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Higher order two-point boundary value problems with asymmetric growth

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  • In this work it is studied the higher order nonlinear equation

    $\u^{( n)} (x)=f(x,u(x),u^{'}(x),\ldots ,u^{( n-1)} (x)) $

    with $n\in \mathbb{N}$ such that $n\geq 2,$ $f:[ a,b] \times \mathbb{R}^{n}\rightarrow \mathbb{R}$ a continuous function, and the two-point boundary conditions

    $u^{(i)}(a) =A_{i},\text{ \ \ }A_{i}\in \mathbb{R},\text{ \ }i=0,\ldots ,n-3$,
    $u^{( n-1) }(a) =u^{( n-1) }(b)=0.$

    From one-sided Nagumo-type condition, allowing that $f$ can be unbounded, it is obtained an existence and location result, that is, besides the existence, given by Leray-Schauder topological degree, some bounds on the solution and its derivatives till order $(n-2)$ are given by well ordered lower and upper solutions.
        An application to a continuous model of human-spine, via beam theory, will be presented.

    Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18,34L30, 34B60.


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