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Smirnov, Sergey

Green’s function and existence of solutions for a third-order boundary value problem involving integral condition. (English) Zbl 1520.34019

Lith. Math. J. 62, No. 4, 509-518 (2022).
In this paper, a third-order boundary value problem of the type \[ \begin{cases} x''' + f(t,x) = 0 \\ x(0) = x'(0) = 0 \\ x(1) = \int_0^1 g(s)x(s)ds \end{cases} \] is considered.
After the introduction of a suitable Green’s function allowing to rewrite the problem as an equivalent integral equation, the existence of a solution is proved, under different conditions on the nonlinearity \(f\), using Schauder’s fixed point theorem.
Reviewer: Alberto Boscaggin (Collegno)

Cited in 1 Document

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
47H10 Fixed-point theorems

Keywords:

third-order nonlinear boundary value problems; integral boundary condition; Green’s function; Schauder’s fixed point theorem
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References:

[1] K. Bingelė, A. Bankauskienė, and A. Štikonas, Spectrum curves for a discrete Sturm-Liouville problem with one integral boundary condition, Nonlinear Anal. Model. Control., 24(5):755-774, 2019, https://doi.org/doi:10.15388/NA.2019.5.5. · Zbl 1426.34038
[2] K. Bingelė, A. Bankauskienė, and A. Štikonas, Investigation of spectrum curves for a Sturm-Liouville problem with two-point nonlocal boundary conditions, Math. Model. Anal., 25(1):53-70, 2020, https://doi.org/doi:10.3846/mma.2020.10787. · Zbl 1476.34079
[3] N. Bouteraa and S. Benaicha, Existence of solution for third-order three-point boundary value problem, Mathematica, 60(83):21-31, 2018, https://doi.org/doi:10.24193/mathcluj.2018.1.03. · Zbl 1438.34099
[4] J.R. Graef and T.Moussaoui, A class of nth-order BVPs with nonlocal conditions, Comput.Math. Appl., 58(8):1662-1671, 2009, https://doi.org/doi:10.1016/j.camwa.2009.07.009. · Zbl 1189.34033
[5] J.R. Graef and J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. Theory Methods Appl., 71(5-6):1542-1551, 2009, https://doi.org/doi:10.1016/j.na.2008.12.047. · Zbl 1189.34034
[6] A. Gritsans and F. Sadyrbaev, A two-point boundary value problem for third order asymptotically linear systems, Electron. J. Qual. Theory Differ. Equ., 28:1-24, 2019, https://doi.org/doi:10.14232/ejqtde.2019.1.28. · Zbl 1438.34089
[7] P.S. Kelevedjiev and T.Z. Todorov, Existence of solutions of nonlinear third-order two-point boundary value problems, Electron. J. Qual. Theory Differ. Equ., 23:1-15, 2019, https://doi.org/doi:10.14232/ejqtde.2019.1.23. · Zbl 1438.34098
[8] R. Ma, Nonlinear periodic boundary value problems with sign-changing Green’s function, Nonlinear Anal. Theory Methods Appl., 74(5):1714-1720, 2011, https://doi.org/doi:10.1016/j.na.2010.10.043. · Zbl 1220.34038
[9] J. Rodriguez and P. Taylor, Multipoint boundary value problems for nonlinear ordinary differential equations, Nonlinear Anal. Theory Methods Appl., 68(11):3465-3474, 2008, https://doi.org/doi:10.1016/j.na.2007.03.038. · Zbl 1151.34012
[10] S. Roman and A. Štikonas, Third-order linear differential equation with three additional conditions and formula for Green’s function, Lith. Math. J., 50(4):426-446, 2010, https://doi.org/doi:10.1007/s10986-010-9097-x. · Zbl 1284.34026
[11] M. Sapagovas, V. Griškonienė, and O. Štikonienė, Application of M-matrices theory to numerical investigation of a nonlinear elliptic equation with an integral condition, Nonlinear Anal. Model. Control, 22(4):489-504, 2017, https://doi.org/doi:10.15388/NA.2017.4.5. · Zbl 1450.65085
[12] M. Sapagovas, O. Štikonienė, K. Jakubėlienė, and R. Čiupaila, Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions, Bound. Value Probl., p. 94, 2019, https://doi.org/doi:10.1186/s13661-019-1202-4. · Zbl 1513.65446
[13] A. Štikonas, A survey on stationary problems, Green’s functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control., 19(3):301-334, 2014, https://doi.org/doi:10.15388/NA.2014.3.1. · Zbl 1314.65105
[14] Štikonas, A.; Sen, E., Asymptotic analysis of Sturm-Liouville problem with nonlocal integral-type boundary condition, Nonlinear Anal. Model. Control, 26, 5, 969-991 (2021) · Zbl 1498.34085 · doi:10.15388/namc.2021.26.24299
[15] J.R.L. Webb, Non-local second-order boundary value problems with derivative-dependent nonlinearity, Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci., 379(2191):20190383, 2021, https://doi.org/doi:10.1098/rsta.2019.0383.
[16] J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA, Nonlinear Differ. Equ. Appl., 15(1-2):45-67, 2008, https://doi.org/doi:10.1007/s00030-007-4067-7. · Zbl 1148.34021
[17] J.R.L. Webb and G. Infante, Non-local boundary value problems of arbitrary order, J. Lond. Math. Soc., II Ser., 79(1):238-258, 2009, https://doi.org/doi:10.1112/jlms/jdn066. · Zbl 1165.34010
[18] Zeidler, E., Nonlinear Functional Analysis and Its Applications I (1986), Springer, New York: Fixed-Point Theorems, Springer, New York · Zbl 0583.47050 · doi:10.1007/978-1-4612-4838-5
[19] J. Zhao, P. Wang, and W. Ge, 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(1):402-413, 2011, https://doi.org/doi:10.1016/j.cnsns.2009.10.011. · Zbl 1221.34053
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