Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian

In this article, we study a class of fractional coupled systems with Riemann-Stieltjes integral boundary conditions and generalized p-Laplacian which involves two different parameters. Based on the Guo-Krasnosel’skii fixed point theorem, some new results on the existence and nonexistence of positive solutions for the fractional system are received, the impact of the two different parameters on the existence and nonexistence of positive solutions is also investigated. An example is then given to illuminate the application of the main results.


Introduction
In this paper, our main research is the existence and nonexistence of positive solutions for the following fractional coupled system with generalized p-Laplacian involving Riemann-Stieltjes integral conditions.

Lemma . ([]) Assume that (H  ) holds. Then
is a continuous function. Through the application of the Guo-Krasnosel'skii fixed point theorem and the Leggett-Williams theorem, sufficient conditions for the existence of positive solutions are received.
In system (),   g  (s)v(s) dA  (s),   g  (s)u(s) dA  (s) denote the Riemann-Stieltjes integrals, and A i is a function of bounded variation, which implies that dA i can be a signed measure. Then, a multipoint boundary value problem and an integral boundary value problem are included in our study, that is to say, system () includes more generalized boundary value conditions. Henderson and Luca in [] considered the following system: where has been studied in many papers, where μ i >  is a constant, for μ i =  as an exceptional case ( see [-] and the references therein). However, these articles only study the existence of positive solutions for the system, and do not relate to the nonexistence of positive solutions.
Up to now, coupled boundary value conditions for a fractional differential system with generalized p-Laplacian like system () have seldom been considered when λ  , λ  are different. Motivated by the results mentioned above, in this paper, we obtain several new existence and nonexistence results for positive solutions in terms of different values of the parameter λ i by using the properties of Green's function and the Guo-Krasnosel'skii fixed point theorem on cone. Especially, paying attention to the nonlinear operator D β  + (φ(D α  + )) with the discussion in (), we can convert it to the linear operator D β  + D α  + , if φ(u) = u, and the additive index law holds under some reasonable constraints on the function u (see []). Therefore, our article promotes, includes and improves the previous results in a certain degree.

Preliminaries and lemmas
For convenience of the reader, we present some necessary definitions about fractional calculus theory.

Definition . ([, ])
The Riemann-Liouville fractional derivative of order α > , n - ≤ α < n, n ∈ N, is defined as where N denotes the natural number set, the function u(t) is n times continuously differentiable on [, +∞).
Similarly to the proof in [], it enables us to obtain the following Lemmas ., . and Remark ..
Lemma . Assume that the following condition (H  ) holds.
From Lemmas . and ., we obtain the following Lemma ..
has a unique integral representation In the rest of the paper, we always suppose that the following assumption holds: where ω is defined as Remark .. It is easy to see that K is a positive cone in X. Under the above conditions (H  )(H  )(H  ), for any (u, v) ∈ K , we can define an integral operator Proof By the routine discussion, we know that T : K → X is well defined, so we only prove On the other hand, Then we have i.e., In the same way as () and (), we can prove that Therefore, we have T(K) ⊆ K . According to the Ascoli-Arzela theorem, we can easily get that T : K → K is completely continuous. The proof is completed.
In order to obtain the existence of the positive solutions of system (), we will use the following cone compression and expansion fixed point theorem.
then A has at least one fixed point in P ∩ (  \  ).

Main results
Denote

Theorem . Assume that (H  )(H  )(H  ) hold and f i∞
, then system () has at least one positive solution for Proof For any λ i satisfying (), there exists ε  >  such that By the definition of f  i , there exists r  >  such that , by the definition of · , we know that Thus, for any (u, v) ∈ ∂K r  , by (), () and (H  ), we have Hence, for any (u, v) ∈ ∂K r  , by Lemmas ., ., . and (), we conclude that Similarly to (), for any (u, v) ∈ ∂K r  , we also have Consequently, we have On the other hand, by the definition of f i∞ , there exist r  , r  >  such that Thus, for any (u, v) ∈ ∂K r  , by (), () and (H  ), we have Hence, for any (u, v) ∈ ∂K r  , by Lemmas ., ., . and (), we have (   ) Therefore, we obtain It follows from the above discussion, (), (), Lemmas . and . that, for any λ i ∈ T has a fixed point (u, v) ∈ K r  \ K r  , so system () has at least one positive solution (u, v); moreover, (u, v) satisfies r  ≤ (u, v) ≤ r  . The proof is completed.
Remark . From the proof of Theorem ., if we choose , the conclusion of Theorem . is valid. Or we choose , the conclusion of Theorem . is valid.

Theorem . Assume that (H  )(H  )(H  ) hold and f i
, then system () has at least one positive solution for The proof of Theorem . is similar to that of Theorem ., and so we omit it.
Remark . Similar to Remark ., if we choose L  as (), then for λ  ∈ ( , the conclusion of Theorem . is valid.
Or we choose L  as (), then for λ  ∈ (, , the conclusion of Theorem . is valid.

Theorem . Assume that (H  )(H  )(H  ) hold and there exist R > r >  such that
Then system () has at least one positive solution (u, v); moreover, (u, v) satisfies r ≤ (u, v) ≤ R.
For any (u, v) ∈ ∂K r , by the definition of · , we have Thus, for any (u, v) ∈ ∂K r , by the first inequality of (), we have Hence, for any (u, v) ∈ ∂K r , by Lemmas ., ., . and (), we have Therefore, we obtain , by the definition of · , we know that Thus, for any (u, v) ∈ ∂K R , by the first inequality of () and (), we have Hence, for any (u, v) ∈ ∂K R , by Lemmas ., ., . and (), we can gain Similarly to (), for any (u, v) ∈ ∂K R , we also have Consequently, we have It follows from the above discussion, (), (), Lemmas . and . that T has a fixed point (u, v) ∈ K R \ K r , so system () has at least one positive solution (u, v); moreover, (u, v) satisfies r ≤ (u, v) ≤ R. The proof is completed.

Remark . From the proof of Theorem ., if we choose
the conclusion of Theorem . is valid. Or we choose Then, for the conclusion of Theorem . is valid.
Remark . From the proof of Theorem ., assume that (H  )(H  )(H  ) hold, if f i = +∞ or f i∞ = +∞, then there exists λ * i >  such that system () has at least one positive solution for λ i ∈ (, λ * i ), i = , .
max{φ(x),φ(y)} }, we have Assume that (u, v) is a positive solution of system (), we will show that this leads to a contradiction. Define λ i = (M  i ) - ϕ  (L -  ), since λ i ∈ (, λ i ), by Lemmas ., . and ., we conclude that Therefore, we conclude Similarly to () (), we also have which is a contradiction. Therefore, system () has no positive solution. The proof is completed.