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RESEARCH

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Systems of differential equations with implicit impulses and fully nonlinear boundary conditions Yawei Song and Bevan Thompson* * Correspondence: [email protected] School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia

Abstract We show that systems of second-order ordinary differential equations, x = f (t, x, x ), subject to compatible nonlinear boundary conditions and impulses, have a solution x such that (t, x(t)) lies in an admissible bounding subset of [0, 1] × Rn when f satisfies a Hartman-Nagumo growth bound with respect to x . We reformulate the problem as a system of nonlinear equations and apply Leray-Schauder degree theory. We compute the degree by homotopying to a new system of nonlinear equations based on the simpler system of ordinary differential equations, x = M0 L(x – v), subject to Picard boundary conditions and impulses and using the Leray index theorem. Our proof is simpler than earlier existence proofs involving nonlinear boundary conditions without impulses and requires weak assumptions on f . MSC: 34A37; 34A34; 34B15 Keywords: boundary value problems; nonlinear boundary conditions; impulses; Leray-Schauder degree

1 Introduction Let q ∈ N, the natural numbers, Q = {t , . . . , tq :  = t < t < · · · < tq < tq+ = }. J = [t , t ] and Jk = (tk , tk+ ] for  ≤ k ≤ q. We call Q a division of the interval [, ]. We consider the system of second-order ordinary differential equations   x = f t, x, x ,

t ∈ [, ] \ Q

()

subject to very general nonlinear boundary conditions of the form   g x(), x(), x (), x () = (, )

()

and very general nonlinear implicit impulses of the form          gk x tk+ , x tk– , x tk+ , x tk– = (, ),

k = , . . . , q,

()

©2013Song and Thompson; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Song and Thompson Boundary Value Problems 2013, 2013:240 http://www.boundaryvalueproblems.com/content/2013/1/240

where f : [, ] × Rn → Rn satisfies f |Jk ×Rn has an extension to fk ∈ C(J¯k × Rn ; Rn ) and   gk = (gk, , gk, ) ∈ C Rn × Rn ; Rn for  ≤ k ≤ q. Our fully nonlinear boundary conditions () include the Picard, periodic, and Neumann boundary conditions as special cases. We establish a general existence result for solutions lying in an admissible bounding set for the system of ordinary differential equations () satisfying boundary conditions () and impulses (). Our result is closely related to those of Thompson [] and of Kongson et al. []. In [] and [], the authors established existence results for systems of second-order ordinary differential equations in more general bounding sets and subject to general boundary conditions () but not subject to impulses. Moreover, the proof in [] is incomplete as it f