General uniqueness and monotone iterative technique for fractional differential equations

By V. Lakshmikantham, Aghalaya S. Vatsala. Applied Mathematics Letters, 2008

329 citations recorded by the source corpus.

In this paper, the general existence and uniqueness result is proved which exhibits the idea of comparison principle. This result is also valid for fractional differential equations in a Banach space. The well-known monotone iterative technique is then extended for fractional differential equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions.

Subfields: Applied Mathematics, Mechanics of Materials, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/3896c5f4c8640d4e7f2cea59c8454c7764c24c89

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In Applied Mathematics

In Mechanics of Materials

In Modeling and Simulation

Citations in the corpus

Positive solutions for boundary value problems of nonlinear fractional differential equation (2009, background): “On the other hand, some basic theory for the initial value problems of fractional differential equations involving the Riemann–Liouville differential operator has been discussed by Lakshmikantham [4,5], A.”
Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences (2012, background): “Recently, an increasing interest in studying the existence of solutions for boundary value problems of fractional order functional differential equations has been observed [1–10].”
Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions (2014, background): “Lakshmikantham and Vatsala [16, 18] obtained local and global existence of solutions of Riemann-Liouville fractional differential equations and uniqueness of solutions.”
A collocation-shooting method for solving fractional boundary value problems (2010, background): “For example, existence and uniqueness of solution have been investigated in [11,19,20,33].”
Stability for Caputo Fractional Differential Systems (2014, background): “[1–5] investigated the basic theory of initial value problems for fractional differential equations involving Riemann-Liouville differential operators of order 0 < q < 1.”
Existence of solutions for fractional differential equations of order q∈(2,3] with anti-periodic boundary conditions (2010, background): “For more details and examples, see [1, 6, 8, 9, 12–20, 22] and the references therein.”
Existence of solutions for nonlinear fractional three-point boundary value problems at resonance (2011, background): “We refer the reader for more applications to references [1–10, 12–20, 22–26].”
The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces (2010, background): “The theory of fractional differential equations has been extensively studied by many authors [5] [6] [7] [8] [9] [10] [11] [12].”