Positive Solutions for Boundary Value Problems with Fractional Order

Abstract

In this paper we investigate the existence of at least one, two positive solutions by using the Krasnoselskii fixed-point theorem in cones for nonlinear boundary value problem with fractional order.

A fully Hadamard and Erdélyi–Kober-type integral boundary value problem of a coupled system of implicit differential equations

“…Conversely, assume that y satisfies the fractional integral equation (4). We have immediately y(0) = φ 2 (y), and using Lemma 15 and Definition 5, an easy computation yields (5). This completes the proof.…”

“…We note here that most of the work on the topic in the literature is based on Riemann-Liouville-and Caputo-type fractional differential equations; for this, we refer the readers to [1,5,6,10,11]. Another kind of fractional derivative that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [13], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of the Hadamard derivative) contains a logarithmic function of arbitrary exponent.…”

Generalized Monotone Method for Sequential Caputo Fractional Boundary Value Problems

“…In literature most of the existence and uniqueness result for Caputo boundary value problems has been obtained using some kind of fixed point theorem. See [1,2,14,15,17,[19][20][21][22][23][24][25]. In this paper, we have developed generalized monotone iterative technique combined with coupled lower and upper solutions to obtain the existence of coupled minimal and maximal solutions.…”