We consider a strongly nonlinear differential equation of the following general type:$$\begin{aligned} (\Phi (a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text {a.e. on }[0,T], \end{aligned}$$where f is a Carathédory function, \(\Phi \) is a strictly increasing homeomorphism (the \(\Phi \)-Laplacian operator), and the function a is continuous and non-negative. We assume that a(t, x) is bounded from below by a non-negative function h(t), independent of x and such that \(1/h \in L^p(0,T)\) for some \(p> 1\), and we require a weak growth condition of Wintner–Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solution method.

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与奇异强非线性方程有关的边值问题的存在性结果

我们考虑以下一般类型的强非线性微分方程：$$ \ begin {aligned}（\ Phi（a（t，x（t））\，x'（t）））'= f（t，x（ t），x'（t）），\ quad \ text {ae on} [0，T]，\ end {aligned} $$其中f是Carathédory函数，\（\ Phi \）是严格增加的同胚性（的\（\披\） -Laplacian运营商），并且函数一个是连续的并且是非负的。我们假设一个（吨，  X）由下面的非负函数界定ħ（吨），独立的X和使得\（以L的1 / h \ ^ P（0，T）\）对于一些\ （p> 1 \），并且我们需要Wintner–Nagumo型的弱增长条件。在这些假设下，我们证明了与上述方程式相关的Dirichlet问题以及不同边界条件的存在结果。我们的方法结合了定点技术和上下解决方法。