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Fixed point theorem of cone expansion and compression of functional type. (English) Zbl 1013.47019
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: \[ \Delta^2(t- 1)+f(x(t)) = 0\text{ for all }t \in [a+ 1, b+ 1],\quad x(a) =x(b+2) =0. \]

47H10 Fixed-point theorems
39A10 Additive difference equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Full Text: DOI
[1] DOI: 10.1016/S0377-0427(97)00201-X · Zbl 1001.39021 · doi:10.1016/S0377-0427(97)00201-X
[2] Avery, R. I. ”Multiple positive solutions to boundary value problems””. University of Nebraska Lincoln. Dissertation · Zbl 0960.34503
[3] Avery R. I., Pan Am. Math. J. 8 pp 1– (1998)
[4] Deimling K., Nonlinear Functional Analysis (1985) · Zbl 0559.47040
[5] Guo D., Nonlinear Problems in Abstract Cones (1988) · Zbl 0661.47045
[6] Kelley W. G., Difference Equations: An Introduction with Applications, 2e (2001) · Zbl 0970.39001
[7] Zeidler E., Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems (1986) · Zbl 0583.47050
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