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 Electronic Journal of Differential Equations , 2003, Abstract: We present an existence result for a nonlinear singular differential equation of fourth order with Navier boundary conditions. Under appropriate conditions on the nonlinearity $f(t,x,y)$, we prove that the problem $$displaylines{ L^{2}u=L(Lu) =f(.,u,Lu)quad hbox{a.e. in }(0,1), cr u'(0) =0,quad (Lu) '(0)=0,quad u(1) =0,quad Lu(1) =0. }$$ has a positive solution behaving like $(1-t)$ on $[0,1]$. Here $L$ is a differential operator of second order, $Lu=frac{1}{A}(Au')'$. For $f(t,x,y)=f(t,x)$, we prove a uniqueness result. Our approach is based on estimates for Green functions and on Schauder's fixed point theorem.
 Physics , 2010, DOI: 10.1007/JHEP02(2011)033 Abstract: A fourth chiral generation, with $m_{t^\prime}$ in the range $\sim (300 - 500)$ GeV and a moderate value of the CP-violating phase can explain the anomalous like-sign dimuon charge asymmetry observed recently by the D0 collaboration. The required parameters are found to be consistent with constraints from other $B$ and $K$ decays. The presence of such quarks, apart from being detectable in the early stages of the LHC, would also have important consequences in the electroweak symmetry breaking sector.
 Yang Liu Electronic Journal of Differential Equations , 2008, Abstract: In this paper, we use the lower and upper solution method to obtain an existence theorem for the fourth-order boundary-value problem $$displaylines{ u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),quad 0 Keywords Fourth-order boundary-value problem --- upper and lower solution --- Riemann-Stieltjies integral --- Nagumo-type condition  Electronic Journal of Differential Equations , 2001, Abstract: We prove the existence of a nonzero solution for the fourth order elliptic equation$$Delta^2u= mu u +a(x)g(u) with boundary conditions $u=Delta u=0$. Here, $mu$ is a real parameter, $g$ is superlinear both at zero and infinity and $a(x)$ changes sign in $Omega$. The proof uses a variational argument based on the argument by Bahri-Lions cite{BL}.