Nonlinear boundary value problems in Hilbert spaces. (English) Zbl 0672.34056

The authors consider the nonlinear two point boundary value problem (*) \(y''=f(t,y,y')\), \(0\leq t\leq 1\), with Sturm-Liouville type boundary conditions. A solution of (*) is a twice continuously differentiable function which takes values in a real Hilbert space H. The nonlinear term f ([0,1]\(\times H\times H\to H)\) is always assumed to be continuous. Subject to various restrictions on f, including a Bernstein-Nagumo type growth condition, the authors prove certain existence theorems for a wide class of problems.
Reviewer: K.S.Miller


34G20 Nonlinear differential equations in abstract spaces
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Banas, J.; Goebel, K., ()
[2] Barbu, V., ()
[3] Bernfeld, S.; Laksmikantham, V., Monotone methods for nonlinear boundary value problems in Banach spaces, Nonlinear anal. TMA, 3, 303-316, (1979) · Zbl 0423.34087
[4] Deimling, K., ()
[5] Dugundji, J.; Granas, A., Fixed point theory, () · Zbl 1025.47002
[6] Granas, A.; Guenther, R.B.; Lee, J.W., Nonlinear boundary value problems for ordinary differential equations, Dissertationes mathematcae, (1985), Warszawa · Zbl 0476.34017
[7] Granas, A.; Guenther, R.B.; Lee, J.W., Topological transversality. II. applications to the Neumann problem for \(y″ = ƒ(t, y, y′)\), Pacific J. math., 104, 53-67, (1983) · Zbl 0534.34006
[8] Guenther, R.B., ()
[9] Lakshmikanthan, V.; Leela, S., ()
[10] Lasota, A.; Yorke, James A., The generic property of existence of solutions of differential equations in Banach spaces, J. differential equations, 13, 1-12, (1973) · Zbl 0259.34070
[11] Martin, R.H., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.