| Summary: | We give a definition of Green’s function of the general boundary value problems for non-self-adjoint second order differential equation with involution. The sufficient conditions for the basis property of system of eigenfunctions are established in the terms of the boundary conditions. Uniform equiconvergence of spectral expansions related to the second-order differential equations with involution:<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msup><mi>y</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><msup><mi>y</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>q</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mi>y</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><mi>λ</mi><mi>y</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>,</mo><mo>−</mo><mn>1</mn><mo><</mo><mi>x</mi><mo><</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> with the boundary conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mfenced separators="" open="(" close=")"><mrow><mo>−</mo><mn>1</mn></mrow></mfenced><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><mi>y</mi><mfenced separators="" open="(" close=")"><mrow><mo>−</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><msup><mi>y</mi><mo>′</mo></msup><mfenced open="(" close=")"><mn>1</mn></mfenced><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub><mi>y</mi><mfenced open="(" close=")"><mn>1</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> is obtained. As a corollary, it is proved that the eigenfunctions of the perturbed boundary value problems form the basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for any complex-valued coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∈</mo><mrow><msub><mi>L</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow></semantics></math></inline-formula>.
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