On the existence of positive solutions for generalized fractional boundary value problems

Abstract

The existence of positive solutions is established for boundary value problems defined within generalized Riemann-Liouville and Caputo fractional operators. Our approach is based on utilizing the technique of fixed point theorems. For the sake of converting the proposed problems into integral equations, we construct Green functions and study their properties for three different types of boundary value problems. Examples are presented to demonstrate the validity of theoretical findings.

Positive solutions for generalized two-term fractional differential equations with integral boundary conditions

“…Proof. The proof of part 1 was done, see [48]. To prove the part 2, we have N(t) = [ψ(t) − ψ(0)] and Υ(t) := N(t) N(1) .…”

Green–Haar wavelets method for generalized fractional differential equations

“…Riemann, Liouville, Caputo, Hadamard, Grunwald and Letinkow are the pioneering researchers who have been contributing and publishing extensively about these applications. Meanwhile, the literature has witnessed the appearance of different types of fractional derivatives that improve and generalize the classical fractional operators defined by the above listed authors [20][21][22]. Recently, Katugampola rediscovered a new type of fractional integral operator which covers both Riemann-Liouville and Hadamard operators and represents them in a single form [23,24].…”

On Hyers–Ulam stability of a multi-order boundary value problems via Riemann–Liouville derivatives and integrals

“…In fact, stability of physical phenomena has an old history, and one can find a lot of works in the literature not only in the last century but also before it [27][28][29][30][31][32][33][34][35][36]. During recent decades, considerable attention has been given to the study of the Hyers-Ulam stability of functional differential and integral equations [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54].…”