| Summary: | We introduce a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mi>ϱ</mi></msub></semantics></math></inline-formula>-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">U</mi><mo>=</mo><mi mathvariant="script">B</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msubsup><mi mathvariant="script">A</mi><mrow><mi>i</mi></mrow><mo>*</mo></msubsup><mi mathvariant="script">G</mi><mrow><mo>(</mo><mi mathvariant="script">U</mi><mo>)</mo></mrow><msub><mi mathvariant="script">A</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> Hermitian positive definite matrix, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">A</mi><mn>1</mn></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">A</mi><mn>2</mn></msub></semantics></math></inline-formula>, … <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">A</mi><mi>m</mi></msub></semantics></math></inline-formula> are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrices, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.
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