On the solution of a boundary value problem associated with a fractional differential equation

Abstract

The problem of the existence and uniqueness of solutions of boundary value problems (BVPs) for a nonlinear fractional differential equation of order 2<α ≤ 3 is studied. The BVP is transformed into an integral equation and discussed by means of a fixed point problem for an integral operator. Conditions for the existence and uniqueness of a fixed point for the integral operator are derived via b‐comparison functions on complete b‐metric spaces. In addition, estimates for the convergence of the Picard iteration sequence are given. An estimate for the Green's function related with the problem is provided and employed in the proof of the existence and uniqueness theorem for the solution of the given problem. Illustrative examples are presented to support the theoretical results.

On the qualitative analysis of the fractional boundary value problem describing thermostat control model via ψ-Hilfer fractional operator

“…Furthermore, electromagnetic waves in a wide range of dielectric media including the susceptibility following a fractional power law are formulated in the framework of integro-differential equations [4]. We can see some recent advances and applications of fractional modelings in several newly published researches such as [5][6][7][8]. Also, in some new papers, the advantages and power of mathematical modeling based on fractional operators are illustrated, and that is why in recent years, many researchers prefer studying real processes and phenomena by applying newly defined versions of fractional operators (see, e.g., [9][10][11][12][13][14][15][16][17][18]).…”

On a time fractional diffusion with nonlocal in time conditions

“…By replacing many differential operators of fractional order with different type of PDEs of integer order, we formulate various types of boundary value problems with fractional order. Let us refer to many papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Proinov type contractions on dislocated b-metric spaces

“…Metric fixed point theory is a field of study that needs an abstract metric framework (see, for instance, [24][25][26][27]). Very recently Proinov [28] proved a fixed point theorem that not only unifies but also generalizes a number of well-known results in the framework of a standard metric space.…”