On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution
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 equic...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/10/1972 |
| _version_ | 1797513057967013888 |
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| author | Abdissalam Sarsenbi Abdizhahan Sarsenbi |
| author_facet | Abdissalam Sarsenbi Abdizhahan Sarsenbi |
| author_sort | Abdissalam Sarsenbi |
| collection | DOAJ |
| description | 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>. |
| first_indexed | 2024-03-10T06:10:21Z |
| format | Article |
| id | doaj.art-53909f6f54dd4a81bcea89c1617f1493 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T06:10:21Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-53909f6f54dd4a81bcea89c1617f14932023-11-22T20:11:50ZengMDPI AGSymmetry2073-89942021-10-011310197210.3390/sym13101972On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with InvolutionAbdissalam Sarsenbi0Abdizhahan Sarsenbi1Department of Mathematics, M. Auezov South Kazakhstan University, Shymkent 160000, KazakhstanDepartment of Mathematics, M. Auezov South Kazakhstan University, Shymkent 160000, KazakhstanWe 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>.https://www.mdpi.com/2073-8994/13/10/1972differential equationinvolutionboundary value problemGreen’s functioneigenvalueeigenfunction |
| spellingShingle | Abdissalam Sarsenbi Abdizhahan Sarsenbi On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution Symmetry differential equation involution boundary value problem Green’s function eigenvalue eigenfunction |
| title | On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution |
| title_full | On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution |
| title_fullStr | On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution |
| title_full_unstemmed | On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution |
| title_short | On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution |
| title_sort | on eigenfunctions of the boundary value problems for second order differential equations with involution |
| topic | differential equation involution boundary value problem Green’s function eigenvalue eigenfunction |
| url | https://www.mdpi.com/2073-8994/13/10/1972 |
| work_keys_str_mv | AT abdissalamsarsenbi oneigenfunctionsoftheboundaryvalueproblemsforsecondorderdifferentialequationswithinvolution AT abdizhahansarsenbi oneigenfunctionsoftheboundaryvalueproblemsforsecondorderdifferentialequationswithinvolution |