Boundary value problems for second order differential equations in convex subsets of a Banach space

Schmitt, Klaus; Volkmann, Peter

doi:10.1090/S0002-9947-1976-0397110-5

Boundary value problems for second order differential equations in convex subsets of a Banach space
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by Klaus Schmitt and Peter Volkmann

Trans. Amer. Math. Soc. 218 (1976), 397-405

DOI: https://doi.org/10.1090/S0002-9947-1976-0397110-5

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Abstract:

Let E be a real Banach space, C a closed, convex subset of E and $f:[0,1] \times E \times E \to E$ be continuous. Let ${u_0},{u_1} \in C$ and consider the boundary value problem \begin{equation}\tag {$\ast $} u'' = f(t,u,u’),\quad u(0) = {u_0},\quad u(1) = {u_1}.\end{equation} We establish sufficient conditions in order that $(\ast )$ have a solution $u:[0,1] \to C$.

References

Paul B. Bailey, Lawrence F. Shampine, and Paul E. Waltman, Nonlinear two point boundary value problems, Mathematics in Science and Engineering, Vol. 44, Academic Press, New York-London, 1968. MR 0230967
James A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. MR 1501880, DOI 10.1090/S0002-9947-1936-1501880-4
E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. MR 1562984, DOI 10.1090/S0002-9904-1934-05978-0
R. M. Redheffer and W. Walter, Flow-invariant sets and differential inequalities in normed spaces, Applicable Anal. 5 (1975), no. 2, 149–161. MR 470401, DOI 10.1080/00036817508839117

Randwertaufgaben für gewöhnliche Differentialgleichungen

K. Schmitt and R. Thompson, Boundary value problems for infinite systems of second-order differential equations, J. Differential Equations 18 (1975), no. 2, 277–295. MR 374594, DOI 10.1016/0022-0396(75)90063-7
Russell C. Thompson, Differential inequalities for infinite second order systems and an application to the method of lines, J. Differential Equations 17 (1975), 421–434. MR 361344, DOI 10.1016/0022-0396(75)90053-4
Peter Volkmann, Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume, Math. Ann. 203 (1973), 201–210 (German). MR 322305, DOI 10.1007/BF01629254
P. Volkman, Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in Banachräumen, Ordinary and partial differential equations (Proc. Conf., Univ. Dundee, Dundee, 1974) Lecture Notes in Math., Vol. 415, Springer, Berlin, 1974, pp. 439–443. MR 0432995
Peter Volkmann, Über die positive Invarianz einer abgeschlossenen Teilmenge eines Banachschen Raumes bezüglich der Differentialgleichung $u’=f(t,u)$, J. Reine Angew. Math. 285 (1976), 59–65 (German). MR 415033, DOI 10.1515/crll.1976.285.59