A fixed point index approach to a third order nonlocal boundary value problem
DOI:
https://doi.org/10.12775/TMNA.2024.046Słowa kluczowe
Boundary value problem, nonlocal boundary conditions, positive solution, coneAbstrakt
Using a topological approach we study the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our method is based on the fixed point index theory in cones. The main results extend and complement some previous works and are illustrated by suitable examples.Bibliografia
H. Amann, Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces, SIAM Review 18 (1976), 620–709.
A. Benmezaı̈ and E.-D. Sedkaoui, Positive solution for singular third-order BVPs on the half line with first-order derivative dependence, Acta Univ. Sapientiae 13 (2021), 105–126.
A. Cabada, L. López-Somoza and F. Minhós, Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem, J. Nonlinear Sci. Appl. 10 (2017), 5445–5463.
F. Cianciaruso, G. Infante and P. Pietramala, Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl. 33 (2017), 317–347.
L. Danziger and G.L. Elmergreen, The thyroid-pituitary homeostatic mechanism, Bull. Math. Biophys. 18 (1956), 1–13.
C.S. Goodrich, Coercive functionals and their relationship to multiplicity of solution to nonlocal boundary value problems, Topol. Methods Nonlinear Anal. 54 (2019), 409–426.
C.S. Goodrich, On a nonlocal BVP with nonlinear boundary conditions, Results Math. 63 (2013), 1351–1364.
C.S. Goodrich, New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green’s function, J. Differential Equations 264 (2018), 236–262.
J. Graef, L. Kong and F. Minhós, Generalized Hammerstein equations and applications, Results Math. 72 (2017), 369–383.
J. Graef and J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. 71 (2009), 1542–1551.
M. Greguš, Third Order Linear Differential Equations, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1987.
D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988.
Ch.P. Gupta, On a third-order three-point boundary value problem at resonance, Differential Integral Equations 2 (1989), 1–12.
B. Hopkins and N. Kosmatov, Third-order boundary value problems with sign-changing solutions, Nonlinear Anal. 67 (2007), 126–137.
G. Infante, Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence, Discrete Contin. Dyn. Syst. B 25 (2020), 691–699.
G. Infante and F. Minhós, Nontrivial solutions of systems of Hammerstein integral equations with first derivative dependence, Mediterr. J. Math. 14 (2017), 242,
W. Jiang and N. Kosmatov, Solvability of a third-order differential equation with functional boundary conditions at resonance, Boundary Value Problems 2017 (2017), 81.
J. Jones, W.C. Troy and A.D. MacGillivary, Steady solutions of the Kuramoto–Sivashinsky equation for small wave speed, J. Differential Equations 96 (1992), 28–55.
K.Q. Lan, Multiple positive solutions of semilinear equations with singularities, J. Lond. Math. Soc. 63 (2001), 690–704.
L. López-Somoza and F. Minhós, Existence and multiplicity results for some generalized Hammerstein equations with a parameter, Adv. Differ. Equ.2019 (2019), 423.
F. Minhós and R. de Sousa, On coupled systems of Hammerstein integral equations, Bound. Value Probl. 2019 (2019), 7.
F. Minhós and R. de Sousa, On the solvability of third-order three point systems of differential equations with dependence on the first derivative, Bull. Braz. Math. Soc. 48 (2017), 485–503.
S. Padhi and S. Pati, Theory of Third-Order Differential Equations, Springer, New Delhi, 2014.
S. Smirnov, Existence of sign-changing solutions for a third-order boundary value problem with nonlocal conditions of integral type, Topol. Methods Nonlinear Anal. 62 (2023), 377–384.
J.-P. Sun and H.-B. Li, Monotone positive solution of nonlinear third-order BVP with integral boundary conditions, Bound. Value Probl., vol. 2010, Article ID 874959, 12 pp.
G. Szajnowska and M. Zima, Positive solutions to a third order nonlocal boundary value problem with a parameter, Opuscula Math. 44 (2024), 267–283.
J.R.L. Webb, Non-local second-order boundary value problems with derivative-dependent nonlinearity, Philos. Trans. Roy. Soc. A 379, 20190383.
J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. (2) 74 (2006), 673–693.
H.-E. Zhang, Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions, Bound. Value Probl. 2014 (2014), 61.
L. Zhao, W. Wang and C. Zhai, Existence and uniqueness of monotone positive solutions for a third-order three-point boundary value problem, Differ. Equ. Appl. 10 (2018), 251–260.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2025 Mirosława Zima, Gabriela Szajnowska
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
