Singular Lidstone boundary value problem with given maximal values for solutions

✍ Scribed by Ravi P. Agarwal; Donal O'Regan; Svatoslav Staněk

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 333 KB
• Volume: 55
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2003.06.001

✦ Synopsis

The singular problem (-1) n x (2n) = f(t; x; : : : ; x (2n-2) ), x (2j) (0)=x (2j) (T )=0 (06j6n -1), max{x(t) : 0 6 t 6 T } = A depending on the parameter is considered. Here the positive Carathà eodory function f may be singular at the zero value of all its phase variables. The paper presents conditions which guarantee that for any A ¿ 0 there exists A ¿ 0 such that the above problem with = A has a solution x ∈ AC 2n-1 ([0; T ]) which is positive on (0; T ). The proofs are based on the regularization and sequential techniques and use the Leray-Schauder degree and Vitali's convergence theorem.