Interval Abel integral equation
The authors study the solvability of the interval Abel integral equation of the form: \[ \frac{1}{\Gamma(\alpha)} \int\limits_a^t (t-s)^{\alpha-1} X(s) \text{d}s = Y(t) ,\, t \in [a, b] \tag{1} \] where \(\alpha \in (0, 1)\), \(K\) is the set of any nonempty compact intervals of the real line \(\mathbb{R}\), \(Y(\cdot) : [a, b] \to K\) being a given interval-valued function and \(X(\cdot) \in L^1([a,b],K)\) is an unknown interval-valued function. The aim of the work is to obtain sufficient conditions for the interval-valued function \(Y(\cdot)\) to ensure that the interval Abel integral equation (1) is uniquely solvable. As first step it is proved that if \(Y(\cdot) \in L^1([a,b],K)\) and the equation (1) has a solution \(X(\cdot) \in L^1([a,b],K)\), then \(Y_{1-\alpha}(\cdot)\) is AC on \([a, b]\), \(Y_{1-\alpha}(a)=\theta\) and \(X(t)=Y_{1-\alpha}'(t)\) for a.e. \(t \in [a, b]\). Here by \(Y_{1-\alpha}(\cdot)\) the interval-valued Riemann-Liouville fractional integral of order \(1-\alpha > 0\) is denoted. The main result about solvability of (1) in the work is proved under the following assumption: Let \(Y(\cdot) \in L^1([a,b],K)\) be an interval-valued function such that \(Y_{1-\alpha}(a)=\theta\), \(Y_{1-\alpha}(\cdot)\) is AC and \(w\)-increasing on \([a, b]\). Then the integral equation (1) has a unique solution \(X(\cdot) \in L^1([a,b],K)\). Several appropriate examples illustrating and clarifying the obtained results are given.
interval-valued Riemann-Liouville fractional integral
scientific article; zbMATH DE number 6844029
scientific article