Generalized monotone iterative technique for set differential equations involving causal operators with memory

Abstract

Using the method of lower and upper solutions and the monotone iterative technique for the linear differential equations involving anticipation and retardation, we develop the generalized monotone iterative technique for the set differential equations involving causal operators with memory.

Lagrange stability in terms of two measures with initial time difference for set differential equations involving causal operators

“…It is clear that (10) and (11) are different in the initial time and position. Moreover, if (P V ) (t) in ( 12) is written as (P V ) (t) = (QV ) (t) + (RV ) (t); Then, we consider (12) as the perturbed form corresponding to the unperturbed equation (11) with the perturbation term (RV ) (t).…”

“…Many researchers were interested in studying set differential equations (SDEs) in the recent decades [2,3,5,[8][9][10]13,14,18,20,23,36,47] due to their unifying properties. Lakshmikantham et al highlighted these properties in one of the most important resources on this topic [23].…”

“…[1, 7-10, 21, 43] SDEs with causal operators unifies the fundamental theory of SDEs, including various corresponding dynamical systems. Some relevant works can be found in [5,[8][9][10][11][12][13][14]47] Although it is never feasible to know the exact solutions of all dynamical systems in practice, their attributes may be determined through a variety of qualitative studies such as stability analysis [2][3][4][5]15,19,20,24,36], initial time difference (ITD) stability analysis [6,29,30,33,34,37,38,[41][42][43][44][45][46][47], practical stability analysis [17,31,40,46], boundedness [2,6,11,16,32,37,38,[40][41][42], etc.…”

Existence Theorems for Solutions to Set-Valued Stochastic Differential Equations and Applications

“…Therefore, set-valued differential equations have been studied by many authors due to many application. Reader can refer to the very good book and papers (see [6,16,5,10,11,13,9,15] and references therein). Beside that, a new class of set-valued differential equations on semilinear Hausdorff space under classic Hukuhara derivative, called setvalued stochastic differential equations (SSDEs) which as far as we known, that the problem of the properties of solutions to set-valued stochastic differential equations is still open.…”

Higher order convergence for a class of set differential equations with initial conditions

“…With this method we can prove that the monotone sequences are convergent uniformly. This fruitful method has been extended to set differential equations in a general setup which includes various known results, and the reader is referred to the papers of Devi and Vatsala [9], Devi [10], Dhaigudel and Naidu [13], McRae, Devi and Drici [26] for details. In [10], the authors introduced a partial ordering in the metric space (K c (R n ), D) and developed a monotone iterative method for set differential equations when the forcing function is the sum of a convex and a concave function.…”