Abstract
This paper presents sufficient conditions for the existence of positive solutions for a class of integral inclusions. Our results are obtained via a new fixed point theorem for multivalued operators developed in the paper, in which some nonnegative function is used to describe the cone expansion and compression instead of the classical norm‐type, and lead to new existence principles.
Highlights
If the function ρ is convex on P, namely, ρ tx 1 − t y ≤ tρ x 1 − t ρ y for all x, y ∈ P and t ∈ 0, 1, the condition h holds provided that ρ θ 0. From this point of view, we extend the corresponding result of 8
Let us start by defining that a function x ∈ C R, R is said to be a solution of 1.4 if it satisfies 1.4
By BC : BC R, R, we mean the Banach algebra consisting of all functions defined, bounded, and continuous on R with the norm x sup{|x t | : t ≥ 0}
Summary
Cone compression and expansion fixed point theorems are frequently used tools for studying the existence of positive solutions for boundary value problems of integral and differential equations. Assume that A : Ω2 → CK P is a u.s.c., k-set contractive (here 0 ≤ k < 1) map and assume one of the following conditions hold: x ∈/ λAx, ∀λ ∈ 0, 1 , x ∈ ∂Ω2 ∩ P, 2.2 there exists a v ∈ P with x ∈/ Ax δv for x ∈ ∂Ω1 ∩ P, δ ≥ 0.
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