Positive solutions to a third order nonlocal boundary value problem with a parameter

• Autorzy: Szajnowska Gabriela , Zima Mirosława

Identyfikatory

DOI

10.7494/OpMath.2024.44.2.267

• Języki publikacji: EN

Abstrakty

EN

We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel’skiĭ-Guo fixed point theorem in cones and the properties of the Green’s function corresponding to the BVP under study. The main results are illustrated by suitable examples.

Słowa kluczowe

EN

boundary value problem; nonlocal boundary conditions; positive solution; cone

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2024
• Tom: Vol. 44, no. 2
• Strony: 267--283
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Szajnowska Gabriela

• University of Rzeszów, Institute of Mathematics, Pigonia 1, Rzeszów, 35-959, Poland

• https://orcid.org/0000-0002-5257-9435

autor

Zima Mirosława

• University of Rzeszów, Institute of Mathematics, Pigonia 1, Rzeszów, 35-959, Poland

• https://orcid.org/0000-0002-6152-4962

Bibliografia

• [1] A. Benmezaï, 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.
• [2] A. Cabada, L. López-Somoza, 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.
• [3] J.A. Cid, G. Infante, M. Tvrdý, M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl. 423 (2015), 1546–1556.
• [4] C.S. Goodrich, On a nonlocal BVP with nonlinear boundary conditions, Results Math. 63 (2013), 1351–1364.
• [5] 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.
• [6] J. Graef, J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. 71 (2009), 1542–1551.
• [7] J. Graef, L. Kong, F. Minhós, Generalized Hammerstein equations and applications, Results Math. 72 (2017), 369–383.
• [8] M. Greguš, Third Order Linear Differential Equations, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1987.
• [9] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988.
• [10] Ch.P. Gupta, On a third-order three-point boundary value problem at resonance, Differential Integral Equations 2 (1989), 1–12.
• [11] B. Hopkins, N. Kosmatov, Third-order boundary value problems with sign-changing solutions, Nonlinear Anal. 67 (2007), 126–137.
• [12] G. Infante, Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence, Discrete Contin. Dyn. Syst. B 25 (2020), 691–699.
• [13] G. Infante, F. Minhós, Nontrivial solutions of systems of Hammerstein integral equations with first derivative dependence, Mediterr. J. Math. (2017) 14:242.
• [14] W. Jiang, N. Kosmatov, Solvability of a third-order differential equation with functional boundary conditions at resonance, Bound. Value Probl. 2017 (2017), Article no. 81.
• [15] I. Kossowski, Radial solustions for nonlinear elliptic equation with nonlinear elliptic equation nonlocal boundary conditions, Opuscula Math. 43 (2023), no. 5, 675–687.
• [16] L. López-Somoza, F. Minhós, Existence and multiplicity results for some generalized Hammerstein equations with a parameter, Adv. Differ. Equ. 2019 (2019), Article no. 423.
• [17] F. Minhós, 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.
• [18] I. Rachůnková, On some three-point problems for third-order differential equations, Math. Bohemica 117 (1992), 98–110.
• [19] J.-P. Sun, H.-B. Li, Monotone positive solution of nonlinear third-order BVP with integral boundary conditions, Bound. Value Probl. 2010 (2010), Article no. 874959.
• [20] J.R.L.Webb, G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. 74 (2006), 673–693.
• [21] J.R.L. Webb, M. Zima, Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear Anal. 71 (2009), 1369–1378.
• [22] H.-E. Zhang, J.-P. Sun, Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions, Electron. J. Qual. Theory Differ. Equ. 2012, no. 18, 1–9.
• [23] H.-E. Zhang, Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions, Bound. Value Probl. 2014 (2014), Article no. 61.
• [24] L. Zhao, W. Wang, 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.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)