Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

Abstract

This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

Analysis of p‐Laplacian Hadamard fractional boundary value problems with the derivative term involved in the nonlinear term

“…For more details, see previous studies. [8][9][10][11][12][13][14][15][16][17][18][19] Except that common fractional differential equations involving Riemann Liouville and Caputo fractional derivatives, there are other types of fractional derivatives such as Hadamard, Hilfer, and so on. We note that generally the Riemann Liouville fractional derivative and the Caputo fractional derivative have been studied by many authors.…”

“…Many diverse methods, such as the Krasnoselskii fixed point theorem on cones, the Legett‐Williams fixed point theorem, the upper and lower solution technique, monotone iterative methods, topological degree methods have been used to establish the existence results of solutions for fractional boundary value problems. For more details, see previous studies 8–19 …”

Existence and multiplicity of solutions for boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity

“…Therefore the delay differential equation has a wide range of applications in control theory, signal processing, biology, finance, and many other fields [4,10,12,14,21]. In addition, unlike the above research problems, the fractional differential equations of two terms are studied in [2,3,17,18].…”

Existence theory and stability analysis to a coupled nonlinear fractional mixed boundary value problem

“…Cabada and Kaithoum [14], presented their work in which the considerable model is all concerned about the existence and uniqueness (EU) of positive solution of implicit nonlinear fractional problem with mixed type boundary conditions, which is as follows;…”