×

A generalization of the cone expansion and compression fixed point theorem and applications. (English) Zbl 1127.47050

The well-known Guo-Krasnosel’skij fixed point theorem of cone compression and expansion of norm type is extended, in replacing the norm by a convex functional, as follows. Theorem. Let \(E\) be a Banach space and \(P\subset E\) be a closed convex cone. Assume that \(\Omega_1\) and \(\Omega_2\) are bounded open subsets with \(0\in \Omega_1\), \(\overline{\Omega}_1\subset \Omega_2\). Let \(A:P\cap(\overline{\Omega}_2\setminus \Omega_1)\to P\) be a completely continuous mapping and \(\rho:P\to [0,\infty)\) be a uniformly continuous convex functional with \(\rho(0)=0\) and \(\rho(x)>0\) for \(x\neq 0\). If either condition of the following is satisfied: (i) \(\rho(Au)\leq \rho(u)\) for \(u\in P\cap\partial \Omega_1\), \(\rho(Au)\geq \rho(u)\) for \(u\in P\cap\partial \Omega_2\) and \(\inf_{u\in P\cap\partial \Omega_2}\rho(x)>0\); (ii) \(\rho(Au)\geq \rho(u)\) for \(u\in P\cap\partial \Omega_1\) and \(\inf_{u\in P\cap\partial \Omega_1}\rho(x)>0\), \(\rho(Au)\leq \rho(u)\) for \(u\in P\cap\partial \Omega_2\), then \(A\) has a fixed point in \(P\cap(\overline{\Omega}_2\setminus \Omega_1)\). An application to the existence of positive solutions for second-order multipoint boundary value problems is given.

MSC:

47H10 Fixed-point theorems
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040
[2] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045
[3] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Semipositone higher-order differential equations, Appl. Math. Lett., 17, 201-207 (2004) · Zbl 1072.34020
[4] Eloe, P.; Henderson, J., Positive solutions for \((n - 1, 1)\) conjugate boundary value problems, Nonlinear Anal. TMA, 28, 1669-1680 (1997) · Zbl 0871.34015
[5] Erbe, L., Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling, 32, 529-539 (2000) · Zbl 0970.34019
[6] Erbe, L. H.; Mathsen, R. M., Positive solutions for singular nonlinear boundary value problems, Nonlinear Anal. TMA, 46, 979-986 (2001) · Zbl 1007.34020
[7] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208, 252-259 (1997) · Zbl 0876.34023
[8] Kaufmann, E. R.; Kosmatov, N., A multiplicity result for a boundary value problem with infinitely many singularities, J. Math. Anal. Appl., 269, 444-453 (2002) · Zbl 1011.34012
[9] Kong, L.; Wang, J., The Green’s function for \((k, n - k)\) conjugate boundary value problems and its applications, J. Math. Anal. Appl., 255, 404-422 (2001) · Zbl 0991.34023
[10] Li, Y., Positive solutions of second-order boundary value problems with sign-changing nonlinear terms, J. Math. Anal. Appl., 255, 404-422 (2001)
[11] Ma, R.; Wang, H., Positive solutions of nonlinear three-point boundary-value problems, J. Math. Anal. Appl., 282, 232-240 (2003)
[12] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019
[13] Anderson, D. R.; Avery, R. I., Fixed point theorem of cone expansion and compression of functional type, J. Difference Equations Appl., 8, 1073-1083 (2002) · Zbl 1013.47019
[14] Guo, Y.; Ge, W., Positive solutions for three-point boundary value problems with dependence on the first order derivative, J. Math. Anal. Appl., 290, 291-301 (2004) · Zbl 1054.34025
[15] Dugundji, J., An extension of Tietze theorem, Pacific J. Math., 1, 353-367 (1951) · Zbl 0043.38105
[16] Zhang, G.; Sun, J., Positive solutions of \(m\)-point boundary value problems, J. Math. Anal. Appl., 291, 406-418 (2004) · Zbl 1069.34037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.