# zbMATH — the first resource for mathematics

Green’s functions for periodic and anti-periodic BVPs to second-order ODEs. (English) Zbl 0795.34013
The paper deals with the second order BVP $$x''+kx= f(t,x,x')$$, $$f\in C([0,T] \times\mathbb{R}^ 2)$$, $$x(0)+ px(T)=0$$, $$x'(0)+ qx'(T)=0$$, where $$p,q\in \{-1,1\}$$, $$k\in\mathbb{R}$$. Using the explicit form of the Green function and the Schauder fixed point theorem, the authors prove the existence of at least one solution of the above BVP.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text:
##### References:
 [1] Hamedani G.G., Mehri B.: Periodic boundary value problem for certain non-linear second order differential equation. Stud. Sci. Math. Hung. 9 (1974), 307-312. · Zbl 0333.34009 [2] Haraux A.: Anti-periodic solutions of some nonlinear evolution equations. Manuscripta Math. 63 (1989), 479-505. · Zbl 0684.35010 [3] Aizicovici S., Pavel N.H.: Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space. J. Funct. Analysis 99 (1991), 387-408. · Zbl 0743.34067 [4] Aftabizadeh A.R., Aizicovici S., Pavel N.H.: Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces. Nonlin. Anal., T.M.A. 18, 3 (1992), 253-267. · Zbl 0779.34054 [5] Erbe L., Palamides P.: Boundary value problems for second order differential systems. J. Math. Anal. Appl. 127 (1987), 80-92. · Zbl 0635.34017 [6] Palamides P.K., Erbe L.H.: Semi-periodic boundary value problems. Diff. Eqns (C. M. Daferemos et al, LNPAM/118, Dekker, Inc., New York, 1989. · Zbl 0711.34035 [7] Erbe L.H., Lin X., Wu J.: Solvability of boundary value problems for vector differential systems. To appear in Proc. Royal-Soc. Edinbourgh. [8] Gaines R.E., Mawhin J.: Ordinary differential equations with nonlinear boundary conditions. J. Diff. Eqns 26, 2 (1977) 200-222. · Zbl 0326.34021 [9] Půža B.: On one class of solvable boundary value problems for ordinary differential equations of n-th order. CMUC 30, 3 (1989), 565-577. · Zbl 0686.34017 [10] Roach G.F.: Green’s functions. Cambridge Univ. Press, Cambridge (1982) · Zbl 0522.65075 [11] Collatz L.: Funkcionální analýza a numerická matematika. SNTL, Praha 1970. [12] Andres J., Vlček V.: On four-point regular BVPs for second-order quasilinear ODEs. Acta UPO, Fac. Rer. Nat., Math. XXXI, Vol. 105 (1992), 37-44. · Zbl 0769.34019 [13] Bihari I.: Notes on a nonlinear integral equation. Stud. Sci. Math. Hung. 2 (1967), 1-6. · Zbl 0147.10302
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.