# A four-point boundary value problem with singular φ-Laplacian

## Abstract

We prove that the four-point boundary value problem

$-\left[ \phi \left( u^{\prime}\right) \right] ^{\prime}=f\left(t,u,u^{\prime}\right) ,\ u\left( 0\right) =\alpha u\left( \xi \right),u\left( T\right) =\beta u\left( \eta \right) ,$

where $$f:\left[ 0,T\right] \times \mathbb{R}_{2}\rightarrow \mathbb{R}\$$ is continuous, $$\alpha,\beta \in \left[ 0,1\right] ,0<\xi<\eta<T,$$ and $$\phi:\left( -a,a\right) \rightarrow \mathbb{R}\left( 0<a<\infty \right) \$$ is an increasing homeomorphism, which is always solvable.

When instead of $$f$$ is some $$g:\left[ 0,T\right] \times \lbrack0,\infty)\rightarrow \lbrack0,\infty)$$, 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.

## Authors

Antonia Chinní
Department of Engineering, University of Messina, Messina, Italy

Beatrice Di Bella
Department of Engineering, University of Messina, Messina, Italy

Petru Jebelean
Department of Mathematics, West University of Timişoara, Timisoara, Romania

Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

Singular φ-Laplace equations; four-point boundary value problems; positive solutions; Schauder’s fixed point theorem; Krasnoselskii’s fixed point theorem

## Paper coordinates

A. Chinni, B. Di Bella, P. Jebelean, R. Precup, A four-point boundary value problem with singular φ-Laplacian, J. Fixed Point Theory Appl. (2019) 21:66, pp 16, https://doi.org/10.1007/s11784-019-0703-1

## About this paper

##### Journal

Journal of Fixed Point Theory and Applications

Springer

1661-7738

##### Online ISSN

1661-7746

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