Existence of Positive Solutions for Second-Order Third-Point Semipositive BVP

In this paper, we study the existence of positive solutions for the following nonlinear second-order third-point semi-positive BVP. We derive an explicit interval of positive parameters, which for any l , μ in this interval, the existence of positive solutions to the boundary value problem is guaranteed under the condition that a ð t , x Þ , b ð t , x Þ are all superlinear (sublinear), or one is superlinear, the other is sublinear.


Introduction
In the applied mathematical field, three-point BVP can describe many phenomena. Moshinsky [1] introduced the vibrations of a guy wire with a uniform cross-section and composed of N parts of different densities using a multipoint BVP. Timoshenko [2] also revealed that the theory of elastic stability can be used by the method of a three-point BVP. Il'in and Moviseev [3] were the first to study this aspect. Since then, more general nonlinear BVP have been studied by several authors .
Suppose operator T can be completely continuous and satisfies one of the following conditions: Then, operator T has at least one fixed point Theorem 2 [4]. We suppose that P ⊂ E is a cone in E, in which E is a real Banach space, the open bounded set Suppose operator A is completely continuous and satisfies the following conditions: Then, operator T has at least two fixed points x * and x * * in P ∩ ð Ω 3 /Ω 1 Þ, and x * ∈ P ∩ ðΩ 2 /Ω 1 Þ and x * * ∈ P ∩ ð Ω 3 / Ω 2 Þ.

Journal of Function Spaces
Obviously, problem (18) is equivalent to Defining the operator T : Obviously TðPÞ ⊂ P and T is completely continuous.
Finally, by (C2) and Lemma 3, we havẽ Thus, x =x − w is the positive solution of problem (2). ☐ Theorem 9. We suppose that condition (C1), (C2), and (C4) hold, and the following condition also holds: (C6) There exist constant D > 0, ρ > 0, and we have Journal of Function Spaces Then, for the small number λ, μ, problem (2) has at least one positive solution.

Remark 12.
We can obtain the results of Theorem 10 if we replace condition (C5) with (C6)

Data Availability
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