| Summary: | In this paper, we analyze the Bohr <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-almost periodic type sequences and the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-almost periodic type sequences of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><mi>I</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∅</mo><mo>≠</mo><mi>I</mi><mo>⊆</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></mrow></semantics></math></inline-formula>, <i>X</i> and <i>Y</i> are complex Banach spaces and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is a general binary relation on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>.</mo></mrow></semantics></math></inline-formula> We provide many structural results, observations and open problems about the introduced classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given.
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