Existence of Solutions to a Class of Fractional Differential Equations (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, the existence of solutions to a class of fractionaldifferential equations $D^α_{0+}u(t) = h(t)f(t, u(t), D^θ_{0+}u(t))$ is obtained by an efficient and simple monotone iteration method. At first, the existence of asolution to the problem above is guaranteed by finding a bounded domain $D_M$ on functions $f$ and $g.$ Then, sufficient conditions for the existence ofmonotone solution to the problem are established by applying monotone iteration method. Moreover, two efficient iterative schemes are proposed, andthe convergence of the iterative process is proved by using the monotonicity assumption on $f$ and $g.$ In particular, a new algorithm which combinesGauss-Kronrod quadrature method with cubic spline interpolation method isadopted to achieve the monotone iteration method in Matlab environment,and the high-precision approximate solution is obtained. Finally, the mainresults of the paper are illustrated by some numerical simulations, and theapproximate solutions graphs are provided by using the iterative method.