Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity

Abstract

In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ⁢ ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ⁢ ( x ) {u(x)} is so-called ( α , p ) {(\alpha,p)} -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) {(\alpha,p)} -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

“…In [21], Zakarya et al provided novel generalizations by considering the generalized conformable fractional integrals for reverse Copson's type inequalities on time scales. For some other applications of fractional calculus, the reader is referred to [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. This paper deals with the following sub-diffusion model with a changing-sign perturbation.…”

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.…”