Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

Abstract

The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which exercises the procedure of upper and lower solutions. Two sequences of extremal solutions are generated in which one converges to the upper and the other to the corresponding lower solution. The method does not need any prior discretization or collocation for generating the aforesaid two sequences for upper and lower solutions. Further, the aforesaid techniques produce a fruitful combination of upper and lower solutions. To demonstrate our results, we provide some pertinent examples.

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“…Many researchers have studied this topic because it has many applications. Related to this matter, we suggest the recent literature [29][30][31][32][33][34][35] and the references therein.…”

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“…Remark 1. A function v(s) is a solution of the inequality (17), if and only if there exists σ(s) ∈ C(J, R) that satisfies the following conditions:…”

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