Positive solutions of a second-order nonlinear Robin problem involving the first-order derivative

where f ∈ C([0, 1]×R+,R+) and α ∈]0, 1[. Based on a priori estimates, we use fixed point index theory to establish some results on existence, multiplicity and uniqueness of positive solutions thereof, with the unique positive solution being the limit of of an iterative sequence. The results presented here generalize and extend the corresponding ones for nonlinearities independent of the first-order derivative.

Notice that our boundary value conditions in Problem (1) correspond to α = 1, β = 0, γ = 1, δ = -α < 0 in (2), so the boundary value conditions in (1) are faux special cases of (2). What is more important, the nonlinearity in (1) involves the first-order derivative, whereas the nonlinearity in (2) does not. This means that (1) is much more difficult to deal with than (2). When tackling the boundary value problems involving the first order derivative, one usually employs the Leggett-Williams fixed point theorem [19] and all kinds of its generalizations [4][5][6], or the coincidence degree theory [11]. The methods mentioned above cannot be applied to generalize the sharp results in [20,23]. Consequently, in this paper, we shall use neither the Leggett-Williams fixed point theorems and its variants, nor the coincidence degree theory to deal with our problem (1), rather utilize the fixed point index theory on cones of Banach spaces to do that. The most important and difficult ingredients in our proofs are the establishing of the a priori estimates of positive solutions for some associated problems, in particular these for the first-order derivatives. In order to facilitate our proofs, we first establish an integral identity and an integral inequality that are of vital importance in the proofs of our main results. For the cases of superlinear nonlinearity at ∞, the Bernstein-Nagumo type condition [7,9,25] is introduced to enable us to obtain the a priori estimate of the first-order derivative for associated boundary value problems. Our results generalize and extend the ones in [20,23], are strikingly different from the ones in [1-3, 8, 12, 14, 15, 17, 21] and also complement the main ones in [16,24]. This paper is also a continuation of [32](see also [30,31]),where we studied (1) with α = 0 by the integro-differential equation argument. In contrast to [32], with the introduction of a cone in C 1 [0, 1] and the function g(x, y) := (1α)x + 2y,we shall tackle Problem (1) using fixed point index theory more directly, thereby rendering the statements and proofs of the main results clearer and more concise.
The remainder of this paper is organized as follows. Some basic lemmas, including two versatile results, i.e. an integral identity and an integral inequality, are stated and proved in Sect. 2. The main results, Theorems 3.1-3.3, are presented and proved in Sect. 3. Also, three extra results parallel with Theorems 3.1-3.3 are given in this section. The uniqueness of positive solutions and its iteration convergence are offered in Sect. 4.

Preliminaries and basic lemmas
For the sake of convenience, we denote β := 1α from now on.
Notice that (1) is equivalent to the integral equation below: where k(t, s) is the Green function defined by Then (E, · ) is a Banach space and P is a normal cone in E. Define A : P → P by Under the condition f ∈ C([0, 1] × R 2 + , R + ), the operator A : P → P is completely continuous. In our setting, the existence of positive solutions for (1) is equivalent to that of positive fixed points of the completely continuous nonlinear operator A : P → P. Lemma 2.1 (see [13]) Let E be a real Banach space and P a cone in E. Suppose that ⊂ E is a bounded open set and that T : ∩ P → P is a completely continuous operator. If there exists w 0 ∈ P \ {0} such that w -Tw = λw 0 , ∀λ ≥ 0, w ∈ ∂ ∩ P, then i(T, ∩ P, P) = 0, where i indicates the fixed point index. [13]) Let E be a real Banach space and P a cone in E. Suppose that ⊂ E is a bounded open set with 0 ∈ and that T : ∩ P → P is a completely continuous operator. If What follows are two versatile lemmas in the proofs of our main resuts.
Proof By integration by parts, we have Combining the preceding two equalities and noting w(0) = w (1)αw(1) = 0, we obtain the desired result.
Proof Integrating by parts, we have and thus 1 0 Then the desired inequality follows immediately.

Existence and multiplicity of positive solutions of (1)
For the sake of convenience, we define the function g by g(x, y) := βx + 2y.
Recall that we have defined β := 1α in the preceding section. We make the following hypotheses in this section.
(H1) f ∈ C([0, 1] × R 2 + , R + ). (H2) There are two constants a > β and c 1 > 0 such that (H5) There are two constants r > 0, d > β such that (H7) f (t, x, y) is increasing in x, y and there is a constant ω > 0 such that Proof Let and there is λ ≥ 0 such that u = Au + λϕ, and equivalently, - Multiply the preceding inequalities by ψ(t) := te βt and integrate over [0, 1] and invoke Lemma 2.3 to obtain Invoking Lemma 2.4 yields thereby establishing the a priori bound of u 0 for M 1 . Now we turn to establish the a priori bound of u 0 for M 1 . Indeed, if u ∈ M 1 ,then u ∈ C 2 [0, 1] (as explained previously) and there is a constant λ ≥ 0 such that u = Au + λψ, which can be equivalently rewritten as Let μ := sup{λ ≥ 0 : u = Au + λϕ for some u ∈ P}.

Now Lemma 2.4 implies
By (H6) again, we have Some basic calculations, along with the boundary value condition u (1) = αu(1), imply This establishes the a priori bound of u 0 for M 3 , which, together with the a priori bound of u 0 , implies the boundedness of M 3 . Taking R > sup{ u : u ∈ M 3 }, we have u = λAu, ∀u ∈ ∂B R ∩ P, λ ∈ [0, 1].
This, along with (5) and (8) Therefore, A has at least two positive fixed points, one on (B R \ B ω ) ∩ P and the other on (B ω \ B r ) ∩ P. This implies (1) has at least two positive solutions, which completes the proof.

Lemma 3.1 If u ∈ P is concave on
The following three results are paralell counterparts of Theorems 3.1-3.3, acquired by replacing (H2) and (H5) with (H2) and (H5) , respectively.  Then Below is a result that is easy to prove.
Recall that we have defined, in the proof of Theorem 3.1, the function ϕ ∈ P \ {0} by It is easy to see that Notice that the cone P defines a partial ordering in E: Proof Let Then, by (H5) and (11), for every ε ∈]0, δ], we have or, equivalently, Au ε u ε .
This, along with Lemmas 4.3 and 4.4, implies that there are two positive constants constant ε and m such that u 0 := εϕ and v 0 := mϕ are a subsolution and a supersolution of