α-, β- and γ-duals of the sequence spaces formed by a regular matrix of Tetranacci numbers

The primary objective of this paper is to create a novel infinite Toeplitz matrix by leveraging Tetranacci numbers. This matrix serves as the foundation for defining new sequence spaces denoted as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>c</m:mi><m:mn>0</m:mn></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {c₀(G)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>c</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {c(G)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="normal">ℓ</m:mi><m:mi mathvariant="normal">∞</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {∞(G)} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="normal">ℓ</m:mi><m:mi>p</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {ₚ(G)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>1</m:mn><m:mo>≤</m:mo><m:mi>p</m:mi><m:mo><</m:mo><m:mi mathvariant="normal">∞</m:mi></m:mrow></m:math> {1≤ p<∞} . By utilizing this newly constructed matrix, the paper also explores and examines various algebraic and topological properties inherent to the sequence spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>c</m:mi><m:mn>0</m:mn></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {c₀(G)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>c</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {c(G)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="normal">ℓ</m:mi><m:mi mathvariant="normal">∞</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {∞(G)} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="normal">ℓ</m:mi><m:mi>p</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {ₚ(G)} for values of p within the range of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>1</m:mn><m:mo>≤</m:mo><m:mi>p</m:mi><m:mo><</m:mo><m:mi mathvariant="normal">∞</m:mi></m:mrow></m:math> {1≤ p<∞} . At last, we also prove existence theorem with example for infinite systems of differential equations in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="normal">ℓ</m:mi><m:mi>p</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>G</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {ₚ(G)} .