| Summary: | This article is concerned with the description of the entire solutions of several Fermat type partial differential-difference equations (PDDEs) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="[" close="]"><mi>μ</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>λ</mi><msub><mi>f</mi><msub><mi>z</mi><mn>1</mn></msub></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>[</mo><mi>α</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>+</mo><mi>c</mi><mo>)</mo></mrow><mo>−</mo><mi>β</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>]</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="[" close="]"><mi>μ</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub><msub><mi>f</mi><msub><mi>z</mi><mn>1</mn></msub></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><msub><mi>f</mi><msub><mi>z</mi><mn>2</mn></msub></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>[</mo><mi>α</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>+</mo><mi>c</mi><mo>)</mo></mrow><mo>−</mo><mi>β</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>]</mo></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><msub><mi>z</mi><mn>1</mn></msub></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><msub><mi>z</mi><mn>2</mn></msub></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mfrac></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><mrow><mo>(</mo><msub><mi>c</mi><mn>1</mn></msub><mo>,</mo><msub><mi>c</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>∈</mo><msup><mi mathvariant="double-struck">C</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, <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><mi>μ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>c</mi><mn>1</mn></msub><mo>,</mo><msub><mi>c</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> are constants in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>. Our theorems in this paper give some descriptions of the forms of transcendental entire solutions for the above equations, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, and Yang. In particular, we exhibit a series of examples to explain that the existence conditions and the forms of transcendental entire solutions with a finite order of such equations are precise.
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