Solution analysis for non-linear fractional differential equations

In this research study, two new types of fractional derivatives, the Caputo-Fabrizio derivative that does not involve a singular kernel (like the Riemann–Liouville derivative), which helps in eliminating some drawbacks, such as non-locality, and the Atangana–Baleanu–Caputo ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>ABC</mml:mi></mml:mstyle></mml:mrow></mml:math> ) derivative that uses a non-singular and non-local kernel, offering different characteristics from other fractional derivatives and often leading to more accurate models in certain physical systems, are used. The primary goal of the research is to analyze a non-linear fractional differential equation involving the fractional derivatives of Caputo–Fabrizio and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>ABC</mml:mi></mml:mstyle></mml:mrow></mml:math> admit at least one solution. Fixed point theory is a fundamental tool in mathematical analysis used to prove the existence of solutions to various equations. Hence, in this study, to achieve the desired results, we employ a novel fixed point theory known as the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>F</mml:mi></mml:mstyle></mml:mrow></mml:math> -contraction type. The study also includes required conditions and inequalities that need to be satisfied to ensure that a solution exists. One of these conditions is based on the Lipschitz hypothesis. Therefore, we provide the required conditions and inequalities, based on the Lipschitz hypothesis, to show that solutions to our problem exist. Furthermore, we provide two illustrative examples to support our primary findings.