Positive solutions for fractional boundary value problems under a generalized fractional operator

Abstract

The work reported here concerns the study of a generalized nonlinear fractional boundary value problem involving ϑ‐fractional derivative in the Riemann–Liouville sense. The existence and uniqueness of positive solutions to the problem at hand are proved. Our discussion relies on the properties of Green's function, the upper and lower solutions method, and the classical fixed point theorems in a cone. Moreover, building upper and lower control functions has an effective role in the analysis. Some examples are given to justify the validity of theoretical results.

Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

“…Therefore, for any u ∈ W α,p ∆,a + \ {0}, ξ ∈ R + , it follows from ( 27), ( 30), (51), and µ > p 2 that…”

“…There have been many results using critical point theory to study boundary value problems of fractional differential equations [46][47][48][49][50][51][52] and dynamic equations on time scales [53][54][55][56][57], but results using critical point theory to study boundary value problems of fractional dynamic equations on time scales are still rare [6]. This section will explain that critical point theory is an effective way to deal with the existence of solutions of (26) on time scales.…”

Left fractional Sobolev space via Riemann$-$Liouville derivatives on time scales and its application to a fractional boundary value problem on time scales

“…There have been many results using critical point theory to study boundary value problems of fractional differential equations ( [37][38][39][40][41][42][43]) and dynamic equations on time scales ( [44][45][46][47][48]), but the results of using critical point theory to study boundary value problems of fractional dynamic equations on time scales are still rare [6]. This section will explain that critical point theory is an effective way to deal with the existence of solutions of (5.1) on time scales.…”

Right Fractional Sobolev Space via Riemann–Liouville Derivatives on Time Scales and an Application to Fractional Boundary Value Problem on Time Scales

“…Although many excellent results have been obtained based on the existence of solutions for fractional boundary value problems [34][35][36][37][38][39][40] and the second-order Hamiltonian systems on time scale T [41][42][43][44][45], it seems that no similar results have been obtained in the literature for FBVP (26) on time scales. The present section seeks to show that the critical point theory is an effective approach to deal with the existence of solutions for FBVP Theorem 26 on time scales.…”