EXISTENCE OF EXTREMAL SOLUTIONS FOR SOME FOURTH ORDER FUNCTIONAL BVPS

TitleEXISTENCE OF EXTREMAL SOLUTIONS FOR SOME FOURTH ORDER FUNCTIONAL BVPS
Publication TypeJournal Article
Year of Publication2011
AuthorsMINHÓS, FELIZ
Volume15
Issue4
Start Page547
Pagination10
Date Published2011
ISSN1083-2564
AMS34B10, 34B15, 34K10, 34K12
Abstract

In this work we present sufficient conditions for the existence of extremal solutions for the fourth order functional problem composed by the equation 

(${\phi}$(u′′′(x) ))′= f (x, u′′(x), u′′′(x), u,u′,u′′ ) ,

for a.a.x ∈ [0, 1] , where ${\phi}$ is an increasing homeomorphism, ${\text { I }}$:= [0, 1], and ${\ f:I }$ ${\times }$ ${\mathbb{R}^2}$ ${\times }$ ${\ ( \ C\ ( I\ ) )^3 \to \mathbb{R} }$ is a L1 -Carathéodory function, and the boundary conditions

0 = L( u(a), u,u′,u′′ )

0 = L2 ( u′(a), u,u′,u′′ )

0 = L3 ( u′′(a), u′′(b), u′′′(a), u′′′(b), u,u′,u′′ )

0 = L4 ( u′′(a), u′′(b) )

where Li , i = 1, 2, 3, 4, are suitable functions with L1 and L2 not necessarily continuous, satisfying some monotonicity assumptions. The arguments make use of lower and upper solutions technique, a version of Bolzano’s theorem and existence of extremal fixed points for a suitable mapping.

URLhttp://www.acadsol.eu/en/articles/15/4/10.pdf
Refereed DesignationRefereed
Full Text

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