Oscillation Theory for Functional Differential Equations (L. H. Erbe, Q. K. Kong, and B. G. Zhang)

Abstract In this paper, we present several new and generalized Hermite-Hadamard type inequalities for s-convex as well as s-concave functions via classical and Riemann-Liouville fractional integrals. As applications, we provide new error estimations for the trapezoidal formula.

Some new inequalities of Hermite-Hadamard type for s-convex functions with applications

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

/pdf/inequalities-for-a-fractional-differentiable-functions-23be05i6px.pdf

Inequalities for α-fractional differentiable functions

In this paper, we introduce the generalized k-fractional integral in terms of a new parameter $k>0$, present some new important inequalities of Polya–Szego and Cebysev types by use of the generalized k-fractional integral. Our consequences with this new integral operator have the abilities to implement the evaluation of many mathematical problems related to real world applications.

/pdf/on-polya-szego-and-cebysev-type-inequalities-via-generalized-2zzobyocor.pdf

On Pólya–Szegö and Čebyšev type inequalities via generalized k-fractional integrals

Euler sums and Stirling sums

We obtain the analytical general solution of the linear fractional differential equations with constant coefficients by Adomian decomposition method under nonhomogeneous initial value condition, which is in the sense of the Caputo fractional derivative.

/pdf/solution-to-the-linear-fractional-differential-equation-28zyj8dsaq.pdf

Solution to the Linear Fractional Differential Equation Using Adomian Decomposition Method

In the paper, we develop an approach to evaluation of Euler sums that involve harmonic numbers and alternating harmonic numbers. We give explicit formulae for several classes of Euler sums in terms of Riemann zeta values and prove that the quadratic sums ${S_{{l^2},l}}$ and cubic sums ${S_{{l^3},l}}$ reduce to linear sums and polynomials in zeta values. The approach is based on constructive Power series and Cauchy product computations.

/pdf/some-results-on-euler-sums-14az0z6fw7.pdf

Some results on Euler sums

We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

/pdf/fractional-difference-equations-with-real-variable-4vzh656oxu.pdf

Fractional Difference Equations with Real Variable

Existence of a nonoscillatory solution of a second-order linear neutral difference equation

This paper presents a kind of new definition of fractional difference, fractional summation, and fractional difference equations and gives methods for explicitly solving fractional difference equations of order (2,𝑞).

/pdf/on-the-fractional-difference-equations-of-order-2-q-3t13kfdlum.pdf

On the Fractional Difference Equations of Order (2,q)

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