The Study of Fractional Quadratic Integral Equations Involves General Fractional Integrals
This paper investigates the well-posedness of analytical solutions to fractional quadratic differential equations, which involve generalized fractional integrals with respect to other functions. The nonlinear components <i>f</i> and <i>h</i> depend on spatial variables and th...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/4/249 |
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| Summary: | This paper investigates the well-posedness of analytical solutions to fractional quadratic differential equations, which involve generalized fractional integrals with respect to other functions. The nonlinear components <i>f</i> and <i>h</i> depend on spatial variables and the general fractional integral, respectively. We use the operator equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mn>1</mn></msub><mi>ω</mi><msub><mi>T</mi><mn>2</mn></msub><mi>ω</mi><mo>+</mo><msub><mi>T</mi><mn>3</mn></msub><mi>ω</mi><mo>=</mo><mi>ω</mi></mrow></semantics></math></inline-formula> to investigate the existence of solutions, marking the first study of its kind. Using an auxiliary function and Boyd and Wang’s fixed-point theorem, the uniqueness and continuous dependence of the solution are obtained. In particular, we applied nonlinear functional analysis to investigate Hyers-Ulam and Hyers-Ulam-Rassias stabilities for fractional quadratic integral equations. New results are provided for specific values of the parameter <i>z</i>, and a fundamental inequality is formulated to ensure the existence of maximal and minimal solutions. Some examples are given to illustrate our results. |
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| ISSN: | 2504-3110 |