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Existence of positive solutions of Neumann boundary value problem via a convex functional compression-expansion fixed point theorem. (English) Zbl 1213.34041

In [D. J. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones. Boston, MA: Academic Press (1988; Zbl 0661.47045)], the cone compression and expansion fixed point theorem is proved. This classical theorem provides the existence of solutions on some cones of Banach spaces provided some compact mapping satisfies two bounds one of which is sublinear. In [G. Zhang and J. Sun, Nonlinear Anal., Theory Methods Appl. 67, No. 2, A, 579–586 (2007; Zbl 1127.47050)], the well-known Guo-Krasnosel’skii fixed point theorem of norm type is extended, in replacing the norm by a convex functional. The aim of the paper under review is to apply the latter theorem to prove the existence of positive solutions to the second-order differential equation
\[ -u''+Mu=h(t)f(t,u),\;t\in(0,1) \]
subject to homogeneous Neumann boundary conditions
\[ u'(0)=u'(1)=0. \]
The coefficient \(h\) is allowed to have a time-singularity at \(t=0\) and \(t=1\) but is integrably bounded. The nonlinearity \(f\) obeys some local growths. Since the corresponding linear problem has a unique solution, the nonlinear problem is formulated as a fixed point problem for an integral operator with the Green function. The authors consider the linear operator \(\rho(t)=\int_0^1h(t)u(t)\,dt\) as suitable convex functional. The existence result is illustrated by means of an example with a nonlinearity having a quadratic growth at positive infinity and behaving as \(\sqrt{u}\) near the origin.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
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