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Multiple positive solutions of some nonlinear heat flow problems. (English) Zbl 1161.34007
The author studies the existence and multiple positive solutions for the weakly singular nonlocal boundary value problem (BVP)
\[ \begin{gathered} u''(t)+g(t)f(t,u(t))=0,\quad t \in (0,1),\\ u'(0)=0,\;\beta u'(1)+u(\eta)=0.\end{gathered} \] This problem has been studied previously by G. Infante and J. R. L. Webb [NoDEA, Nonlinear Differ. Equ. Appl. 13, No. 2, 249–261 (2006; Zbl 1112.34017)], who were motivated by some earlier work of P. Guidotti and S. Merino [Differ. Integral Equ. 13, No. 10–12, 1551–1568 (2000; Zbl 0983.35013)], and arises in the study of the steady states of a heated bar of length one, where a controller at \(t=1\) adds or removes heat according to the temperature detected by a sensor at \(t=\eta\).
The author studies this BVP by seeking fixed points of the Hammerstein integral operator
\[ Tu(t):=\int^1_0 k(t,s)g(s)f(s,u(s))ds. \]
A nice feature of this paper is that the author, utilizing previous results of J. R. L. Webb and K. Q. Lan [Topol. Methods Nonlinear Anal. 27, No. 1, 91–115 (2006; Zbl 1146.34020)], proves the existence of positive solutions also by studying the relationship between the behaviour of the nonlinearity \(f\) near \(0\) and \(\infty\) and the principal characteristic value \(\mu_1\) of the compact linear integral operator
\[ Lu(t):=\int^1_0 k(t,s)g(s)u(s)ds. \]
The author provides a careful analysis of constants that occur in his theory, obtaining optimal values of these constants, and improves the results of [NoDEA, Nonlinear Differ. Equ. Appl. 13, No. 2, 249–261 (2006; Zbl 1112.34017)].

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)