The Solvability of the Discrete Boundary Value Problem on the Half-Line
This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ&...
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/f0be19e74b154aeea8eaec1dd582a2d6 |
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| Sumario: | This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mfenced separators="" open="(" close=")"><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>Δ</mo><mi>x</mi><mo>(</mo><mi>n</mi><mo>)</mo></mfenced><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mo>Δ</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>β</mi><mi>a</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>Δ</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="1.em"></mspace><mi>x</mi><mo>(</mo><mo>∞</mo><mo>)</mo><mo>=</mo><mi>d</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem. |
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