Positive Solutions for a Singular Superlinear Fourth-Order Equation with Nonlinear Boundary Conditions

(1) where λ > 0 is a small positive parameter, f: (0,∞)⟶ R is continuous, h: [0, 1]⟶ [0, ∞) is continuous. If f(0) > 0, then (1) is called a positone problem; if f(0) < 0, then (1) is called a semipositone problem. ,e study of semipositone problems was formally introduced by Castro and Shivaji [1]. From an application viewpoint, one is usually interested in the existence of positive solutions for semipositone problems. Significant processes on second-order semipositone problems have taken place in the last 10 years, see [1–5] and the references therein. Fourth-order boundary value problems modeling bending equilibria of elastic beams have been considered in several papers [6–9]. Most of them are concerned with nonlinear equations with null boundary conditions. When the boundary conditions are nonzero or nonlinear, fourthorder equations can model beams resting on elastic bearings located in their extremities. See for instance, [10–12] and the references therein. For instance, Cabrera et al. [10] studied the existence of positive solutions for the fourth-order positone problem


Introduction
Consider the singular superlinear fourth-order nonlinear boundary value problem u ⁗ (x) � λh(x)f(u(x)), x ∈ (0, 1), e study of semipositone problems was formally introduced by Castro and Shivaji [1]. From an application viewpoint, one is usually interested in the existence of positive solutions for semipositone problems. Significant processes on second-order semipositone problems have taken place in the last 10 years, see [1][2][3][4][5] and the references therein.
Fourth-order boundary value problems modeling bending equilibria of elastic beams have been considered in several papers [6][7][8][9]. Most of them are concerned with nonlinear equations with null boundary conditions. When the boundary conditions are nonzero or nonlinear, fourth-order equations can model beams resting on elastic bearings located in their extremities. See for instance, [10][11][12] and the references therein.
For instance, Cabrera et al. [10] studied the existence of positive solutions for the fourth-order positone problem x, y) is increasing in x and decreasing in y, for fixed t ∈ [0, + ∞]. For convenience, we denote c(s) � c(s)s in this article. In [10][11][12], the authors studied positive solutions of fourth-order nonlinear boundary value problems in the positone case based on a mixed monotone operator method and a well-known fixed-point theorem in cones.
It should be noted that nonlinear part f is either bounded or positive states in [1,[10][11][12]. In this paper, we prove the existence of positive solution to (1) by assuming that f: (0, ∞) ⟶ R is continuous and is allowed to be singular at 0; in other words, f may be unbounded from below and satisfies the superlinear condition. Moreover, we prove a useful lemma (Lemma 3) in this paper which plays a key role to guarantee the positivity of solution. It can be obtained by concavity and convexity of solution or calculation for the second-order boundary value problem, but for the fourthorder boundary value problem, this becomes complicated.
In addition, we will replace h(x)f(u(x)) by m(x) and c by c(s), and we perform a study of the sign of Green's function of the corresponding linear problems: In detail, we construct Green's function G(x, s) by disconjugacy theory and give a sufficient condition to make ensure that G(x, s) is either positive or negative. is fact is crucial for our arguments.
Motivated by the above facts, in this paper, we shall obtain the existence of positive solutions for a singular superlinear fourth-order equation with nonlinear boundary conditions via the fixed-point result of Krasnoselskii type in a Banach space.
Our main result is as follows.
Theorem 1. Let (H1)-(H4) hold. en, there exists a constant λ 0 > 0 such that (1) has a positive solution u λ for λ < λ 0 with u λ ⟶ ∞ as λ ⟶ 0 + uniformly on compact subsets of (0, 1). e paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, we prove the main result that the existence of a large positive solution to (1) for λ small.

Notation and Preliminaries
Suppose that E is a real Banach space which is partially ordered by a cone P ⊂ E, i.e., x ≤ y if and only if y − x ∈ P. If x ≤ y and x ≠ y, then we denote x < y or y > x. By θ, we denote the zero element of E. Recall that a nonempty closed convex We first recall the following fixed-point result of Krasnoselskii type in a Banach space (see e.g., [1] and the Banach space X � L 1 (0, 1) equipped with the norm ‖ϕ‖ X � 1 0 |ϕ(t)|dt.

Proof of the Main Results
Proof of eorem 1.
c v � c(|v(1)|),and q(x) is defined in Lemma 2. en, where Green's function G v (x, s) is given by Note that G v (x, s) ≤ 1 for all x, s. By (H3), there exists a constant M > 0 depending on ‖v‖ ∞ such that It follows from the Lebesgue dominated convergence theorem that u ∈ B i.e. L λ : B ⟶ B. (31) Define the cone P in B by ‖u‖ ∞ q(x), x ∈ (τ, 1) .
We next show that L λ : P ⟶ P is a completely continuous operator. Now, we show L λ : P ⟶ P is continuous. To this end, let (v n ) ∈ C[0, 1] be such that v n ⟶ v in P and let Fix x, s ∈ (0, 1), and define en, H ′ (z) � (1/(6 − 2z) 2 )x 2 s 2 (x − 3)(s − 3), and there exists a constant N such that |H′(z)| ≤ N for 0 ≤ z ≤ 2, and the mean value theorem gives Notice that