The main goal of this paper is to give an answer to the main problem of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we give necessary and sufficient conditions which guarantee that nonautonomous superposition operators map that space into itself and are locally bounded. Moreover, special attention is drawn to nonautonomous superposition operators generated by locally bounded mappings as well as to superposition operators generated by functions with separable variables.
We discuss the existence or the existence and uniqueness of global and local Λ-bounded variation (ΛBV) solutions as well as continuous ΛBV-solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations formulated in terms of the Lebesgue integral. Since the space of functions of bounded variation in the sense of Jordan is a proper subspace of functions of Λ-bounded variation and for some class of functions φ, the space of functions of bounded φ-variation in the sense of Young is also a proper subspace of the space under consideration, our results extend known results in the literature.
In the paper we introduce the new concept of variation which allows to work in the spaces of functions measurable in the Lebesgue sense. We define the Banach space ΛBV [0, 1] and we provide some of its basic geometric and topological properties. We also define the useful notion of good representatives of functions generating the suitable equivalence classes from the space ΛBV [0, 1] and raise the question of their existence as well as their properties. Moreover, we investigate convolution and superposition operators acting in ΛBV [0, 1] and give some applications to linear differential and nonlinear integral equations.
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