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

Abstract

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.

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

“…In addition to the above information, numerous different forms of IDEs with and without delay have been taken under consideration in previous studies [28][29][30][31][32][33]. Fundamental properties of solutions of those IDEs have been investigated by the LKM or some the other methods.…”

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

“…Furthermore, several attractive outcomes concerning applications of fixed point methods, various fixed point theorems, and EUSs, several other qualitative properties with regard to certain IEs and IDEs can be found in the books of Abbas et al [18] and Burton [19], and the papers of Chauhan et al [20], Deep et al [21], Graef et al [22], Lungu and Rus [23], Khan et al [24,25], Tunç and Tunç [26,27], Tunç et al [27,28], and their references.…”

Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel

“…It also includes relevant theorems as well as numerical applications. In the paper [17], Tunç et al consider a class of scalar nonlinear integro-differential equations with fading memory. The article is devoted to examining concepts such as convergence, asymptotic stability, uniform stability, boundedness and square integrability according to considered equations.…”