On the Enhanced New Qualitative Results of Nonlinear Integro-Differential Equations

In this article, a class of scalar nonlinear integro-differential equations of first order with fading memory is investigated. For the considered fading memory problem, we discuss the effects of the memory over all the values of the parameter in the kernel of the equations. Using the Lyapunov-Krasovski functional method, we give various sufficient conditions of stability, asymptotic stability, uniform stability of zero solution, convergence and boundedness, and square integrability of nonzero solutions in relation to the considered scalar nonlinear integro-differential equations for various cases. As the novel contributions of this article, the new scalar nonlinear integro-differential equation with the fading memory is firstly investigated in the literature, and seven theorems, which have novel sufficient qualitative conditions, are provided on the qualitative behaviors of solutions called boundedness, convergence, stability, integrability, asymptotic stability and uniform stability of solutions. The novel outcomes and originality of this article are that the considered integro-differential equations are new mathematical models, they include former mathematical models in relation to the mathematical models of this paper as well as the given main seven qualitative results are also new. The outcomes of this paper enhance some present results and provide new contributions to the relevant literature. The results of the article have complementary properties for the symmetry of integro-differential equations.