Remarks on the existence of solutions of \(x''+x+\arctan(x')=p(t)\); \(x(0)=x(\pi)=0\). (English) Zbl 0802.34021

Of concern is the boundary value problem listed in the title of the paper, where \(p\) is a continuous given function. The authors use the contraction mapping principle to prove an existence and uniqueness result when \(\| p\|_ \infty\) is sufficiently small. Other conditions and approaches for existence are also discussed.


34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Kannan, R.; Ortega, R., Existence of solutions of x″ + x + g(x) = p(t), x(0) = 0 = x(π), Proc. am. math. soc., 96, 67-70, (1986) · Zbl 0585.34001
[2] Cesari, L., Functional analysis, nonlinear differential equations and the alternative method, (), 1-197
[3] Nagle, K., Nonlinear boundary value problems for ordinary differential equations with a small parameter, SIAM J. math. analysis, 9, 719-729, (1978) · Zbl 0389.34017
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.