Opuscula Math. 37, no. 5 (2017), 705-724
http://dx.doi.org/10.7494/OpMath.2017.37.5.705

Opuscula Mathematica

# Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems

Z. Denton
J. D. Ramírez

Abstract. In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.

Keywords: Riemann Liouville derivative, integro-differential equation, monotone method.

Mathematics Subject Classification: 26A33, 34A08, 34A45, 45J05.

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• Z. Denton
• North Carolina A&T State University, Department of Mathematics, Greensboro, NC, 27411 USA
• J. D. Ramírez
• Savannah State University, Department of Mathematics, Savannah, GA 31404, USA
• Communicated by Theodore A. Burton.
• Revised: 2016-12-21.
• Accepted: 2016-12-23.
• Published online: 2017-07-05. 