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Generalization of Fredholm alternative for nonlinear differential operators. (English) Zbl 0623.34031

The boundary value problem \[ -(| u'(t)|^{p- 2}u'(t))'=f(t,u(t))+g(t),\quad t\in (0,\pi), \] u(0)\(=u(\pi)=0\), is considered. The authors prove that if \(\liminf_{u\to \pm \infty}f(t,u)/| u|^{p-2}u,\limsup_{u\pm \infty}f(t,u)/| u|^{p-2}u\) lies between the eigenvalues of the eigenvalue problem \(-(| u'|^{p-2}u')'=\lambda | u|^{p-2}u\) on \((0,\pi)\), \(u(0)=u(\pi)=0\), then the considered problem is solvable for any \(g\in L_ 1\). Certain more general assumptions on f, under which the considered boundary problem is solvable, are formulated using properties of the problem \(-(| u'|^{p-2}u')'=\mu | u|^{p- 2}u^+-\nu | u|^{p-2}u^-\) on (0,\(\pi)\), \(u(0)=u(\pi)=0\). The authors prove under similar assumptions on f the solvability of the equations asymptotically close to the considered boundary problem. Certain weaker results are obtained for partial differential equations of the analogous type.
Reviewer: I.Onciulescu

MSC:

34L99 Ordinary differential operators
47E05 General theory of ordinary differential operators
47H99 Nonlinear operators and their properties
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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