Abstract
We show that systems of second-order ordinary differential equations, , subject to compatible nonlinear boundary conditions and impulses, have a solution x such that lies in an admissible bounding subset of when f satisfies a Hartman-Nagumo growth bound with respect to . 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, , 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.
Highlights
Let q ∈ N, the natural numbers,Q = {t, . . . , tq : = t < t < · · · < tq < tq+ = }.J = [t, t ] and Jk =
We consider the system of second-order ordinary differential equations x = f t, x, x, t ∈ [, ] \ Q
Where f : [, ] × R n → Rn satisfies f |Jk×R n has an extension to fk ∈ C(Jk × R n; Rn) and gk = ∈ C R n × R n; R n for ≤ k ≤ q
Summary
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 ( ). 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. In Section , we present the Nagumo-type condition that we use in our existence result to a priori bound the derivative of solutions.
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