On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

In this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

Boundary value problems with p-Laplacian operator arise in the modeling of many natural phenomena like in unsteady flow of fluid through a semi-infinite porous medium.
Boundary value problem (1.1) is said to be at resonance if the corresponding homogeneous problem ⎧ ⎨ ⎩ (ϕ p (D a 0+ u(t))) = 0, t ∈ (0, +∞), has nontrivial solutions. p-Laplacian resonant boundary value problems can be expressed in the abstract form Lu = Nu, where L is a non vertible fractional-order differential operator. When p = 2, the differential operator L is linear, and Mawhin's coincidence degree theory [13] can be applied, for example, see [2,3,8,9,15]. However, when p > 2, Mawhin's coincidence degree can no longer be applied directly, hence the Ge and Ren extension of the coincidence degree [4] becomes an efficient tool for the investigation, see [6,11,[17][18][19] and the references therein.
In [15] the authors obtained existence conditions for the fractional-order p-Laplacian boundary value problem at resonance Chen et al. [2] obtained the existence of solution for the fractional-order p-Laplacian boundary value problem In [19], the authors studied the following fractional-order boundary value problem: where 0 < a, b ≤ 1, 1 < a + b ≤ 2, p = 2, f is a continuous function, D a 0+ and D b 0+ are Caputo fractional derivatives.
Motivated by the above results, we study the solvability for the fractional-order p-Laplacian boundary value problem at resonance on the half-line with integral boundary conditions. We also note that p-Laplacian fractional boundary value problems with integral boundary conditions have not been given much attention in literature. In Sect. 2 of this work, the required lemmas, theorem, and definitions are given; Sect. 3 is dedicated to stating and proving the conditions for existence of solutions. An example is given in Sect. 4 to illustrate the results obtained.

Preliminaries
In this section, we give definitions, lemmas, and theorems that will be used in this work. Let U 1 = ker M and U 2 be the complement of Suppose that the following hold:

Then at least one solution exists for the abstract equation Mu
Then T is said to be relatively compact if all functions from T are bounded, equicontinuous on any compact subinterval of [0, +∞), and equiconvergent at ∞.
) Let a > 0, the Riemann-Liouville fractional-order integral of a function z : (0, +∞) → R is defined by provided that the right-hand side is pointwise defined on (0, +∞).

Lemma 2.1 ([12]
) Let a ∈ (0, +∞), then the general solution of the fractional differential equation . . , n, and n = [a] + 1 is the smallest integer greater than or equal to a.
Remark 2.1 ([4]) We will use the following properties of ϕ p . For d, e ≥ 0, Then the spaces (U, · ) and (Z, · Z ) by the standard argument are Banach Spaces. We Then boundary value problem (1.1) in an abstract form is Mu = N k u. Throughout this paper, we assume that and M is a quasi-linear operator.
Proof By simple calculation, we can see that ker As t → +∞, then while, when t = 0, we have Hence, For u ∈ dom M, it is clearly seen that dim ker L = 2 and Im M, a subset of Z is closed. Hence, M is a quasi-linear operator.
We define the projector E : U → U 1 as and F : Z → Z 1 as where D 1 z = If w is v-Caratheodory, then P is M-compact.
Proof of Theorem 3. 1 We have previously proved that M is quasi-linear and N k is Mcompact on . Also, from Lemma 3.1 and Lemma 3.2 we proved that (τ 1 ) and (τ 1 ) of Theorem 2.1 hold. Finally, we will prove that (τ 3 ) of Theorem 2.1 also holds. Let J : Im F → ker M be defined as If (3.1) holds for any u ∈ dom ∂ ∩ ker M, where u = b 1 t a-1 + b 2 t a = 0. We define the homeomorphism by Then H(u, 1) = -u = 0 and H(u, 0) = JFNu = 0 since Nu / ∈ Im M. For k ∈ (0, 1) and by way of contradiction, we assume H(u, k) = 0, then Since D = 0, we have , Hence, which contradicts |b 1 | + |b 2 | ≥ 0. If (3.2) holds, then we define Then which is also a contradiction. Hence, by the homotopy property of Brouwer degree, we have Therefore, at least one solution of (1.1) exists in .

Conclusion
Fractional differential equations are an efficient tool for describing the memory of different substances and have become popular recently. In order to further enrich this subject area, this work considers existence results for fractional-order p-Laplacian boundary value problem on the half-line at resonance where the differential operator is nonlinear and has a kernel dimension equal to two. The proof of the main result is based on the Ge and Ren coincidence degree theory, and the results obtained are new and extend some current results to the two-dimensional kernel. Examples were given to demonstrate the practicability and validity of our main results.