| Summary: | We discuss the solvability of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-difference equation of fractional order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>, equipped with anti-periodic boundary conditions involving the first-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-difference operator. The desired results are accomplished with the aid of standard fixed point theorems. Examples are presented for illustrating the obtained results.
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