Existence and Multiplicity of Positive Solutions for a Third-Order Two-Point Boundary Value Problem

Sergey Smirnov

doi:10.2478/tmmp-2021-0027

.blurhash-client-img { display: none !important; }

Existence and Multiplicity of Positive Solutions for a Third-Order Two-Point Boundary Value Problem

Tatra Mountains Mathematical Publications

Volume 79 (2021): Issue 2 (December 2021)

By: Sergey Smirnov

Open Access

|Jan 2022

Abstract

We study the existence and multiplicity of positive solutions for a third-order two-point boundary value problem by applying Krasnosel’skii’s fixed point theorem. To illustrate the applicability of the obtained results, we consider some examples.

References

[1] AGARWAL, R. P.—O’REGAN, D.—WONG, P.J.Y.: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht, 1998.10.1007/978-94-015-9171-3
Search in Google Scholar Back to article
[2] ANDERSON, D. R.—DAVIS, J.M.: Multiple solutions and eigenvalues for third-order right focal boundary value problem, J. Math. Anal. Appl. 267 (2002), 135–157.10.1006/jmaa.2001.7756
Search in Google Scholar Back to article
[3] BAXLEY, J,—HAYWOOD, L. J.: Multiple positive solutions of nonlinear boundary value problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), 157–168.
Search in Google Scholar Back to article
[4] CABADA, A.—DIMITROV, N.D.: Third-order differential equations with three-point boundary conditions, Open Mathematics 19 (2021), 11–31.10.1515/math-2021-0007
Search in Google Scholar Back to article
[5] ERBE, L. H.—WANG, H.: On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), 743–748.10.1090/S0002-9939-1994-1204373-9
Search in Google Scholar Back to article
[6] GRAEF, J. R.—QIAN, C.—YANG, B.: A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl. 287 (2003), 217–233.10.1016/S0022-247X(03)00545-6
Search in Google Scholar Back to article
[7] GRAEF, J. R.—YANG, B.: Multiple positive solutions to a three point third order boundary value problem, Discrete Contin. Dyn. Syst. (2005), suppl., 337–344.
Search in Google Scholar Back to article
[8] GRAEF, J. R.—YANG, B.: Positive solutions of a third order nonlocal boundary value problem, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), 89–97.
Search in Google Scholar Back to article
[9] GRAEF, J. R.—YANG, B.: Existence and nonexistence of positive solutions of a nonlinear third order boundary value problem, Electron. J. Qual. Theory Differ. Equ. 9 (2008), 1–13.10.14232/ejqtde.2008.1.31
Search in Google Scholar Back to article
[10] GRITSANS, A.—SADYRBAEV, F.: A two-point boundary value problem for third order asymptotically linear systems, Electron. J. Qual. Theory Differ. Equ. 28 (2019), 1–24.10.14232/ejqtde.2019.1.28
Search in Google Scholar Back to article
[11] GUO, D.—LAKSHMIKANTHAM, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego, 1988.
Search in Google Scholar Back to article
[12] GUO, D.—LAKSHMIKANTHAM, V.: Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. Math. Anal. Appl. 129 (1988), 211–222.10.1016/0022-247X(88)90243-0
Search in Google Scholar Back to article
[13] HENDERSON, J.—THOMPSON, H.B.: Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000), 2373–2379.10.1090/S0002-9939-00-05644-6
Search in Google Scholar Back to article
[14] INFANTE, G.—WEBB, J.R. L.: Positive solutions of some nonlocal boundary value problems, Abstract and Applied Analysis. 18 (2003), 1047–1060.10.1155/S1085337503301034
Search in Google Scholar Back to article
[15] KELEVEDJIEV, P. S.—TODOROV, T. Z.: Existence of solutions of nonlinear third-order two-point boundary value problems, Electron. J. Qual. Theory Differ. Equ. 23 (2019), 1–15.10.14232/ejqtde.2019.1.23
Search in Google Scholar Back to article
[16] KRASNOSEL’SKII, M. A.: Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964.
Search in Google Scholar Back to article
[17] WEBB, J.R.L. : Optimal constants in a nonlocal boundary value problem, Nonlinear Anal. 63 (2005), 672–685.10.1016/j.na.2005.02.055
Search in Google Scholar Back to article
[18] WEBB, J.R.L.—INFANTE, G.: Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear Differential Equations and Appl. (NoDEA) 15 (2008), 45–67.10.1007/s00030-007-4067-7
Search in Google Scholar Back to article
[19] WEBB, J.R.L.—INFANTE, G.: Nonlocal boundary value problems of arbitrary order, Journal of the London Mathematical Society 79 (2009), no. 1, 238–258.
Search in Google Scholar Back to article
[20] ZHAO, J.—WANG, P.— GE, W.: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 402–413.10.1016/j.cnsns.2009.10.011
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2021-0027 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

Journal RSS Feed

Language: English

Page range: 199 - 212

Submitted on: Sep 24, 2021

Published on: Jan 1, 2022

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Nonlinear boundary value problems,

existence and multiplicity of positive solutions,

Green’s function,

Krasnosel’skii’s fixed point theorem

Related subjects:

Mathematics,

General mathematics

© 2022 Sergey Smirnov, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 79 (2021): Issue 2 (December 2021)