## Positive solutions of even order system periodic boundary value problems.(English)Zbl 1195.34036

The authors consider the nonlinear boundary value problem
$(AD^2+ B)^n u= W(t) f(t,u),\quad t\in (0,\omega),\tag{1}$
$u^{(k)}(0)= u^k(\omega),\quad k= 0,1,\dots, 2n-1,\tag{2}$
where $$n\in\mathbb{N}$$, $$D={d\over dt}$$, $$u=(u_1,\dots, u_m)^T\in \mathbb{R}^m_+$$, $$A= \text{diag}(a_1,\dots, a_m)$$, $$B= \text{diag}(b_1,\dots, b_m)$$, with $$a_i= \pm1$$, $$b_i> 0$$; $$i= 1,\dots, m$$;
$W= (w_{ij})_{m\times m}\in C([0, \omega], \mathbb{R}^{m\times m}_+)$
such that $$\sum^m_{j=1} w_{i_j}(t)\not\equiv 0$$ on $$[0,\omega]$$ for all $$i= 1,\dots, m$$, and
$f= (f_1,\dots, f_m)^T\in C([0,\omega]\times \mathbb{R}^m_+, \mathbb{R}^m_+).$
The existence of one, two, any arbitrary number of positive solutions for problem (1), (2) is proved under some assumptions on the function $$f$$.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations
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### References:

  Clark, S.; Henderson, J., Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations, Proc. Amer. Math. Soc., 134, 3363-3372 (2006) · Zbl 1120.34010  Erbe, L., Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling, 32, 529-539 (2000) · Zbl 0970.34019  Ge, W.; Xue, C., Some fixed point theorems and existence of positive solutions of two-point boundary-value problems, Nonlinear Anal., 70, 16-31 (2009) · Zbl 1361.34023  Graef, J. R.; Yang, B., Positive solutions to a multi-point higher order boundary value problem, J. Math. Anal. Appl., 316, 409-421 (2006) · Zbl 1101.34004  Henderson, J.; Karna, B.; Tisdell, C. C., Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc.., 133, 1365-1369 (2005) · Zbl 1061.34009  Kong, L.; Kong, Q., Positive solutions of higher-order boundary-value problems, Proc. Edinb. Math. Soc., 48, 445-464 (2005) · Zbl 1084.34023  Kong, L.; Kong, Q., Second-order boundary value problems with nonhomogeneous boundary conditions, Math. Nachr., 278, 173-193 (2005) · Zbl 1060.34005  Kong, L.; Kong, Q., Multi-point boundary value problems with nonhomogeneous boundary conditions, Math. Nachr., 58, 909-931 (2004) · Zbl 1066.34012  Kwong, M. K., The topological nature of Krasnoselskii’s cone fixed point theorem, Nonlinear Anal., 69, 891-897 (2008) · Zbl 1142.47336  Kong, Q., Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear Anal., 66, 2635-2651 (2007) · Zbl 1119.34024  Kong, L.; Kong, Q., Nodal solutions of second order nonlinear boundary value problems, Math. Proc. Cambridge. Philos. Soc., 146, 3, 747-763 (2009) · Zbl 1189.34043  Kwong, M. K.; Wong, J. S.W., The shooting method and non-homogeneous multi-point BVPs of second-order ODE, Bound. Value Probl., 16 (2007), (Art. ID 64012)  Ma, R., Nodal solutions of second-order boundary value problems with superlinear or sublinear nonlinearities, Nonlinear Anal., 66, 950-961 (2007) · Zbl 1113.34011  Rynne, B. P., Spectral properties and nodal solutions for second-order, $$m$$-point, boundary value problems, Nonlinear Anal., 67, 3318-3327 (2007) · Zbl 1142.34010  Atici, F. M.; Guseinov, G. Sh., On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. Comput. Appl. Math., 132, 341-356 (2001) · Zbl 0993.34022  Graef, J. R.; Kong, L., Existence results for nonlinear periodic boundary value problems, Proc. Edinb. Math. Soc., 52, 79-95 (2009) · Zbl 1178.34024  Graef, J. R.; Kong, L.; Wang, H., Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. Differential Equations, 245, 1185-1197 (2008) · Zbl 1203.34028  Jiang, D.; Chua, J.; O’Regan, D.; Agarwal, R., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. Math. Anal. Appl., 286, 563-576 (2003) · Zbl 1042.34047  Lan, K. Q., Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems, Appl. Math. Comput., 154, 531-542 (2004) · Zbl 1055.45005  Li, Y., Positive solutions of fourth-order periodic boundary value problems, Nonlinear Anal., 54, 1069-1078 (2003) · Zbl 1030.34025  Li, Y., Existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions, J. Math. Anal. Appl., 322, 530-539 (2006) · Zbl 1131.34015  Li, F.; Li, Y.; Liang, Z., Existence and multiplicity of solutions to $$2 m th$$-order ordinary differential equations, J. Math. Anal. Appl., 331, 958-977 (2007) · Zbl 1119.34014  Rachu̇nková, I.; Tvrdý, M.; Vrkoč, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differential Equations, 176, 445-469 (2001) · Zbl 1004.34008  Torres, P. J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190, 643-662 (2003) · Zbl 1032.34040  Yao, Q., Positive solutions of nonlinear second-order periodic boundary value problems, Appl. Math. Lett., 20, 583-590 (2007) · Zbl 1131.34303  Zhang, Z.; Wang, J., On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations, J. Math. Anal. Appl., 281, 99-107 (2003) · Zbl 1030.34024  Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer: Springer Berlin, New York · Zbl 0676.58017  Zhao, F.; Wu, X., Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity, Nonlinear Anal., 60, 325-335 (2005) · Zbl 1087.34022  Wang, H., On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl., 281, 287-306 (2003) · Zbl 1036.34032  Infante, G.; Pietramala, P., Existence and multiplicity of non-negative solutions for system of perturbed Hammerstein integral equations, Nonlinear Anal., 71, 1301-1310 (2009) · Zbl 1169.45001  Yang, Z., Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl., 321, 751-765 (2006) · Zbl 1106.34014  O’Regan, D.; Wang, H., Positive periodic solutions of systems of second order ordinary differential equations, Positivity, 10, 285-298 (2006) · Zbl 1103.34027  Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0559.47040  Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press, Inc. · Zbl 0661.47045
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