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A four-point boundary value problem with singular \(\phi \)-Laplacian

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We prove that the four-point boundary value problem

$$\begin{aligned} -\left[ \phi (u') \right] ^{\prime }=f(t,u, u'), \quad u(0)=\alpha u(\xi ), \quad u(T)=\beta u(\eta ), \end{aligned}$$

where \(f:[0,T] \times \mathbb {R}^2 \rightarrow \mathbb {R}\) is continuous, \(\alpha , \; \beta \in [0,1)\), \(0<\xi< \eta <T \), and \(\phi :(-a,a) \rightarrow \mathbb {R}\) (\(0<a<\infty \)) is an increasing homeomorphism, which is always solvable. When instead of f is some \(g:[0,T] \times [0, \infty ) \rightarrow [0, \infty )\), we obtain existence, localization, and multiplicity of positive solutions. Our approach relies on Schauder and Krasnoselskii’s fixed point theorems, combined with a Harnack-type inequality.

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The authors express their gratitude to the anonymous referee for the careful reading of manuscript.

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Correspondence to Antonia Chinní.

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Antonia Chinní, Beatrice Di Bella and Petru Jebelean have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INDAM).

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Chinní, A., Di Bella, B., Jebelean, P. et al. A four-point boundary value problem with singular \(\phi \)-Laplacian. J. Fixed Point Theory Appl. 21, 66 (2019).

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