An application of Schauder’s fixed point theorem with respect to higher order BVPs

Wong, Fu-Hsiang

doi:10.1090/S0002-9939-98-04709-1

An application of Schauder’s fixed point theorem with respect to higher order BVPs
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Proc. Amer. Math. Soc. 126 (1998), 2389-2397 Request permission

Abstract:

We shall provide conditions on the function $f(t,u_{1},\cdots , u_{n-1})$. The higher order boundary value problem \begin{equation*}\begin {cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\mathrm {~for~}~~~~~t\in (0,1)~~~~\mathrm {and}~~~~~~n\ge 2,\ (BC)~~ \begin {cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag {{BVP}}\end{equation*} has at least one solution.

References

R. P. Agarwal and F. H. Wong, Existence of positive solutions for higher order boundary value problems, Nonlinear Studies, to appear.
R. P. Agarwal and F. H. Wong, Existence of positive solutions for non-positive higher order BVP’s, to appear in Computational and Applied Mathematics.
R. P. Agarwal and F. H. Wong, An application of topological transervality with respect to non-positive higher order BVP’s, to appear.
V. Anuradha, D. D. Hai, and R. Shivaji, Existence results for superlinear semipositone BVP’s, Proc. Amer. Math. Soc. 124 (1996), no. 3, 757–763. MR 1317029, DOI 10.1090/S0002-9939-96-03256-X
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
C. Bandle and Man Kam Kwong, Semilinear elliptic problems in annular domains, Z. Angew. Math. Phys. 40 (1989), no. 2, 245–257 (English, with French and German summaries). MR 990630, DOI 10.1007/BF00945001
Y. S. Choi and G. S. S. Ludford, An unexpected stability result of the near-extinction diffusion flame for nonunity Lewis numbers, Quart. J. Mech. Appl. Math. 42 (1989), no. 1, 143–158. MR 992796, DOI 10.1093/qjmam/42.1.143
Chuan Jen Chyan and Johnny Henderson, Positive solutions for singular higher order nonlinear equations, Differential Equations Dynam. Systems 2 (1994), no. 2, 153–160. MR 1386045
E. N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations 37 (1980), no. 3, 404–437. MR 590000, DOI 10.1016/0022-0396(80)90107-2
H. Dang and K. Schmitt, Existence of positive solutions for semilinear elliptic equations in annular domains, Differential Integral Equations 7 (1994), no. 3-4, 747–758. MR 1270101
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
James Dugundji and Andrzej Granas, Fixed point theory. I, Monografie Matematyczne [Mathematical Monographs], vol. 61, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982. MR 660439
L. H. Erbe and Haiyan Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), no. 3, 743–748. MR 1204373, DOI 10.1090/S0002-9939-1994-1204373-9
Xabier Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), no. 1, 69–92. MR 904816, DOI 10.1016/0022-0396(87)90169-0
A. Granas, R. B. Guenther, and J. W. Lee, On a theorem of S. Bernstein, Pacific J. Math. 74 (1978), no. 1, 67–82. MR 470310, DOI 10.2140/pjm.1978.74.67
A. Granas, R. B. Guenther, and J. W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math. 10 (1980), no. 1, 35–58. MR 573860, DOI 10.1216/RMJ-1980-10-1-35
Johnny Henderson, Singular boundary value problems for difference equations, Dynam. Systems Appl. 1 (1992), no. 3, 271–282. MR 1182649
Petio Kelevedjiev, Existence of solutions for two-point boundary value problems, Nonlinear Anal. 22 (1994), no. 2, 217–224. MR 1258957, DOI 10.1016/0362-546X(94)90035-3
Petio Kelevedjiev, Nonexistence of solutions for two-point boundary value problems, Nonlinear Anal. 22 (1994), no. 2, 225–228. MR 1258958, DOI 10.1016/0362-546X(94)90036-1
M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
J. W. Lee and D. O’Regan, Nonlinear boundary value problems in Hilbert spaces, J. Math. Anal. Appl. 137 (1989), no. 1, 59–69. MR 981923, DOI 10.1016/0022-247X(89)90273-4
J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, NSFCB Regional Conference Series in Math., vol. No. 40, Amer. Math. Soc., Providence. R. I., 1979.
N. Vasilev and I. Klokov, Bases of the theory of boundary value problems for ordinary differential equations, Izdat “Zinatne.” Riga, 1978.
Haiyan Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations 109 (1994), no. 1, 1–7. MR 1272398, DOI 10.1006/jdeq.1994.1042
Fu Hsiang Wong, Existence of positive solutions of singular boundary value problems, Nonlinear Anal. 21 (1993), no. 5, 397–406. MR 1237131, DOI 10.1016/0362-546X(93)90083-5