Abstract
In this paper, the existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces are established under more general conditions by constructing a special Banach space and using cone theory and the Banach contraction mapping principle. The results obtained herein improve and generalize some well-known results.
Highlights
1 Introduction Nonlinear integro-differential equations in abstract spaces arise in different fields of physical sciences, engineering, biology, and applied mathematics
There has been a significant development in nonlinear integro-differential equations
⎧ ⎨u (t) = f (t, u(t), (Tu)(t), (Su)(t)), ⎩u(t0) = u0, t ∈ I = [t0, t0 + a], in Banach space E, where u0 ∈ E, f : I × E × E × E → E, for any u ∈ C[I, E], g(t) = f (t, u(t), (Tu)(t), (Su)(t)) : I → E is continuous, and T is a Volterra integral operator defined by t (Tu)(t) = k(t, s)u(s) ds, t0
Summary
Nonlinear integro-differential equations in abstract spaces arise in different fields of physical sciences, engineering, biology, and applied mathematics. Using the upper and lower solutions method and monotone iterative technique, Guo, Liu and Zheng et al [2,3,4] studied the existence and uniqueness of solutions for the first order integro-differential equations In [5], the authors studied the second order integro-differential equations
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