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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: $\Delta^2(t- 1)+f(x(t)) = 0\text{ for all }t \in [a+ 1, b+ 1],\quad x(a) =x(b+2) =0.$

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