Abstract
This paper presents a new type of Gronwall-Bellman inequality, which arises from a class of integral equations with a mixture of nonsingular and singular integrals. The new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order on (0, 1). This result generalizes the existence and uniqueness theorem related to fractional order (1/2 1) appearing in [1]. Finally, the fractional type Fokker-Planck-Kolmogorov equation associated to the solution of the fractional SDE is derived using It^o's formula.
Highlights
It is well known that integral inequalities are instrumental in studying the qualitative analysis of solutions to differential and integral equations (Ames & Pachpatte, 1997)
For a mathematical model which arises from a class of integral equations with a mixture of nonsingular and singular integrals, there is lack of a powerful Gronwall-Bellman inequality to help researchers on this case
To derive such a Gronwall-Bellman inequality, the new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order 0 < α < 1
Summary
It is well known that integral inequalities are instrumental in studying the qualitative analysis of solutions to differential and integral equations (Ames & Pachpatte, 1997). The first goal of this paper, presented, is to derive a new type of Gronwall-Bellman inequality which is applicable to study the qualitative behaviors of the solution to the fractional SDE (Equation (1)) or the stochastic integral equation (Equation (2)). Since the Mittag-Leffler function E (t ) is an entire function in t , see Gorenflo, Loutchko, Luchko, and Mainardi (2002), the exponential function exp(t) is uniformly continuous in t, and both t −1 and a(t) are locally integrable over 0 ≤ t < T, the integral ∫0t dL(M;t−s) a(s) ds is finite This implies that the RHS of dt Equation (4) is finite.
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