Solvability of singular dirichlet boundary-value problems with given maximal values for positive solutions
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2005-02-01Author
Agarwal, Ravi P.
O’Regan, Donal
Staněk, Svatoslav
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Agarwal, Ravi P. O’Regan, Donal; Staněk, Svatoslav (2005). Solvability of singular dirichlet boundary-value problems with given maximal values for positive solutions. Proceedings of the Edinburgh Mathematical Society 48 , 1-19
Abstract
The singular boundary-value problem (g(x'(t)))' = muf (t, x(t), x'(t)), x(0) = x(T) = 0 and max {x(t) : 0 less than or equal to t less than or equal to T} = A is considered. Here p is the parameter and the negative function f (t, u, v) satisfying local Caratheodory conditions on [0, T] x (0, infinity) x (R \ {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 muA > 0 such that the above problem with mu = muA 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.