Existence and Nonexistence of Positive Solutions for Mixed Fractional Boundary Value Problem with Parameter and 𝑝 -Laplacian Operator

This paper mainly studies a class of mixed fractional boundary value problem with parameter and 𝑝 -Laplacian operator. Based on the Guo-Krasnosel’skii fixed point theorem, results on the existence and nonexistence of positive solutions for the fractional boundary value problem are established. An example is then presented to illustrate the effectiveness of the results.

Lu et al. in [29] investigated the BVP 0 + ( ( 0 + ( ))) = ( , ( )) , 0 < < 1, where 1 < ≤ 2, 2 < ≤ 3, 0 + and 0 + are Riemann-Liouville derivative operators, is the -Laplacian operator defined by ( ) = | | −2 , and : [0,1] × In this paper, we study the existence and nonexistence of positive solutions for the mixed fractional boundary value problem as BVP (1), which leads to lots of difference and new features. On one hand, compared to the papers mentioned above, which only involve one derivative, our study involves both the Riemann-Liouville fractional derivative and the Caputo fractional derivative, which making the studied problems difficult. On the other hand, under different combinations of superlinearity and sublinearity of the function , results on the existence and nonexistence of positive solutions are received and the impact of the parameter on the existence and nonexistence of positive solutions is also obtained.
and we now consider the following BVP: Lemma 4 (see [32]). If ∈ [0, 1], then BVP (7) has a unique solution where For any given ∈ [0, 1], consider the following BVP: By analysis, we know that (11) can be decomposed into the BVP (7) and the BVP has a unique solution where Proof. By Lemma 3, BVP (13) is equivalent to the following integral equation: By (1) = 0, we get Combining (16) and (18), we can obtain (14). The proof is completed.
Lemma 6. The Green functions ( , ) and ( , ) defined separately by (9) and (15)  ( Proof. Obviously, (1) holds, in the following, and we proof (3). From the definition of ( , ), for 0 ≤ ≤ ≤ 1, we know that (3) holds. For 0 ≤ ≤ ≤ 1, we have − ≥ − and then We know that is a positive solutions of BVP (1) if and only if is a fixed point of in .
By the similar proof as Theorem 9, the following Theorem 10 holds.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that they have no conflicts of interest.