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Boundary Value Problems with Compatible Boundary Conditions

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Abstract

If Y is a subset of the space ℝn × ℝn, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝn into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 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 δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece

    G. L. Karakostas & P. K. Palamides

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  1. G. L. Karakostas
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  2. P. K. Palamides
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Karakostas, G.L., Palamides, P.K. Boundary Value Problems with Compatible Boundary Conditions. Czech Math J 55, 581–592 (2005). https://doi.org/10.1007/s10587-005-0047-4

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  • DOI: https://doi.org/10.1007/s10587-005-0047-4

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