On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces

Abstract

We provide and prove some new fundamental properties of the Erdélyi–Kober (EK) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces (Lp) and Orlicz spaces (Lφ). We employ these properties with the concept of the measure of noncompactness (MNC) associated with the fixed-point hypothesis (FPT) in solving a quadratic integral equation of fractional order in Lp,p≥1 and Lφ. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems.

Keywords

• MNC )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> measure of noncompactness (<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm11111111111"> <mml:semantics> <mml:mi mathvariant="script">MNC</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>)
• EK ) fractional operator%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> Erdélyi–Kober’s (<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm11111111111112"> <mml:semantics> <mml:mi mathvariant="script">EK</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>) fractional operator
• Orlicz spaces
• FPT )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> fixed-point theorem (<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm1111111111111113"> <mml:semantics> <mml:mi mathvariant="script">FPT</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>)