In this paper, the existence and localization result will be proven for multivalued vector Dirichlet problem with an upper-Carathéodory right-hand side by using bound sets approach. Since Scorza-Dragoni type technique will be furthermore applied, the conditions for bounding functions can be required directly on the boundaries of bound sets and not at some vicinity of them.

In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

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