Corollary to the five functionals fixed point theorem (Q2783617)

This article deals with the following modification of the five functionals fixed point theorem: Let \(P\) be a cone in a real Banach space \(E, \alpha,\psi\) nonnegative continuous concave and \(\gamma,\beta,\theta\) nonnegative continuous convex functionals on \(P\) with \(\alpha(x) \leq\beta(x)\) and \(\|x\|\leq M\gamma(x)\) for all \(x\in\overline{\{x\in P:\gamma (x)<c \}}\), and a completely continuous operator \(A:\overline {\{x\in P:\gamma (x)< c\}}\to \overline {\{x\in P:\gamma (x)<c\}}\) satisfy, for some \(h,a,k,b\), \(0<a <b\), the following conditions:NEWLINENEWLINENEWLINE(i) \(\{x\in\{x\in P:b< \alpha(x)\), \(\theta(x)\leq k\), \(\gamma (x)\leq c\}\neq\emptyset\) and \(\alpha(Ax) >b\) for \(x\in\{x\in P:b\leq \alpha (x)\), \(\theta(x) \leq k\), \(\gamma(x)\leq c\}\);NEWLINENEWLINENEWLINE(ii) \(\{c\in P:h\leq \psi (x)\), \(\beta(x) <a\), \(\gamma(x)\leq c\}\neq \theta\) and \(\beta(Ax) <a\) for \(x\in \{x\in P:h\leq \psi(x)\), \(\beta(x)\leq a\), \(\gamma(x)\leq c\}\);NEWLINENEWLINENEWLINE(iii) \(\alpha (Ax) <b\) for \(x\in\{x\in P:b\leq \alpha(x)\), \(\gamma(x)\leq c\}\) with \(\theta (Ax) >k\); (iv) \(\beta (Ax)<a\) for \(x\in \{\beta (x)\leq a,\gamma (x)\leq c\}\) with \(\psi (Ax) <h\).NEWLINENEWLINENEWLINEThen \(A\) has at least three fixed points \(x_1, x_2, x_3 \in \overline {\{x\in P:\gamma (x)<c\}}\) such that \(\beta(x_1)<a\), \(b< \alpha (x_2) \), and \(a<\beta (x_3)\), \(\alpha(x_3) <b\).NEWLINENEWLINENEWLINEThis modification covers some of M. A. Krasnosel'skij's results for compression-expansion operators, the Leggett Williams multiple fixed point theorem, and Avery's five functionals fixed point theorem, and allows the authors to analyze the second-order boundary value problem NEWLINE\[NEWLINEy''+f(y)=0,\quad y(0)=y (1)=0,NEWLINE\]NEWLINE where \(f: \mathbb{R} \to [0,\infty)\) is a continuous function, satisfying the weakened growth conditions.