## A monotone method for constructing extremal solutions to second-order periodic boundary value problems.(English)Zbl 0993.34011

Summary: The authors describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem $u''(t)= f(t,u,u'(t)),\quad t\in [0,2\pi],\quad u(0)= u(2\pi),\quad u'(0)= u'(2\pi),$ in the presence of an upper solution $$\beta$$ and a lower solution $$\alpha$$ with $$\beta\leq \alpha$$.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

extremal solutions; periodic boundary value problem
Full Text:

### References:

 [1] Cabada, A., The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. math. anal. appl., 185, 302-320, (1994) · Zbl 0807.34023 [2] A. Cabada, The monotone method for boundary value problems, Doctoral Thesis, Universidad de Santiago de Compostela, 1992 (in Spanish). [3] Cabada, A.; Nieto, J.J., A generalization of the monotone iterative technique for linear second order periodic boundary value problems, J. math. anal., 151, 181-189, (1990) · Zbl 0719.34039 [4] Cabada, A.; Nieto, J.J., Extremal solutions of second order nonlinear periodic boundary value problems, Appl. math. comput., 40, 135-145, (1990) · Zbl 0723.65056 [5] Gao, W.J.; Wang, J.Y., On a nonlinear second order periodic boundary value problem with caratheodory functions, Ann. polon. math., 62, 283-291, (1995) · Zbl 0839.34031 [6] Ladde, G.S.; Lakshmikantham, V.; Vatsala, A.S., Monotone iterative techniques for nonlinear differential equations, (1985), Pitman Boston · Zbl 0658.35003 [7] Nieto, J.J., Nonlinear second-order periodic boundary value problems, J. math. anal. appl., 30, 22-29, (1988) · Zbl 0678.34022 [8] Nieto, J.J., Nonlinear second-order periodic boundary value problems with caratheodory functions, Appl. anal., 34, 111-128, (1989) · Zbl 0662.34022 [9] Jiang, D.Q.; Wang, J.Y., A generalized periodic boundary value problem for the one-dimensional p-Laplacian, Ann. polon. math., 65, 265-270, (1997) · Zbl 0868.34015 [10] Nieto, J.J.; Cabada, A., A generalized upper and lower solution method for nonlinear second order ordinary differential equations, J. appl. math. stochastic anal., 5, 157-166, (1992) · Zbl 0817.34016 [11] Omari, P., A monotone method for constructing extremal soltions of second order scalar boundary value problems, Appl. math. comput., 18, 257-275, (1986) · Zbl 0625.65075 [12] Omari, P., Nonordered lower and upper solutions and solvability of the periodic problem for the lienard and the Rayleigh equations, Rend. istit. mat. univ. trieste, 20, 54-64, (1991) [13] Rudolf, B.; Kubacek, Z., Remarks on J. J. Niteo’s paper: nolinear second-order periodic boundary value problems, J. math. anal. appl., 146, 203-206, (1990) · Zbl 0713.34015 [14] Seda, V.; Nieto, J.J.; Gera, M., Periodic boundary value problems for nonlinear higher order ordinary differential equations, Appl. math. comput., 48, 71-82, (1992) · Zbl 0748.34014 [15] Wang, M.X.; Cabada, A.; Nieto, J.J., Monotone method for nonlinear second-order periodic boundary value problemws with caratheodory functions, Ann. polon. math., 58, 221-235, (1993) · Zbl 0789.34027 [16] Wang, C.G., Generalized upper and lower solution method for the forced Duffing equation, Proc. amer. math. soc., 125, 397-406, (1997) · Zbl 0866.34037 [17] Rachunkova, I., Upper and lower solutions satisfying the inverse inequality, Ann. polon. math., 65, 235-244, (1997) · Zbl 0868.34014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.