| Summary: | This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mspace width="0.166667em"></mspace><msup><mi>u</mi><mo>″</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> on the interval [0, 1] with the boundary condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>u</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><msup><mi>u</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mi>u</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mrow></semantics></math></inline-formula> which models a statically bending elastic beam whose two ends are simply supported. Without assuming that the nonlinearity <i>f</i>(<i>x</i>, <i>u</i>, <i>v</i>) is nonnegative, an existence result of positive solutions is obtained under the inequality conditions that |(<i>u</i>, <i>v</i>)| is small or large enough. The discussion is based on the method of lower and upper solutions.
|
|---|