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Solvability of singular Dirichlet boundary-value problems with given maximal values for positive solutions. (English) Zbl 1066.34017
Summary: The singular boundary value problem \((g(x'(t)))' =\mu f(t,x(t),x'(t))\), \(x(0)= x(T)= 0\) and \(\max\{x(t): 0\leq t\leq T\}= A\) is considered. Here, \(\mu\) is the parameter and the negative function \(f(t,u,v)\) satisfying local Carathéodory conditions on \([0, T]\times(0, \infty)\times(\mathbb{R}\setminus\{0\})\) may be singular at the values \(u= 0\) and \(v= 0\) of the phase variables \(u\) and \(v\).
The paper presents conditions which guarantee that for any \(A> 0\) there exists \(\mu_A> 0\) such that the above problem with \(\mu= \mu_A\) has a positive solution on \((0, T)\). The proofs are based on the regularization and sequential techniques and use the Leray-Schauder degree and Vitali’s convergence theorem.

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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