| Summary: | In this paper, we introduce and systematically analyze the classes of (pre-)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">B</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mrow><mo>(</mo><msub><mi>t</mi><mi>k</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-piecewise continuous almost periodic functions and (pre-)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">B</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mrow><mo>(</mo><msub><mi>t</mi><mi>k</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>t</mi><mi>k</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions.
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