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Global continuum of positive solutions for discrete p -Laplacian eigenvalue problems

Dingyong Bai, Yuming Chen (2015)

Applications of Mathematics

We discuss the discrete p -Laplacian eigenvalue problem, Δ ( φ p ( Δ u ( k - 1 ) ) ) + λ a ( k ) g ( u ( k ) ) = 0 , k { 1 , 2 , ... , T } , u ( 0 ) = u ( T + 1 ) = 0 , where T > 1 is a given positive integer and φ p ( x ) : = | x | p - 2 x , p > 1 . First, the existence of an unbounded continuum 𝒞 of positive solutions emanating from ( λ , u ) = ( 0 , 0 ) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum 𝒞 is a monotone continuous curve globally defined for all λ > 0 .

Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections

Yukihiko Nakata, Gergely Röst (2015)

Mathematica Bohemica

We describe the global dynamics of a disease transmission model between two regions which are connected via bidirectional or unidirectional transportation, where infection occurs during the travel as well as within the regions. We define the regional reproduction numbers and the basic reproduction number by constructing a next generation matrix. If the two regions are connected via bidirectional transportation, the basic reproduction number R 0 characterizes the existence of equilibria as well as...

Global existence and stability of some semilinear problems

Mokhtar Kirane, Nasser-eddine Tatar (2000)

Archivum Mathematicum

We prove global existence and stability results for a semilinear parabolic equation, a semilinear functional equation and a semilinear integral equation using an inequality which may be viewed as a nonlinear singular version of the well known Gronwall and Bihari inequalities.

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