Abstract
Not considering the Green’s function, the present study starts to construct a cone formed by a nonlinear term in Banach spaces, and through the cone creates a convex closed set. We obtain the existence of solutions for the boundary values problems of n th-order impulsive singular nonlinear integro-differential equations in Banach spaces by applying the Monch fixed point theorem. An example is given to illustrate the main results. MSC:45J05, 34G20, 47H10.
Highlights
Introduction and preliminaries By using theSchauder fixed point theorem, Guo [ ] obtained the existence of solutions of initial value problems for nth-order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with infinite number of impulsive times in a Banach space
In [ ], by using the fixed point theorem in a cone, Chen and Qin investigated the existence of multiple solutions for a class of boundary value problems of singular nonlinear integro-differential equations of mixed type in Banach spaces
On the new convex closed set, we apply the Mönch fixed point theorem to investigate the existence of solutions for the boundary value problems of nthorder impulsive singular nonlinear integro-differential equations in Banach spaces
Summary
Let B , B ⊂ PCn– [J, E] be two countable sets. U ∈ PCn– [J, E] ∩ Cn[J , E] is a solution of SBVP ( ), if and only if u ∈ Q is a fixed point of the operator A defined by ( ). If u ∈ Q is a fixed point of the operator A, i.e., u is a solution of the following impulsive integro-differential equation: u(t) = (Au)(t). Since {Aul}∞ l= is a relatively compact set in PC(n– )[J, E], there exists a subsequence of {Aulj }∞ j= which converges to y ∈ PC(n– )[J, E]. It is easy to see that A(B) ⊂ PCn– [J, E] is bounded and the elements of (A(B))(n– ) are equicontinuous on each Jk
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