INVESTIGATING A COUPLED SYSTEM WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE BY MODIFIED MONOTONE ITERATIVE TECHNIQUE

In recent time, the area of arbitrary order differential equations (AODEs)
has been considered very well. Different aspects have been investigated for the said
area. One of the important and most warm area is devoted to study multiplicity
results along with existence and uniqueness of solutions for the said equations. In this
regard various techniques have been utilized to investigate the said area. Monotone
iterative technique (MIT) coupled with the method of extremal solutions has been
used recently to investigate multiplicity of solutions to some AODEs. In this research
work, we deal a coupled system of nonlinear AODEs under boundary conditions (BCs)
involving Riemann-Liouville fractional derivative by using fixed point theorems due
to Perov’s and Schuader’s to study existence and uniqueness results. Using Perove’s
fixed point theorem ensures uniqueness of solution to systems of equations, while
existence of at least one solution is achieved by Schauder’s fixed point theorem.
Then we come across the multiplicity of solutions and establish some criteria for the
iterative solutions via using updated type MIT together with the method of upper and
lower solutions for the considered system of AODEs. Corresponding to multiplicity
results of solutions, we first establish two sequences of extremal solutions. One of the
sequence is monotonically decreasing and converging to lower solution. On the other
hand, the other sequence is monotonically increasing and converging to the upper
solution. In last we give suitable examples to illustrate the main results.