| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with unity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula> be a triangular algebra over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>. Let a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>=</mo><msub><mrow><mo>{</mo><msub><mi>δ</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></msub></mrow></semantics></math></inline-formula> of nonlinear mappings <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>n</mi></msub><mo>:</mo><mi mathvariant="script">T</mi><mo>→</mo><mi mathvariant="script">T</mi></mrow></semantics></math></inline-formula> is a Lie triple higher derivation by local actions satisfying the equation. Under some mild conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula>, we prove in this paper that every Lie triple higher derivation by local actions on the triangular algebras is proper. As an application, we shall give a characterization of Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras, respectively.
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