Abstract We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type ( Φ ( k ( t ) x ′ ( t ) ) ) ′ + f ( t , G x ( t ) ) ρ ( t , x ′ ( t ) ) = 0 , $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}}_x(t))\,\rho(t, x'(t)) = 0,$$ on a compact interval [a, b]. These equations are quite general due to the presence of a strictly increasing homeomorphism Φ, the so-called Φ-Laplace operator, of a non-negative function k, which may vanish on a set of null measure, and moreover of a functional term Gx. We look for solutions, in a suitable weak sense, which belong to the Sobolev space W1,1([a, b]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.

中文翻译：

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与具有函数项的奇异强非线性方程相关的边值问题

摘要 我们研究与具有 ( Φ ( k ( t ) x ′ ( t ) ) ) ′ + f ( t , G x ( t ) ) ρ ( t , x ) 类型函数项的奇异、强非线性微分方程相关的边值问题′ ( t ) ) = 0 , $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}} _x(t))\,\rho(t, x'(t)) = 0,$$ 在紧凑区间 [a, b] 上。由于非负函数 k 的严格递增同胚 Φ 的存在，即所谓的 Φ-拉普拉斯算子，这些方程是非常通用的，它可能在一组空测度上消失，而且还有一个函数项 Gx . 我们在适当的弱意义上寻找属于 Sobolev 空间 W1,1([a, b]) 的解。在存在一对有序的上下解和合适的 Nagumo 型生长条件的假设下，