Computation of Positive Solutions for Nonlinear Impulsive Integral Boundary Value Problems with p-Laplacian on Infinite Intervals
نویسندگان
چکیده
and Applied Analysis 3 2. Preliminaries and Several Lemmas Definition 1. LetE be a real Banach space. A nonempty closed set P ⊂ E is said to be a cone provided that (1) au + bv ∈ P for all u, v ∈ P and all a ≥ 0, b ≥ 0, (2) u, −u ∈ P implies that u = 0. Definition 2. A map α : P → [0, +∞) is said to be concave on P, if α(tu + (1 − t)v) ≥ tα(u) + (1 − t)α(v) for all u, v ∈ P and t ∈ [0, 1]. Let PC[J, R] = {x : x is a map from J into R such that x(t) is continuous at t ̸ = t k , left continous at t = t k and x(t+ k ) exists for k = 1, 2, . . . , }, PC1[J, R] = {x ∈ PC[J, R] : x(t) exists and is continuous at t ̸ = t k , left continous at t = t k and x (t + k ) exists for k = 1, 2, . . . , } FPC [J, R] = {x ∈ PC [J, R] : sup t∈J |x (t)| 1 + t < ∞} ,
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