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Solvability of boundary value problems for vector differential systems*

  • L. H. Erbe (a1), Xinzhi Liu (a1) and Jianhong Wu (a1)

Synopsis

Brouwer topological degree theory, the shooting type method, the disconjugacy theory of Hamiltonian systems and the Liapunov-Razumikhin technique of Volterra integrodifferential equations are employed to establish some solvability results for the 2n-dimensional differential system

subject to one of the following boundary conditions:

(i) x(0) = Qx(l), Qg(l x(1), qy(1)= g(0, x(0), y(0)),

(ii) Blx(O) = B2g(O, x(O), y(O)), C1x(l) = −C2g(l, x(l), y(l)),

where Q, Bi, Ci, i = 1, 2, are n x n real matrices. An application is given to the second order equation xn = h(t, x, x') subject to certain nonlinear boundary conditions.

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References

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1Bernfeld, S. and Lakshmikantham, V.. An Introduction to Nonlinear Boundary Value Problems (New York: Academic Press, 1974).
2Burton, T. A.. Volterra Integral and Differential Equations (New York: Academic Press, 1983).
3Coppel, W. A.. Disconjugacy. Lecture Notes in Mathematics. 200 (Berlin: Springer, 1971).
4Erbe, L. H.. Existence of solutions to boundary value problem for second order differential equations. Nonlinear Anal. 6 (1982), 11551162.
5Erbe, L. H. and Knobloch, H. W.. Boundary value problems for systems of second order differential equations. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 6176.
6Erbe, L. H. and Liu, Xinzhi. Existence results for boundary value problems of second order impulsive differential equations. J. Math. Anal. Appl. (to appear).
7Erbe, L. H. and Schmitt, K.. On solvability of boundary value problems for systems of differential equations. J. Appl. Math. Phys. 38 (1987), 184192.
8Habets, P. and Schmitt, K.. Nonlinear boundary value problems for systems of differential equation. Arch. Math. (Basel) 40 (1983), 441446.
9Hartman, P.. Ordinary Differential Equations (New York: Interscience, 1964).
10Hartman, P.. On N-parameter families and interpolation problems for nonlinear ordinary differential equations. Trans. Amer. Math. Soc. 154 (1971), 201226.
11Kelley, W. G.. Second order systems with nonlinear boundary conditions. Proc. Amer. Math. Soc. 62 (1977), 287292.
12Knobloch, H. W.. Boundary value problems for systems of nonlinear differential equations. In Proc. EquadifflV, 1977, pp. 197–204. Lecture Notes in Mathematics 703 (Berlin: Springer 1978).
13Knobloch, H. W. and Schmitt, K.. Nonlinear boundary value problems for systems of differential equations. Proc. Roy. Soc. Edinburgh Sect. A 78(1977), 139158.
14Lakshmikantham, V. and Leela, S.. Differential and Integral Inequalities (New York: Academic Press, 1969).
15Lasota, A. and Yorke, J. A.. Existence of solutions of two-point boundary value problems for nonlinear systems. J. Differential Equations 11 (1972), 509518.
16Mawhin, J. and Schmidt, K.. Upper and lower solutions and semilinear second order elliptic equations with nonlinear boundary conditions. Proc. roy. Soc. Edinburgh Sect. A 97 (1984), 199207.
17Miller, R. K.. Nonlinear Volterra Integral Equations (New York: Benjamin, 1971).
18Schmitt, K.. Periodic solutions of systems of second order differential equations. J. Differential Equations 11 (1972), 180192.
19Wu, Jianhong. Periodic solutions of nonconvolution neutral integrodifferential equations. Ada Math. Sci. (English Ed.) 8:3 (1988), 307314.

Solvability of boundary value problems for vector differential systems*

  • L. H. Erbe (a1), Xinzhi Liu (a1) and Jianhong Wu (a1)

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