Study of Nonlocal Multiorder Implicit Differential Equation Involving Hilfer Fractional Derivative on Unbounded Domains

Abstract

This paper aims to study the existence and uniqueness of the solution for nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains a , ∞ , a ≥ 0 , in an applicable Banach space by utilizing the Banach contraction principle. Furthermore, we discuss various types of stability such as Ulam–Hyers–Rassias (UHR), Ulam–Hyers (UH), and semi-Ulam–Hyers–Rassias (sUHR) for nonlocal boundary value problem. Absolutely, our results cover that several outcomes have existed in the literature.

The Iterative Properties for Positive Solutions of a Tempered Fractional Equation

“…Since fractional order differential equations can describe many natural phenomena with long-time behavior such as abnormal dispersion, analytical chemistry, biological sciences, artificial neural network, time-frequency analysis, and so on, the theories of fractional calculus have attracted the attention of a large number of mathematical researchers [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The classical fractional integrals and derivatives are only convolutions by using a power law, such as the Riemann-Liouville fractional derivatives [21][22][23], the Caputo fractional derivatives [24,25], the Hilfer fractional derivative [26], the Atangana-Baleanu-Caputo fractional derivative [27], Hadamard fractional derivatives [28,29], and so on, which fail to model the limits of random walk if they have an exponentially tempered jump distribution [30] exhibiting the semi-heavy tails or semi-long range dependence. Thus, to describe semi-heavy tails or semi-long range dependence, it is necessary to multiply by an exponential factor leading to tempered fractional integrals and derivatives [31,32].…”