On generalizations of certain nonlinear integral inequalities and their applications

The aim of this paper is to generalize some integral inequalities of Gronwall–Bellman type. We generalize the results presented by Pachpatte in [Inequalities for Differential and Integral Equations (Academic Press, New York, 1998)] and Abdeldaim in [On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation, J. Egypt. Math. Sci. Lett. 23 (2015) 470–475] and also establish some new forms.

Oscillation of differential equations with non-monotone retarded arguments

Consider the first-order retarded differential equation $$\begin{eqnarray}x^{\prime }(t)+p(t)x({\it\tau}(t))=0,\quad t\geqslant t_{0},\end{eqnarray}$$ where $p(t)\geqslant 0$ and ${\it\tau}(t)$ is a function of positive real numbers such that ${\it\tau}(t)\leqslant t$ for $t\geqslant t_{0}$, and $\lim _{t\rightarrow \infty }{\it\tau}(t)=\infty$. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $\liminf$, is established when the well-known oscillation condition $$\begin{eqnarray}\liminf _{t\rightarrow \infty }\int _{{\it\tau}(t)}^{t}p(s)\,ds>\frac{1}{e}\end{eqnarray}$$ is not satisfied. An example illustrating the result is also given.

Retarded Integral Inequality and its Application in Retarded Differential Equation

Differential equations are important tools in studying of natural science, engineering technology, and the laws of social economic development. It is necessary to seek some new inequalities in order to study of boundedness, uniqueness, stability and boundary value problem of a differential equation. Motivated by Abdeldaim integral inequalities, in this paper, we establish a class of generalized retarded nonlinear Gronwall-Bellman-Type integral inequalities and give upper bound estimation of the unknown function by analysis skills. Finally we give an example to illustrate the effectiveness of our results in estimation of solutions of some differential equations with the initial conditions.

Approximation of solutions to retarded differential equations with applications to population dynamics

We consider a retarded differential equation with applications to population dynamics. We establish the convergence of a finite-dimensional approximations of a unique solution, the existence and uniqueness of which are also proved in the process.

Existence, uniqueness, and regularity of solutions to semilinear retarded differential equations

In the present work, we consider a semilinear retarded differential equation in a Banach space. We first establish the existence and uniqueness of a mild solution and then prove its regularity under different additional conditions. Finally, we consider some applications of the abstract results.

Method of semidiscretization in time to nonlinear retarded differential equations with nonlocal history conditions

This paper deals with the applications of the method of semidiscretization in time to a nonlinear retarded differential equation with a nonlocal history condition. We establish the existence and uniqueness of a strong solution. Finally, we consider some applications of the abstract results.