Abstract
In this article, we employ the Tarski's fixed point theorem to establish the existence of extremal solutions for fractional differential equations with maxima.
Highlights
Fractional calculus has become an exciting new mathematical method of solution of diverse problems in mathematics, science, and engineering
The theory and applications of fractional differential equations received in recent years considerable interest both in pure mathematics and in applications
Whereas in mathematical treatises on fractional differential equations the Riemann-Liouville approach to the notion of the fractional derivative is normally used [3,4,5], the Caputo fractional derivative often appears in applications [6], Erdèlyi-Kober fractional derivative [7] and The WeylRiesz fractional operators [8]
Summary
Fractional calculus has become an exciting new mathematical method of solution of diverse problems in mathematics, science, and engineering. There are some advantages in studying the extremal solution for fractional differential equations, because some boundary conditions are automatically fulfilled and due to lower order differential requirements (see [9]). We establish the extreme solutions (maximal and minimal solutions) for fractional differential equation with maxima in sense of Riemann-Liouville fractional operators, by using the Tarski’s fixed point theorem. We extend the existence of extremal solutions from initial value problems to boundary value problems for infinite quasi-monotone functional systems of fractional differential equations.
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