Difference equations involving causal operators with nonlinear boundary conditions

Abstract

In this paper, we investigate nonlinear boundary problems for difference equations with causal operators. Our boundary condition is given by a nonlinear function, and more general than ones given before. By using the method of upper and lower solutions coupled with the monotone iterative technique, criteria on the existence of extremal solutions are obtained, an example is also presented.

Impulsive Evolution Equations with Causal Operators

“…Applications of differential equations with causal operators in optimal control, adaptive control or hysteresis phenomena can be found in the papers [13][14][15][16][17][18][19][20]. Theoretical aspects regarding existence, stability or periodicity of solutions of differential equations with causal operators in finite or infinite dimensional spaces were presented in a series of works, such as: [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].…”

Causal difference equations with upper and lower solutions in the reverse order

“…With the development of boundary value problems (BVPs) for differential equations and for difference equations [18,19,25,26], and the theory of causal differential equations [6-9, 14, 21, 23], many authors have focused their attention on BVPs for causal difference equations [11,12,24]. In particular, in 2011, Jankowski [11] investigated first-order BVPs of difference equations with causal operators and developed the monotone iterative technique.…”

“…In 2006, Atici, Cabada, and Ferreiro [2] considered the difference equations with functional boundary value conditions. Inspired by this paper, in 2015, Wang and Tian [24] established some existence criteria for the following difference equations involving causal operators with nonlinear boundary conditions: To obtain existence results of causal difference equations for problem (1) and 2, we use the method of lower and upper solutions coupled with the monotone iterative technique. This method is well known not only for the continuous case but also for the discrete case, see [1,10,13,15,17,20,22].…”

Quasilinearization method for nonlinear differential equations with causal operators

“…Moreover, MIT has been generally applied to the basic results concerning the existence problems (see [7,10,11,12,13,28] and the references therein). For example, in [8], the notion of a causal operator has been extended to periodic BVPs…”

“…Finally, the same authors of the paper [27] investigated nonlinear BVPs for difference equations with causal operators in [28] by using the method of upper and lower solutions coupled with the monotone iterative technique.…”