Abstract
Homotopy methods are used to find sufficient conditions for the solvability of nonlinear boundary value problems of the form
where (α, β) = (0, 1), (1, 0), (0, 0) or (1, 1), \({\phi}\) is a homeomorphism from the open ball \({B(a) \subset \mathbb{R}^n}\) onto \({\mathbb{R}^n}\), f is a Carathéodory function, \({g : \mathbb{R}^n \times \, \mathbb{R}^n \rightarrow \mathbb{R}^m}\) is continuous and m ≤ 2n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bereanu C., Mawhin J.: Boundary value problems for some nonlinear systems with singular \({\phi}\) -Laplacian. J. Fixed Point Theory Appl. 4, 57–75 (2008)
H. Freudenthal, Über die Klassen von Sphärenabbildungen I. Compos. Math. 5 (1937), 299–314.
Gaines R. E., Mawhin J.: Ordinary differential equations with nonlinear boundary conditions. J. Differential Equations 26, 200–222 (1977)
K. Gȩba, Algebraic topology methods in the theory of compact fields in Banach spaces. Fund. Math. 54 (1964), 177–209.
K. Gȩba, Sur les groupes de cohomotopie dans les espaces de Banach. Constructions générales. C. R. Acad. Sci. Paris 254 (1962), 3293–3295.
K. Gȩba and A. Granas, Algebraic topology in linear normed spaces. I. Basic categories. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 287–290.
K. Gȩba and A. Granas, Algebraic topology in linear normed spaces. II. The functor π(X, U). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 341–345.
K. Gȩba and A. Granas, Algebraic topology in linear normed spaces. III. The cohomology functors \({\bar{H}^{\infty-n}}\). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 137–143
K. Gȩba and A. Granas, Algebraic topology in linear normed spaces. IV. The Alexander-Pontriagin invariance theorem in E. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 145–152.
K. Gȩba and A. Granas, Algebraic topology in linear normed spaces. V. Cohomology theory on C(E). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 123–130.
K. Gȩba and A. Granas, Infinite dimensional cohomology theories. J. Math. Pures Appl. 52 (1973), 145–270.
A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. Rozprawy Mat. 30 (1962), 1–91.
A. Granas and J. Dugundji, Fixed Point Theory. Springer, New York, 2003.
H. Hopf, Über die Abbildungen der 3-Sphäre auf die Kugenfläche. Math. Ann. 104 (1931), 637–665.
J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations. In: Stability and Bifurcation Theory for Non-Autonomous Differential Equations (CIME, Cetraro 2011), R. Johnson, M.P. Pera (eds.), Lecture Notes in Math. 2065, Springer, Berlin, 2013, 103–184.
J. Mawhin, Stable homotopy and ordinary differential equations with nonlinear boundary conditions. Rocky Mountain J. Math. 7 (1977), 417–424.
L. Nirenberg, An application of generalized degree to a class of nonlinear problems. In: Troisième Colloque CBRM d’Analyse fonctionnelle (Liège 1970), Vander, Louvain, 1971, 57–74.
L. Nirenberg, Topics in Nonlinear Functional Analysis. Courant Institute of Mathematical Sciences, New York University, New York, 1974.
Author information
Authors and Affiliations
Corresponding author
Additional information
Cordially dedicated to Kazimierz Gȩba
Rights and permissions
About this article
Cite this article
Mawhin, J. Homotopy and nonlinear boundary value problems involving singular \({\phi}\)-Laplacians. J. Fixed Point Theory Appl. 13, 25–35 (2013). https://doi.org/10.1007/s11784-013-0112-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-013-0112-9