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In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes are based on one or another additive splitting of the operator into "simpler" operators that are more convenient/easier for the computer implementation and use inhomogeneous (explicitly-implicit) time approximations. In this paper, a new class of splitting schemes is proposed that is characterized by an additive representation of the solution instead of the operator corresponding to the problem (called problem operator). A specific feature of the proposed splitting is that the resulting coupled equations for individual solution components consist of the time derivatives of the solution components. The proposed approaches are motivated by various applications, including multiscale methods, domain decomposition, and so on, where spatially local problems are solved and used to compute the solution. Unconditionally stable splitting schemes are constructed for a first-order evolution equation, which is considered in a finite-dimensional Hilbert space. In our splitting algorithms, we consider the decomposition of both the main operator of the system and the operator at the time derivative. Our goal is to provide a general framework that combines temporal splitting algorithms and spatial decomposition and its analysis. Applications of the framework will be studied separately.