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Existence theorems for semilinear equations in cones. (English) Zbl 1015.47042

This article is a continuation of the author’s previous article [C. T. Cremins, Nonlinear Anal., Theory Methods Appl. 46 A, 789–806 (2001; Zbl 1015.47041)]. It deals with positive and non-negative solutions for operator equations of type \(Lx=Nx\) with Fredholm linear operator \(L\) of zero index in a Banach space \(X\) ordered by a cone \(K\). Based on the fixed-point index theory introduced by the author and the notion of a quasi-normal cone, the author presents a series of solvability results for semilinear equations with \(P_\gamma\)-compact maps. These results are close to some results by K. Deimling, R. E. Gaines and J. Santanilla, Guo Dajun B. Lafferriere and W. V. Petryshyn, J. Mawhin, J. Mawhin and K. P. Rybakovski, J. J. Nieto, W. V. Petryshyn, J. Santanilla, J. R. L. Webbs. As an application, the Picard boundary value problem \[ -x''(t)= f\bigl(t,x(t), x'(t),x''(t) \bigr),\quad x(0)= x(1)= 0 \] in the cone \(K=\{x\in X=C^2 [0,1]: -x''(t)\geq 0\), \(x(0)= x(1)=0\}\) is considered.

MSC:

47H11 Degree theory for nonlinear operators
47J05 Equations involving nonlinear operators (general)
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47J25 Iterative procedures involving nonlinear operators
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 1015.47041
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References:

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