# zbMATH — the first resource for mathematics

Nagumo conditions for systems of second-order differential equations. (English) Zbl 0604.34002
The existence of solutions for the boundary value problem $(1)\quad x''(t)+f(t,x(t),x'(t))=0,\quad t\in [a,b],\quad x(a)=A,\quad x(b)=B$ where x takes values in a Hilbert space H and f:[a,b]$$\times H\times H\times H\to H$$ is a compact mapping, is studied. By transforming the problem (1) into a fixed point problem for an operator $$T:{\bar \Omega}_{\phi,\rho}\to C^ 1([a,b],H),$$ where $\Omega_{\phi,\rho}=\{x\in C^ 1([a,b],H);\quad | x(t)| <\phi (t),\quad \| x'\| <\rho \}$ the author obtains an a priori estimate for $$| x'(t)|$$, when x(t) has a known bound. For this aim, the author proves two results in the Nagumo’s conditions on the $$<x,x''>$$, $$<x',x''>.$$
Theorem 1 is a generalization of the results obtained by Hartman, Opial, Schmidt and Thomson. The results established by Theorem 2 are strongly connected to those obtained by Lasota, Yorke, Fabry and Habets. The main result is the proof of the existence of the solutions of problem (1) (Theorem 3). Also the restrictive conditions, but easy to verify, for the Theorem 3, are indicated.
Reviewer: N.Luca

##### MSC:
 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text:
##### References:
  Nagumo, M, Über die differentialgleichung y″ = f(t, y, y′), (), 861-866 · JFM 63.1021.04  Heinz, E, On certain nonlinear elliptic differential equations and univalent mappings, J. anal. math., 5, 197-272, (1956-1957) · Zbl 0085.08701  Hartman, P, Ordinary differential equations, (1964), Wiley-Interscience New York · Zbl 0125.32102  Opial, Z, Sur la limitation des dérivées des solutions bornées d’un système d’équations différentielles du second ordre, Ann. polon. mat., 10, 73-79, (1961) · Zbl 0097.29401  Schmitt, K; Thompson, R, Boundary value problems for infinite systems of second order differential equations, J. differential equations, 18, 277-295, (1975) · Zbl 0302.34081  Fabry, C; Habets, P, The Picard boundary value problem for nonlinear second order vector differential equations, J. differential equations, 42, 186-198, (1981) · Zbl 0439.34018  Mawhin, J, The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations, () · Zbl 0497.34020  Mawhin, J, Two-point boundary value problems for nonlinear second order differential equations in Hilbert spaces, Tôhoku math. J., 32, 225-233, (1980) · Zbl 0436.34057  Lasota, A; Yorke, J.A, Existence of solutions of two-point boundary value problems for nonlinear systems, J. differential equations, 11, 509-518, (1972) · Zbl 0263.34016  Bernfeld, S.R; Lakshmikantham, V, An introduction to nonlinear boundary value problems, () · Zbl 0403.34055  Gaines, R.E; Mawhin, J.L, Coincidence degree and nonlinear differential equations, () · Zbl 0326.34020  Granas, A; Guenther, R.B; Lee, J.W, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky mountain J. math., 10, 35-58, (1980) · Zbl 0476.34017  Granas, A; Guenther, R.B; Lee, J.W, Topological transversality II—applications to the Neumann problem for y″ = f(t, y, y′), Pacific J. math., 104, 95-109, (1983) · Zbl 0534.34006  Bailey, P; Shampine, L.F; Waltman, P, Nonlinear two-point boundary value problems, () · Zbl 0145.11104
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.