Maximal and minimal iterative positive solutions for singular infinite-point p-Laplacian fractional differential equations

Abstract. The existence of maximal and minimal positive solutions for singular infinite-point p-Laplacian fractional differential equation is investigated in this paper. Green’s function is derived, and some properties of Green’s function are obtained. Based upon these properties of Green’s function, the existence of maximal and minimal positive solutions is obtained, and iterative schemes are established for approximating the maximal and minimal positive solutions.

In [18], the authors considered the following fractional differential equation: Motivated by the results above, in this paper, we investigate the existence of positive solutions for a class of infinite-point singular p-Laplacian fractional differential equations. p-Laplacian fractional differential equation is a type of equation that is very wide, and the general equation are special cases of p-Laplacian equation. Compared with [24,29], the fractional-order derivatives are involved in the nonlinear term and boundary condition, and at the same time, iterative solutions are obtained by iterative sequences. Compared with [19], values at infinite points are involved in the boundary conditions of the boundary value problem (1), and the nonlinearity is singular, that is, f (t, u, v) is allowed to be singular at t = 0. Compared with [7], we do not only obtain the existence of positive solutions, but we also establish iterative sequences to approximate the maximal and minimal positive solutions.

Preliminaries and lemmas
Some basic definitions and lemmas, which will be used in the proof of our results and can also be found in the recent literature such as [9,20], we omit some here. Now we list a condition below to be used later in the paper.
can be expressed by where in which Proof. By means of Lemma 1, we reduce (2) to an equivalent integral equation . By some properties of the fractional integrals and fractional derivatives, we have On the other hand, D p1 , and combining with (5), we get Hence, Therefore, It is easy to check that G(t, s) and Lemma 4. Let ∆ > 0, then the Green function (4) has the following properties: By direct calculation, we get P (s) 0, s ∈ [0, 1], and so, P (s) is nondecreasing with respect to s. For by (4) and (10), we have by (9) and (11), we have Clearly, ∆Γ(α)G(t, s) Γ(α)t α−1 P (s)(1−s) α−p1−1 . So, the proof of (7) is completed. Similarly, (8) also holds.
and E is endowed with an order relation . Moreover, we define a cone of E by and define an operator Problems (1)  Proof. First, for u ∈ P , by the continuity of G(t, s), s σ φ q (f (s, u(s), D µ 0 + u(s))), and the integrability of s −σ , is well defined on K. Thus, it follows from the uniform continuity of G(t, s) on [0, 1] × [0, 1] and that Au ∈ C[0, 1], u ∈ K. Furthermore, by the uniform continuity of D µ 0 + G(t, s), for t, s ∈ [0, 1], we get On the other hand, since u n → u in C 1 [0, 1], there exists A > 0 such that u n A (n = 1, 2, . . . ), and then u A. Furthermore, Hence, for any ε > 0, there exists δ > 0 such that for any s 1 , s 2 ∈ [0, 1], By u n − u → 0, for the above δ > 0, there exists N such that for all n > N , we get Hence, for any t ∈ [0, 1], n > N , by (13), we derive Thus, for n > N , t ∈ [0, 1], by (14), we have and hence, we get Au n − Au 0 → 0, , hence A(K) ⊂ K. Now we will prove that AV is relatively compact for bounded V ⊂ K. Since V is bounded, there exists D > 0 such that for any u ∈ V , u D, and by the which shows that AV is bounded in E. Next, we will verify that D µ Thus, we obtain From above, the uniform continuity of t α−µ−σ , t α−µ−1 , and together with Lemma 2, we can derive that AV is relatively compact in E, and so, we get that A : K → K is completely continuous.
If f (t, 0) ≡ 0, 0 t 1, then the zero function is not the solution of BVP (1). Hence, v * is a positive solution of BVP (1).
Since each fixed point of A in K is a solution of BVP (1), by above proof, we get that u * and v * are positive solutions of the BVP (1) on [0, 1]. Remark 1. The iterative sequences in Theorem 1 begins with a simple function, which is useful for computational purpose.
Remark 2. u * and v * are the maximal and minimal solutions of the BVP (1), respectively, but u * and v * may be coincident, and when u * and v * are coincident, the boundary value problem (1) will have a unique solution in K d .