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Solvability of higher order three-point iterative systems. (English) Zbl 1463.34100

Ufim. Mat. Zh. 12, No. 3, 109-124 (2020) and Ufa Math. J. 12, No. 3, 107-122 (2020).
Summary: In this paper, we consider an iterative system of nonlinear \(n^{\text{th}}\) order differential equations: \[ y_i^{(n)}(t)+\lambda_i p_i(t)f_i(y_{i+1}(t))=0,\quad 1\leq i\leq m,\quad y_{m+1}(t)= y_1(t),\quad t\in[0,1], \] with three-point non-homogeneous boundary conditions \[ \begin{aligned} y_i(0)={y_i}'(0)=\cdots=y_i^{(n-2)}(0)=0, \\ \alpha_iy_i^{(n-2)}(1)-\beta_i y_i^{(n-2)}(\eta)=\mu_i,\quad 1\leq i\leq m, \end{aligned} \] where \(n\geq 3\), \(\eta\in (0,1)\), \(\mu_i\in (0, \infty)\) is a parameter, \(f_i:\mathbb{R}^+ \rightarrow \mathbb{R}^+\) is continuous, \(p_i:[0,1] \rightarrow \mathbb{R}^+\) is continuous and \(p_i\) does not vanish identically on any closed subinterval of \([0,1]\) for \(1\leq i\leq m\). We express the solution of the boundary value problem as a solution of an equivalent integral equation involving kernels and obtain bounds for these kernels. By an application of Guo-Krasnosel’skii fixed point theorem on a cone in a Banach space, we determine intervals of the eigenvalues \(\lambda_1,\lambda_2,\cdots,\lambda_m\) for which the boundary value problem possesses a positive solution. As applications, we provide examples demonstrating our results.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
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References:

[1] K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985 · Zbl 0559.47040
[2] J. R. Graef, J. Henderson, B. Yang, “Positive solutions to a fourth order three-point boundary value problem”, Discr. Contin. Dyn. Syst., 2009, Special (2009), 269-275 · Zbl 1198.34024
[3] J. R. Graef, B. Yang, “Multiple positive solutions to a three-point third order boundary value problem”, Discr. Contin. Dyn. Syst., 2005, Special (2005), 337-344 · Zbl 1152.34325
[4] D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, Orlando, 1988 · Zbl 0661.47045
[5] J. Henderson, S. K. Ntouyas, “Positive solutions for systems of \(n^{th}\) order three-point nonlocal boundary value problems”, Electron. J. Qual. Theory Differ. Equ., 2007 (2007), 1-12 · Zbl 1182.34029
[6] J. Henderson, S. K. Ntouyas, I. K. Purnaras, “Positive solutions for systems of generalized threepoint nonlinear boundary value problems”, Comment. Math. Univ. Carolin., 49:1 (2008), 79-91 · Zbl 1212.34058
[7] J. Henderson, S. K. Ntouyas, I. K. Purnaras, “Positive solutions for systems of second order four-point nonlinear boundary value problems”, Comm. Appl. Anal., 12:1 (2008), 29-40 · Zbl 1166.34006
[8] M. A. Krasnosel’skii, Positive solutions of operator equations, P. Noordhoff Ltd, Groningen, 1964 · Zbl 0121.10604
[9] A. G. Lakoud, L. Zenkoufi, “Existence of positive solutions for a fourth order three-point boundary value problem”, J. Appl. Math. Comput., 50 (2016), 139-155 · Zbl 1334.34057
[10] X. Lin, Z. Zhao, “Iterative technique for a third order differential equation with three-point nonlinear boundary value conditions”, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-10 · Zbl 1363.34039
[11] Z. Liu, H. Chen, C. Liu, “Positive solutions for singular third order non-homogeneous boundary value problems”, J. Appl. Math. Comput., 38 (2012), 161-172 · Zbl 1303.34019
[12] D. Liu, Z. Ouyang, “Solvability of third order three-point boundary value problems”, Abstr. Appl. Anal., 2014, 793639 · Zbl 1474.34129
[13] A. P. Palamides, G. Smyrlis, “Positive solutions to a singular third order three-point boundary value problem with an indefinitely signed Green”s function”, Nonlinear Anal. TMA, 68:7 (2008), 2104-2118 · Zbl 1153.34016
[14] K. R. Prasad, N. Sreedhar, K. R. Kumar, “Solvability of iterative systems of three-point boundary value problems”, TWMS J. APP. Eng. Math., 3:2 (2013), 147-159 · Zbl 1307.34036
[15] Y. Sun, “Positive solutions for third order three-point non-homogeneous boundary value problems”, Appl. Math. Lett., 22 (2009), 45-51 · Zbl 1163.34313
[16] Y. Sun, C. Zhu, “Existence of positive solutions for singular fourth order three-point boundary value problems”, Adv. Difference Equ., 2013:51 (2013), 1-13 · Zbl 1380.34044
[17] C. X. Wang, H. R. Sun, “Positive solutions for a class of singular third order three-point nonhomogeneous boundary value problem”, Dynam. Syst. Appl., 19 (2010), 225-234 · Zbl 1225.34034
[18] 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., 10:3 (2018), 251-260 · Zbl 1411.34041
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