Abstract
We study a first order differential system subject to a nonlocal condition. Our goal in this paper is to establish conditions sufficient for the existence of positive solutions when the considered problem is at resonance. The key tool in our approach is Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. We conclude the paper with several examples illustrating the main result.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
D.R. Anderson, Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions. J. Math. Anal. Appl. 408, 318–323 (2013)
O. Bolojan, R. Precup, Implicit first order differential systems with nonlocal conditions. Electron. J. Qual. Theor. Differ. Equ. 2014(69), 1–13 (2014)
O. Bolojan-Nica, G. Infante, R. Precup, Existence results for systems with coupled nonlocal initial conditions. Nonlinear Anal. 94, 231–242 (2014)
A. Boucherif, First-order differential inclusions with nonlocal initial conditions. Appl. Math. Lett. 15, 409–414 (2002)
C.T. Cremins, A fixed point index and existence theorems for semilinear equations in cones. Nonlinear Anal. 46, 789–806 (2001)
D. Franco, G. Infante, M. Zima, Second order nonlocal boundary value problems at resonance. Math. Nachr. 284, 875–884 (2011)
D. Franco, J.J. Nieto, D. O’Regan, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions. Appl. Math. Comput. 153, 793–802 (2004)
R.E. Gaines, J. Santanilla, A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations. Rocky Mt. J. Math. 12, 669–678 (1982)
J.R. Graef, S. Padhi, S. Pati, Periodic solutions of some models with strong Allee effects. Nonlinear Anal. Real World Appl. 13, 569–581 (2012)
G. Infante, M. Zima, Positive solutions of multi-point boundary value probelms at resonance. Nonlinear Anal. 69, 2458–2465 (2008)
T. Jankowski, Boundary value problems for first order differential equations of mixed type. Nonlinear Anal. 64, 1984–1997 (2006)
W. Jiang, C. Yang, The existence of positive solutions for multi-point boundary value problem at resonance on the half-line. Bound. Value Probl. 2016, 13 (2016)
N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 68, 2158–2171 (2008)
N. Kosmatov, A singular non-local problem at resonance. J. Math. Anal. Appl. 394, 425–431 (2012)
V. Lakshmikantham, S. Leela, Existence and monotone method for periodic solutions of first order differential equations. J. Math. Anal. Appl. 91, 237–243 (1983)
B. Liu, Existence and uniqueness of solutions to first-order multipoint boundary value problems. Appl. Math. Lett. 17, 1307–1316 (2004)
Y. Liu, Multiple solutions of periodic boundary value problems for first order differential equations. Comput. Math. Appl. 54, 1–8 (2007)
B. Liu, Z. Zhao, A note on multi-point boundary value problems. Nonlinear Anal. 67, 2680–2689 (2007)
J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces. J. Differ. Equ. 12, 610–636 (1972)
O. Nica, Nonlocal initial value problems for first order differential systems. Fixed Point Theory 13, 603–612 (2012)
J.J. Nieto, R. Rodríguez-López, Greens function for first-order multipoint boundary value problems and applications to the existence of solutions with constant sign. J. Math. Anal. Appl. 388, 952–963 (2012)
D. O’Regan, M. Zima, Leggett-Williams norm-type theorems for coincidences. Arch. Math. 87, 233–244 (2006)
W.V. Petryshyn, On the solvability of x ∈ Tx + λFx in quasinormal cones with T and F k-set contractive. Nonlinear Anal. 5, 585–591 (1981)
R. Precup, D. Trif, Multiple positive solutions of non-local initial value problems for first order differential systems. Nonlinear Anal. 75, 5961–5970 (2012)
J. Santanilla, Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 105, 357–371 (1985)
J. Santanilla, Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Anal. Appl. 126, 397–408 (1987)
C.C. Tisdell, Existence of solutions to first-order periodic boundary value problems. J. Math. Anal. Appl. 323, 1325–1332 (2006)
M. Zima, P. Drygaś, Existence of positive solutions for a kind of periodic boundary value problem at resonance. Bound. Value Probl. 2013, 19 (2013)
Acknowledgements
This work was completed with the partial support of the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów. The author wishes to express her thanks to the referees for careful reading of the manuscript and constructive comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Zima, M. (2018). Positive Solutions for a Nonlocal Resonant Problem of First Order. In: Drygaś, P., Rogosin, S. (eds) Modern Problems in Applied Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72640-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-72640-3_14
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-72639-7
Online ISBN: 978-3-319-72640-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)