| Summary: | The double-controlled metric-type space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">D</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a metric space in which the triangle inequality has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>≤</mo><msub><mi>ζ</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>ζ</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. The maps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ζ</mi><mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>≤</mo><mi>α</mi><mfenced separators="" open="(" close=")"><mi mathvariant="script">D</mi><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mfenced><mo>+</mo><mi>β</mi><mfenced separators="" open="(" close=")"><mi mathvariant="script">D</mi><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mfenced></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. In our new space, the control functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula> are composed of the metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula> in the triangle inequality, where the control functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ζ</mi><mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in a double-controlled metric-type space are multiplied with the metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula>. We establish some fixed-point theorems along with the examples and applications.
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