In the paper, we introduce the generalized convex function on fractal sets (0 1)real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

In this paper, we establish the Hermite-Hadamard-type inequalities for the generalized s-convex functions in the second sense on real linear fractal set R a ð0\a\1Þ:

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with Gaussian kernel bound, and let L −α/2 be the fractional integral of L for 0 < α < n. Suppose that b = (b1, b2, . . . , bm) is a finite family of locally integral functions, then the multilinear commutator generated by b and L −α/2 is defined bywhere m ∈ Z + . When b1, b2, . . . , bm ∈ BMO or bj ∈Λ β j (0 < βj < 1) for 1 ≤ j ≤ m, the authors study the boundedness of L −α/2 b .

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The (2+1)-dimensional Davey-Stewartson-like equations with variable coefficients have the applications in the ultra-relativistic degenerate dense plasmas and Bose-Einstein condensates. Via the Bell polynomials and symbolic computation, the bilinear form, Bäcklund transformation and Lax pair for such equations are obtained. Based on the Hirota method, we construct the soliton solutions, analyze the elastic collisions with the constant and variable coefficients, and observe that solitons no longer keep rectilinear propagation and display different shapes because of the variable coefficients. Besides, localized excitations are derived through the variable separation.