Positive solution for system of nonlinear first-order PBVPs on time scales. (English) Zbl 1071.34017

Sun, Jian-Ping; Li, Wan-Tong

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62, No. 1, 131-139 (2005).

Summary: We are concerned with the following system of nonlinear first-order periodic boundary value problems on time scale \(\mathbb{T}\) \[ \begin{gathered} x^\Delta_i(t)+ f_i(t, x_1(\sigma(t)), x_2(\sigma(t)),\dots, x_n(\sigma(t)))= 0,\quad t\in [0,T],\\ x_i(0)= x_i(\sigma(T)),\quad i= 1,2,\dots, n,\end{gathered} \] where \(f_i: [0, T]\times [0,+\infty)^n\to\mathbb{R}\) is continuous and there exists a constant \(M_i> 0\) such that \[ M_i x_i- f_i(t, x_1,x_2,\dots, x_n)\geq 0\quad\text{for }(x_1, x_2,\dots, x_n)\in [0,+\infty)^n,\quad t\in [0,T]. \] Some existence criteria for a positive solution are established by using a fixed-point theorem for operators on cone.

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

time scale; periodic boundary value problems; positive solution; fixed-point; cone

References:

[1]	Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 3-22 (1999) · Zbl 0927.39003
[2]	Agarwal, R. P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear Anal., 44, 527-535 (2001) · Zbl 0995.34016
[3]	Bohner, M.; Peterson, A., Dynamic Equations on Time Scales (2001), Springer: Springer New York · Zbl 1021.34005
[4]	Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differ. Eq. Dynamical Syst., 1, 223-246 (1993) · Zbl 0868.39007
[5]	Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. Comput. Model., 32, 5-6, 571-585 (2000) · Zbl 0963.34020
[6]	Gulsan Topal, S., Second-order periodic boundary value problems on time scales, Comput. Math. Appl., 48, 637-648 (2004) · Zbl 1068.34016
[7]	Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045
[8]	Haddock, J. R.; Nkashama, M. N., Periodic boundary value problems and monotone iterative methods for functional differential equations, Nonlinear Anal., 22, 267-276 (1994) · Zbl 0804.34062
[9]	Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[10]	Kaymakcalan, B.; Lakshmikanthan, V.; Sivasundaram, S., Dynamic Systems on Measure Chains (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0869.34039
[11]	Lakshmikantham, V., Periodic boundary value problems of first and second order differential equations, J. Appl. Math. Simulat., 2, 131-138 (1989) · Zbl 0712.34058
[12]	Lakshmikantham, V.; Leela, S., Existence and monotone method for periodic solutions of first-order differential equations, J. Math. Anal. Appl., 91, 237-243 (1983) · Zbl 0525.34031
[13]	Lakshmikantham, V.; Leela, S., Remarks on first and second order periodic boundary value problems, Nonlinear Anal., 8, 281-287 (1984) · Zbl 0532.34029
[14]	Leela, S.; Oguztoreli, M. N., Periodic boundary value problem for differential equations with delay and monotone iterative method, J. Math. Anal. Appl., 122, 301-307 (1987) · Zbl 0616.34062
[15]	Liz, E.; Nieto, J. J., Periodic boundary value problems for a class of functional differential equations, J. Math. Anal. Appl., 200, 680-686 (1996) · Zbl 0855.34080
[16]	Peng, S., Positive solutions for first order periodic boundary value problem, Appl. Math. Comput., 158, 345-351 (2004) · Zbl 1082.34510