Multiple positive solutions of some nonlinear heat flow problems.

*(English)*Zbl 1161.34007The 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)].

\[ \begin{gathered} u''(t)+g(t)f(t,u(t))=0,\quad t \in (0,1),\\ u'(0)=0,\;\beta u'(1)+u(\eta)=0.\end{gathered} \] 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)].

Reviewer: Gennaro Infante (Arcavata di Rende)

##### MSC:

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, Nemytskiĭ, Uryson, etc.) |