ∈ Ω.

5) and
(3.6) Then the boundary value problem (1.2) has at least one solution.