Perturbation technique for a class of nonlinear implicit semilinear impulsive integro-differential equations of mixed type with noncompactness measure

Abstract

By using the Arzela-Ascoli theorem, the Bellman inequality, and a monotone perturbation iterative technique in the presence of lower and upper solutions, we discuss the existence of mild solutions for a class of nonlinear first-order implicit semilinear impulsive integro-differential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique mild solution between lower and upper solutions.

Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces

“…By using the method of lower and upper solutions coupled with the monotone iterative technique, Chen and Li considered the existence of mild solutions for nonlinear impulsive evolution equation where f ( t , u , u )= f 1 ( t , u )+ f 2 ( t , u ), f 1 ( t , u ) is nondecreasing in u and f 2 ( t , u ) is nonincreasing in u . By using a monotone perturbation iterative technique, Lan and Cui discussed the existence of mild solutions for a class of nonlinear first‐order implicit semilinear impulsive integro‐differential equations under monotone conditions and the noncompactness measure conditions, they also obtained the existence of extremal solutions and a unique mild solution. In , Chen and Mu established the existence of extremal mild solutions and ...…”

“…uj tDtk denotes the jump of u.t/ at t D t k , that is, uj tDtk D u.t C k / u.t k /, where u.t C k / and u.t k / represent the right and left limits of u.t/ at t D t k , respectively. Semilinear impulsive evolution equations as an important branch of modern mathematics and applied mathematics have been studied by many authors (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein). When A D Â, the corresponding semigroup G.t/ D I, then the impulsive evolution equation (1) becomes the following first-order impulsive integro-differential equations in Banach space E: The existence of solutions of the previous problem and its special and related cases have been studied under several kinds of conditions by many authors, for example, [19][20][21][22][23][24][25].…”

“…By using a monotone perturbation iterative technique, Lan and Cui [14] discussed the existence of mild solutions for a class of nonlinear first-order implicit semilinear impulsive integro-differential equations 8 <…”

“…Impulsive differential equations arise naturally from a wide variety of applications, such as spacecraft control, electrical engineering, medicine biology, echoing, and so on. For more detail on this theory and its applications, we refer to and references therein.…”

Approximate solutions of impulsive integro-differential equations

“…Such type of problems can be modelled with impulsive integro-differential equations. Thus, many researchers [2,4,5,[8][9][10]13] have opted for this research area and contributed to the development of the theory of impulsive differential equations. More information related to this can be found in monographs of Bainov and Simeonov [2] and Lkashmikantham et al [8].…”