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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.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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References:

[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
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