### An extension of Leggett-Williams norm-type theorem for coincidences and its application

#### Abstract

In this paper, several versions extension of

Leggett-Williams norm-type theorem for coincidences are given and

proved to obtain the positive solutions of the operator equation

$Mx=Nx$, where $M$ is a quasi-linear operator and $N$ is nonlinear.

Moreover, as an application, the existence of positive solutions for

multi-point boundary value problem with a $p$-Laplacian is obtained

by one of those theorems.

Leggett-Williams norm-type theorem for coincidences are given and

proved to obtain the positive solutions of the operator equation

$Mx=Nx$, where $M$ is a quasi-linear operator and $N$ is nonlinear.

Moreover, as an application, the existence of positive solutions for

multi-point boundary value problem with a $p$-Laplacian is obtained

by one of those theorems.

#### Keywords

Boundary value problem; resonance; cone; positive solution; coincidence

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