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Keywords:
differential equations of second order; two-point boundary value problems
differential equations of second order; two-point boundary value problems
Summary:
If $Y$ is a subset of the space $\mathbb{R}^{n}\times {\mathbb{R}^{n}}$, we call a pair of continuous functions $U$, $V$ $Y$-compatible, if they map the space $\mathbb{R}^{n}$ into itself and satisfy $Ux\cdot Vy\ge 0$, for all $(x,y)\in Y$ with $x\cdot y\ge {0}$. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer’s fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points $P_{\delta }$ is obtained. Then passing to the limits as $\delta $ tends to zero the so-obtained accumulation points are solutions of the problem.
References:
[1.] J. W. Bebernes and K. Schmitt: Periodic boundary value problems for systems of second order differential equations. J. Differential Equations 13 (1973), 32–47. DOI 10.1016/0022-0396(73)90030-2 | MR 0340700
[2] W. A. Coppel: Stability and Asymptotic Behavior of Differential Equations. Heath Mathematical Monographs. Heat and Company, Boston, 1965. MR 0190463
[3] L. H. Erbe and P. K. Palamides: Boundary value problems for second order differential equations. J. Math. Anal. Appl. 127 (1987), 80–92. DOI 10.1016/0022-247X(87)90141-7 | MR 0904211
[4] J. K. Hale: Ordinary Differential Equations. Krieger Publ., Malabar, 1980. MR 0587488 | Zbl 0433.34003
[5] L. Jackson and P. K. Palamides: An existence theorem for a nonlinear two-point boundary value problem. J. Differential Equations 53 (1984), 48–66. DOI 10.1016/0022-0396(84)90025-1 | MR 0747406
[6] G. Karakostas and P. K. Palamides: A boundary value problem for operator equations in Hilbert spaces. J. Math. Anal. Appl. 261 (2001), 289–294. DOI 10.1006/jmaa.2001.7523 | MR 1850974
[7] J. Mawhin: Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS Regional Conf. Series No. 40 AMS, Providence, 1979. MR 0525202 | Zbl 0414.34025
[8] J. Mawhin: Continuation theorems and periodic solutions of ordinary differential equations. In: Topological Methods in Differential equations and inclusions. NATO ASI Series Vol. 472, A. Granas (ed.), Kluwer Acad. Publ., Dordrecht, 1995. MR 1368675 | Zbl 0834.34047
[9] Dong Yujun: On solvability of second-order Sturm-Liouville boundary value problems at resonance. Proc. Amer. Math. Soc. 126 (1998), 145–152. DOI 10.1090/S0002-9939-98-04212-9 | MR 1443173
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