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Positive solutions of a certain type of two-point boundary value problems. (English) Zbl 0753.34011
Summary: The paper gives sufficient and necessary conditions for the existence of positive solutions for two point boundary value problems of the type \(- u''=(f_ a(x)+g(u))\cdot u-s(u)\cdot v\), \(-v''=(a+r(u))\cdot v-v^ 2\), \(u(0)=u(\pi)=v(0)=v(\pi)=0\), which depend on a parameter \(a\in\mathbb{R}\).
MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:
[1] BLAT J., BROWN K. J.: Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal., 17, 1986, 1339-1353. · Zbl 0613.35008
[2] SMOLLER J.: Shock Waves and Reaction-Diffusion Equations. A Series of Comprehensive Studies in Mathematics, 258. Springer-Verlag, New York, Berlin 1983. · Zbl 0508.35002
[3] PROTTER M. R., WEINBERGER H. F.: Maximum Principle in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ 1967. · Zbl 0153.13602
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