Abstract
We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.
Highlights
Differential equations of fractional order have recently been proven to be valuable tools in the modeling of many physical phenomena [1,2,3]
The basic theory for initial value problems for fractional differential equations involving the Riemann–Liouville and Liouville–Caputo differential operator was discussed by Diethelm [9]
We study the existence of solutions of the following implicit fractional differential equation with initial condition:
Summary
Differential equations of fractional order have recently been proven to be valuable tools in the modeling of many physical phenomena [1,2,3]. We study the existence of solutions of the following implicit fractional differential equation with initial condition:. = y0 where J = [0, b], 0 < α < 1 and f : J × R × R → R is a continuous function This problem is motivated by the importance of implicit ordinary differential equations of the form:. Our intention is to extend the results to implicit differential equations of fractional order. Some existence results for an implicit fractional differential equation on compact intervals were investigated [27,28,29]. Our goal in this work is to give some existence and uniqueness results for implicit fractional differential equations
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