Fixed point theorem of cone expansion and compression of functional type. (English) Zbl 1013.47019

Anderson, Douglas R.; Avery, Richard I.

J. Difference Equ. Appl. 8, No. 11, 1073-1083 (2002).

This article deals with a modification of the classical Krasnosel’skij fixed point theorem about compressions and expansions of a cone \(P\). The authors formulate conditions of compression and expansion in terms of two nonnegative continuous functionals \(\alpha,\gamma : P\to [0,\infty)\) and the sets \[ P(\gamma,R) = \{x \in P : \gamma(x) < R\},\quad P(\gamma,\alpha,r,R) = \{x\in P: r < \alpha(x)\text{ and }\gamma(x) < r\} \] under the assumptions that \(\overline{P(\alpha,R)}\subset P(\gamma,R)\) and \(\inf_{x\in\partial P(\gamma,\alpha,r,R)}\|Ax\|> 0\), the operator \(A: \overline{P(\gamma,\alpha, r, R)}\to P\) has at least one positive fixed point \(x^*\) such that \(r\leq \alpha(x_*)\) and \(\gamma(x^*)\leq R\), if one of the two conditions is satisfied:
(H1) \(\alpha(Ax)\leq r\) for \(x\in \partial P(\alpha, r)\), \(\gamma(Ax)\geq R\) for \(x\in \partial P(\gamma, R)\), and, in addition, \(\alpha(\lambda y)\geq \lambda\alpha(y)\), \(\gamma(\mu z)\leq \mu\gamma(z)\) \((y\in \partial P(\alpha,r)\), \(z\in \partial P(\gamma,R)\), \(\lambda\geq 1\), \(\mu\in (0,1])\), \(\alpha(0)=0\);
(H2) \(\alpha(Ax)\geq r\) for \(x\in \partial P(\alpha,r)\), \(\gamma(Ax)\leq R\) for \(x\in\partial P(\gamma,R)\), and, in addition, \(\alpha(\lambda y)\leq \lambda\alpha(y)\), \(\gamma(\mu z)\geq \mu\gamma(z)\) \((y\in \partial P(\alpha,r)\), \(z\in \partial P(\gamma,R)\), \(\lambda\in (0,1]\), \(\mu\geq 1)\), \(\gamma(0)=0\).
As applications, the authors consider the existence problem of a positive solution to the following discrete second-order conjugate boundary value problem: \[ \Delta^2(t- 1)+f(x(t)) = 0\text{ for all }t \in [a+ 1, b+ 1],\quad x(a) =x(b+2) =0. \]

Reviewer: Peter Zabreiko (Minsk)

MSC:

47H10	Fixed-point theorems
39A10	Additive difference equations
47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

Keywords:

existence of a positive solution; discrete boundary value problem; Krasnosel’skij fixed point theorem; compressions and expansions of a cone

References:

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