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

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.

Forced oscillation of fractional differential equations via conformable derivatives with damping term

“…The books [1][2][3][4][5][6] summarize and organize much of fractional calculus and many of theories and applications of fractional differential equations. Many authors have studied the existence and uniqueness of solutions for different types of fractional boundary value problems; see the papers [7][8][9][10][11][12][13][14][15][16][17][18] and the references cited therein.…”

Multiple positive solutions for mixed fractional differential system with p-Laplacian operators

“…In the mathematical context, many mathematicians and applied scholars have studied the fractional differential equation or system in recent years [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In addition, by applying the functional analysis methods such as the lower and upper solutions, monotone iterative techniques, fractional integro-differential equations or singular equations are researched by Dumitru et al [16], Denton et al [17], Lyons and Neugebauer [18], Ambrosio [19], Zhou and Qiao [20]. There are also related books [21,22].…”

Comparison principles for fractional differential equations with the Caputo derivatives

“…Meanwhile, they presented the Gronwall inequalities for the Riemann-Liouville-type fractional differential equations. Many other researches can be seen in [9][10][11][12][13][14][15][16] and related references therein. However, for the Riemann-Liouville case, one would have to specify the values of certain fractional derivatives (and integrals) of the unknown solution at the initial point.…”