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The stability of the resonance set for a problem with jumping nonlinearity. (English) Zbl 0704.34010

Let I be the open unit interval in \({\mathbb{R}}\), let \(Q\in L^ 2(I)\) and denote by \(\Gamma \subset {\mathbb{R}}^ 2\) the set of all pairs (\(\alpha\),\(\beta\)) for which the equation \[ \int^{1}_{0}(u'(x)v'(x)- (u(x)v(x))'Q(x))dx=\int^{1}_{0}(\alpha u^+(x)-\beta u^-(x))v(x)dx \] for all \(v\in H^ 1_ 0(I)\) has a solution \(u\in H^ 1_ 0(I)\) which has exactly one zero in I and is positive near \(x=0\). The author establishes the stability, in a certain sense, of \(\Gamma\) under small perturbations of Q, and illustrates the dependence of \(\Gamma\) on Q by consideration of the case in which \(Q(x)=0\) for \(x\in [0,a]\), \(Q(x)=p\) for all \(x\in (a,1]\).

MSC:

34A34 Nonlinear ordinary differential equations and systems
34D99 Stability theory for ordinary differential equations
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[1] DOI: 10.1007/BF01789470 · Zbl 0475.35046 · doi:10.1007/BF01789470
[2] Gallouët, C.R. Acad. Sci. Paris 291 pp 193– (1980)
[3] DOI: 10.5802/afst.568 · Zbl 0495.35001 · doi:10.5802/afst.568
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