In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H∈(0,1/2)∪(1/2,1). We prove the well-posedness type equations, and then establish general result on Bismut formula for Lions derivative using Malliavin calculus. As applications, provide formulas kind both non-degenerate degenerate cases, obtain e...

This study outlines the local fractional integro-differential equations carried out by the local fractional calculus. The analytical solutions within local fractional Volterra and Abel’s integral equations via the Yang-Laplace transform are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application t...

The aim of the present paper is to introduce and investigate a new subclass of Bazilevi'{c} functions in the punctured unit disk $mathcal{U}^*$ which have been described through using of the well-known fractional $q$-calculus operators, Hadamard product and a linear operator. In addition, we obtain some sufficient conditions for the func...

We present a class of nonlinear adaptive image restoration filters which may be steered to preserve sharp edges and contrasts in the restorations. From a theoretical point of view we discuss the associated variational problems and prove existence of solutions in certain Sobolev spaces W 1,p or in a BV -space. The degree of regularity of the solution may be understood as a mathematical explanati...

The aim of this paper is to establish the existence of solutions of boundary value problems of nonlinear fractional integro-differential equations involving Caputo fractional derivative by using the techniques such as fractional calculus, H"{o}lder inequality, Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type. Examples are exhibited to illustrate the main resu...

material, was first recorded in ‘Shennong Bencao Jing’ and has been used clinically in China for 2000 years. It is the dry gallstone of Bos taurus domesticus GMELIN and, according to the theory of traditional Chinese medicine (TCM), has the effects of sedation, anti-hyperspasmia, relieving fever, diminishing inflammation and normalizing function of the gallbladder. Because of its scarcity of na...