In this research paper, we investigate generalized fractional integrals to obtain midpoint type inequalities for the co-ordinated convex functions. First of all, we establish an identity for twice partially differentiable mappings. By utilizing this equality, some midpoint type inequalities via generalized fractional integrals are proved. We also show that the main results reduce some midpoint inequalities given in earlier works for Riemann integrals and Riemann-Liouville fractional integrals. Finally, some new inequalities for $k$-Riemann-Liouville fractional integrals are presented as special cases of our results.
We focus on a new type of nonlinear integro-differential equations with nonlocal integral conditions. The considered problem has one nonlinearity with time variable singularity. It involves also some convergent series combined to Riemann-Liouville integrals. We prove a uniqueness of solutions for the proposed problem, then, we provide some examples to illustrate this result. Also, we discuss the Ulam-Hyers stability for the problem. Some numerical simulations, using Rung Kutta method, are discussed too. At the end, a conclusion follows.
In this work, we are concerned with a sequential nonlinear random differential equation of fractional order with nonlocal conditions. This is the first time in the literature where sequential problems and random ones are combined and considered. An existence and uniqueness of solutions for the problem is obtained by means of an appropriate random fixed point theorem. Then, new concepts on the sequential continuous and fractional derivative dependence are introduced. At the end, some results of stability on random, as well for deterministic, data dependence are discussed.
With respect to the non-integro-fractional derivative, in previous studies, the non-integro-fractional derivative of non-negative real numbers can be calculated. However, by previous denitions, the non-integro-fractional de- rivative of negative values can not be calculated due to t; 2 (0; 1). For example, (2)12 =2 R for t = 2 and = 1 2 : So what should we do for the non-integro-fractional derivative of “negative” real numbers? The pur- pose of this paper is to introduce more general derivative denition and we claim that we will obtain non-integro-fractional derivative of “all” real num- bers. Classic derivative, q-derivative, (p; q)-derivative, comformable fractional derivative, Katugampola fractional derivative and backward-forward dierence operator in Time Scale are the special cases of these general derivative deni- tions. These new denitions of ours must give us derivatives on both discrete and continuous calculus.