In this paper we consider the following nth-order neutral delay differential equation:By employing the contraction mapping principle, we obtain several existence results of nonoscillatory solutions for the above equation, construct a few Mann-type iterative approximation schemes for these nonoscillatory solutions and establish several error estimates between the approximate solutions and the nonoscillatory solutions. In addition, we obtain some sufficient conditions for the existence of infinitely many nonoscillatory solutions. These results presented in this paper extend, improve and unify many known results due to Cheng and Annie [J.F. Cheng, Z. Annie, Existence of nonoscillatory solution to second order linear neutral delay equation, differential equations with positive and negative coefficients, Math. Bohem. 124 (1999) 87-102], Kulenović and Hadžiomerspahić [M.R.S. Kulenović, S. Hadžiomerspahić, Existence of nonoscillatory solution of second order linear neutral delay equation, J. Math. Anal. Appl. 228 (1998) 436-448; M.R.S. Kulenović, S. Hadžiomerspahić, Existence of nonoscillatory solution for linear neutral delay equation, Fasc. Math. 32 (2001) 61-72], Zhang and Yu [B.G. Zhang, J.S. Yu, On the existence of asymptotically decaying positive solutions of second order neutral differential equations, Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients, Appl. Math. Lett. 15 (2002) 867-874] and others. Some nontrivial examples are given to illustrate the advantages of our results.

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In this article, we have established the Hadamard inequalities for strongly convex functions using generalized Riemann–Liouville fractional integrals. The findings of this paper provide refinements of some fractional integral inequalities. Furthermore, the error bounds of these inequalities are given by using two generalized integral identities.

We introduce and study a new class of generalized nonlinear variational-like inequalities and prove an existence theorem of solutions for this kind of generalized nonlinear variational-like inequalities. By using the auxiliary principle technique, we construct a new iterative scheme for solving the class of the generalized nonlinear variational-like inequalities. The convergence of the sequence generated by the iterative algorithm is also discussed. Our results extend and unify the corresponding results due to Ding, Liu, Ume, Kang, Yao, and others.

Fractional integral operators are useful tools for generalizing classical integral inequalities. Convex functions play very important role in the theory of mathematical inequalities. This paper aims to investigate the Hadamard type inequalities for a generalized class of functions namely strongly (α,h−m)-p-convex functions by using Riemann–Liouville fractional integrals. The results established in this paper give refinements of various well-known inequalities which have been published in the recent past.