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We study a class of boundary value problems of the impulsive differential equations on whole lines with a non-Carathéodory nonlinearity. Sufficient conditions to guarantee the existence of solutions are established. A new Banach function space X and its relatively compact property of subset of X is proved. An example is given to illustrate the main results.

Solvability of boundary value problems for second order impulsive differential equations on whole line with a non-Carathédory nonlinearity