The phrase "a infinite interval" is not correct in written English.
The correct form is "an infinite interval" due to the vowel sound at the beginning of "infinite."
Example: "The mathematician studied an infinite interval on the number line to understand its properties."
Alternatives: "a boundless interval" or "an unbounded interval."
In [31], Guo investigated the minimal nonnegative solution of the following initial value problem for a second order nonlinear impulsive integrodifferential equation of Volterra type on an infinite interval with an infinite number of impulsive times in a Banach space : (1.4).
Now, in this paper, we will use the cone theory and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times.
We notice that there has been increasing interest in studying nonlinear differential equation and impulsive integro differential equation on an infinite interval with an infinite number of impulsive times to identify a few, see [22 26] and the references therein.
At the same time, we also notice that there has been increasing interest in studying nonlinear differential equation and impulsive integrodifferential equation on an infinite interval with an infinite number of impulsive times; to identify a few, we refer the reader to Guo and Liu [29], Guo [30 32], and Li and Shen [33].
For the region of engineering interest, mapping is isometrical, and for far field, shrink mapping which transforms an infinite interval into a finite interval is adopted.
Firstly, the positive solutions on J are obtained, which expands the domain of definition of t from a finite interval to an infinite interval.
Recently, in [27], Li and Nieto obtained some new results of the case that impulse effects on an infinite interval with a finite number of impulsive times.
Motivated by above papers, we consider the following singular -point boundary value problem on an infinite interval in a Banach space (1.2).
Solve the singular integral equation of the first kind, involving an infinite interval with a gap, as given by ∫ − ∞ − k ψ ( t ) dt t − x + ∫ k ∞ ψ ( t ) dt t − x = g ( x ), x ∈ ( − ∞, − k ) ∪ ( k, ∞ ).
One of the key tools or ideas used in the proofs of Theorem 3 is the following simple but specially successful conclusion: If f ( x ) is a function defined on an infinite interval I ⊆ R and it satisfies lim x → ∞ f ( x ) = δ and f ( x ) − f ( x + ε ) > 0 for x ∈ I and some fixed number ε > 0, then f ( x ) > δ on I.
