Monotone iterative method for fractional p-Laplacian differential equations with four-point boundary conditions

A four-point boundary problem for a fractional p-Laplacian differential equation is studied. The existence of two positive solutions is established by means of the monotone iterative method. An example supporting the abstract result is given.

Our paper is distinguished from [22,25,28] in the following three aspects. Firstly, the boundary condition (1.3) links the values of derivatives of the same order. At the same time, condition x(1) = aD γ 0+ x(ξ ) in (1.4) links the derivatives of different order (as usual, we regard x(1) as the derivative of order 0 of x at t = 1). Finally, in (1.2), the authors imposed the boundary condition D γ 0+ x(1) = ax(ξ ), where 1 + γ ≤ β, ξ ∈ (0, 1), and applied the Leggett-Williams fixed-point theorem. This is in a sharp contrast with the boundary condition , which allows us to use a monotone iterative method. Summing up: although the methodology that we use rests on the one developed in [28], our setting is different from the ones considered in [22,25,28].

Preliminaries
In this section, we present some preliminary results including estimates for Green functions and solvability of non-homogeneous fractional p-Laplacian BVPs. These results constitute key ingredients of the proof of our main result (Theorem 3.1). All the function spaces considered below consist of scalar functions. The two lemmas following below are well known.

Lemma 2.3 Let G and M be defined by
Proof (i) This statement follows immediately from (2.1) and (2.2).
On the other hand, ∀ξ ∈ (0, 1 2 Obviously, Therefore, The remaining three cases 0 can be treated using the similar method, so that we omit the obvious modifications. Thus, G(t, z) > 0 for all t, z ∈ (0, 1). Similarly, to prove that M(t, z) > 0, for all t, z ∈ (0, 1), consider, first, the case Put One can apply a similar argument in order to treat the remaining three cases 0 Thus, M(t, z) > 0 for t, z ∈ (0, 1).
Similarly, consider four cases for the function ν.

The proof of Lemma 2.3 is complete.
The next statement provides the existence and uniqueness result for the non-homogeneous problems of our interest.

M(z, r)y(r) dr dz
The proof of Lemma 2.4 is complete.
We complete this section with the following simple observation. Proof Since G, M and h are nonnegative and continuous, one has T(P) ⊂ P and T is continuous. To prove the complete continuity of T, one needs to use the standard argument based on the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem (see, for example, [23]).

Main result
We are now in a position to formulate our main result. To this end, denote where μ and ν are provided by Lemma 2.3(iii).
and (S 2 ) it follows immediately that We claim that T( ) ⊆ . In fact, for any x ∈ , we have Tx ∈ P, and by Lemma 2.3, one has
Hence, there exists a subsequence {y n i } ∞ i=1 of {y n } ∞ n=1 convergent to y * ∈ . Since {y n } ∞ n=1 is monotone, one has y n → x * . Combining the continuity of T with Ty n = y n+1 → y * yields Ty * = y * . It remains to observe that, by assumption (S 3 ), the zero function is not a solution of problem (1.4). So x * > 0, and y * > 0. The proof is completed.