An upper semicontinuous multivalued map F:X→Y is said to be ε -small if the diameter of F(x) is less than ε for each x∈X . F and G are ε -homotopic if there is an ε -small homotopy H:X×I→Y joining F and G . F:X×[0,∞)→Y is a fine multivalued map if for each ε>0 there is m≥0 such that F|X×[m,∞) is ε -small. Fine multivalued maps F,G:X×[0,∞)→Y are homotopic provided for each ε>0 there is m≥0 such that F|X×[m,∞) is ε -homotopic to G|X×[0,∞) . The main result of the paper is to show a bijective correspondence between shape morphisms from X to Y , X and Y being compact metrizable, and homotopy classes of fine multivalued maps from X×[0,∞) to Y . \u
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