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Multiple solutions for a system of \((n_i p_i)\) boundary value problems. (English) Zbl 1160.34313
Summary: We consider the system of boundary value problems \[ \begin{cases} u^{(n_i)}_i(t)+f_i(t,u_1(t),\dots,u_m(t))=0\\ u^{(j)}_i(0)=0,\quad u^{(p_i)}_i(1)=0\end{cases} \] for \(t\in[0,1]\), \(i=1,\dots,m\) and \(0\leq j\leq n_i-2\) where \(n_i\geq 2\) and \(1\leq p_i\leq n_i-1\). Several criteria are offered for the existence of single and twin solutions of the system that are of fixed signs.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:
[1] Agarwal, R. P.: Difference Equations and Inequalities. New York: Marcel Dekker 1992. · Zbl 0925.39001
[2] Agarwal, R. P.: Focal Boundary Value Problems for Differential and Difference Equations. Dordrecht: Kluwer Acad. Publ. 1998. · Zbl 0914.34001
[3] Agarwal, R. P. and D. O’Regan: A coupled system of difference equations. Appl. Math. Comp. (to appear).
[4] Agarwal, R. P., O’Regan, D. and P. J. Y. Wong: Positive Solutions of Differential, Dif- ference and Integral Equations. Dordrecht: Kluwer Acad. Publ. 1999.
[5] Agarwal, R. P. and P. J. Y. Wong: Advanced Topics in Difference Equations. Dordrecht: Kluwer Acad. Publ. 1997. · Zbl 0878.39001
[6] Agarwal, R. P. and P. J. Y. Wong: Existence criteria for a system of two-point boundary value problems. Appl. Anal. (to appear). · Zbl 1031.34020 · doi:10.1080/00036810008840878
[7] Aronson, D., Crandall, M. G. and L. A. Peletier: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlin. Anal. 6 (1982), 1001 - 1022. · Zbl 0518.35050 · doi:10.1016/0362-546X(82)90072-4
[8] Choi, Y. S. and G. S. Ludford: An unexpected stability result of the near-extinction dif- fusion flame for non-unity Lewis numbers. Q.J. Mech. Appl. Math. 42 (1989), 143 -158. · Zbl 0682.76076 · doi:10.1093/qjmam/42.1.143
[9] Cohen, D. C.: Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory. SIAM J. Appl. Math. 20 (1971), 1 - 13. P. J. Y Wong and R. P. Agarwal · Zbl 0219.34027 · doi:10.1137/0120001
[10] Dancer, E. N.: On the structure of solutions of an equation in catalysis theory when a parameter is large. J. Diff. Equ. 37 (1980), 404 - 437. · Zbl 0417.34042 · doi:10.1016/0022-0396(80)90107-2
[11] Eloe, P. W. and J. Henderson: Positive solutions for (n - 1, 1) conjugate boundary value problems. Nonlin. Anal. 28 (1997), 1669 - 1680. · Zbl 0871.34015 · doi:10.1016/0362-546X(95)00238-Q
[12] Eloe, P. W. and J. Henderson: Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Electron. J. Diff. Equ. 3 (1997), 1 - 11. · Zbl 0888.34013 · emis:journals/EJDE/Volumes/1997/03/abstr.html · eudml:119392
[13] Eloe, P. W., Henderson, J. and E. R. Kaufmann: Multiple positive solutions for difference equations. J. Diff. Equ. Appl. 3 (1998), 219 - 229. · Zbl 1005.39502 · doi:10.1080/10236199808808098
[14] Fujita, H.: On the nonlinear equations \Delta u + eu = 0 and \partial v = \Delta v + ev. Bull. Amer. \partial t Math. Soc. 75 (1969), 132 - 135. · Zbl 0216.12101 · doi:10.1090/S0002-9904-1969-12175-0
[15] Gel’fand, I. M.: Some problems in the theory of quasilinear equations. Uspehi Mat. Nauka 14 (1959), 87 - 158; Engl. transl. in: Trans. Amer. Math. Soc. 29 (1963), 295 - 381.
[16] Henderson,J. and E. R. Kaufmann: Multiple positive solutions for focal boundary value problems. Comm. Appl. Anal. 1 (1997), 53 - 60. · Zbl 0887.34018
[17] Krasnosel’skii, M. A.: Positive Solutions of Operator Equations. Groningen: Noordhoff 1964. · Zbl 0121.10604
[18] Leggett, R. W. and L. R. Williams: A fixed point theorem with application to an infectious disease model. J. Math. Anal. Appl. 76 (1980), 91 - 97. · Zbl 0448.47044 · doi:10.1016/0022-247X(80)90062-1
[19] Parter, S.: Solutions of differential equations arising in chemical reactor processes. SIAM J. Appl. Math. 26 (1974), 687 - 716. · Zbl 0285.34013 · doi:10.1137/0126063
[20] Wong, P. J. Y.: Solutions of constant signs of a system of Sturm-Liouville boundary value problems. Mathl. Comp. Modelling 29 (1999), 27 - 38. · Zbl 1041.34015 · doi:10.1016/S0895-7177(99)00079-5
[21] Wong, P. J. Y.: A system of (ni, pi) boundary value problems with positive/nonpositive nonlinearities. J. Math. Anal. Appl. (to appear). · Zbl 0953.34013 · doi:10.1006/jmaa.1999.6671
[22] Wong, P. J. Y. and R. P. Agarwal: Fixed-sign solutions of a system of higher order difference equations. J. Comp. Appl. Math. 113 (2000), 167 - 181. · Zbl 0940.39003 · doi:10.1016/S0377-0427(99)00251-4
[23] Wong, P. J. Y. and R. P. Agarwal: Existence theorems for a system of difference equations with (n, p) type conditions (submitted). · Zbl 1025.39002 · doi:10.1016/S0096-3003(00)00078-3
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