Eigenvalue Intervals of Multivalued Operator and its Application for a Multipoint Boundary Value Problem
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Eigenvalue Intervals of Multivalued Operator and its Application for a Multipoint Boundary Value Problem Vo Viet Tri1 · Shahram Rezapour2,3 Received: 25 May 2020 / Accepted: 18 August 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we establish a general existence result on positive eigenvalue interval for the multivalued operators in cone. By using the main result, we investigate the existence of positive solutions for a multi-point boundary value problem. We study this problem via a parametric multivalued inclusion. Keywords Eigenvalue intervals · Eigenvalue · Eigenvector · Global continua · Multivalued equations Mathematics Subject Classification 47H07 · 47H08 · 47H10 · 35P30
1 Introduction The studies of parametric equations of the form x = A(λ, x) in ordered space have earned profound results (see [2,3,5,8,10,11,14,15,17,18] and references therein). Particularly, in Krasnoselskii [10] used the monotone minorant method in conjunction with the theory of fixed point index to prove that the set S = {x : ∃λ, x = A(λ, x)} forms an unbounded continuous branch emanating from zero, i.e., S ∩ G = ∅ for any bounded open neighborhood G of zero. In the special case, when (X , .) is a Banach space ordered by a cone K he obtained the existence of an interval of λ for
Communicated by Mohammad B. Asadi.
B
Shahram Rezapour [email protected] Vo Viet Tri [email protected]
1
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
2
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
3
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
123
Bulletin of the Iranian Mathematical Society
the equation x = λA (x) to have a positive solution which is expressed in Proposition 1.1. Proposition 1.1 [10] Let A : K → K be a compact mapping. Assume that the following conditions are satisfied: (i) A(x) = 0 for x ∈ K \{0}, (ii) S = {x ∈ K \{0} : ∃λ = λ(x) > 0, x = λA(x)} forms an unbounded continuous branch emanating from θ and (iii) there exist real numbers a and b satisfying either limx→0 λ(x) = a < b = limx→∞ λ(x) or limx→∞ λ(x) = a < b = limx→0 λ(x). Then the equation x = λA(x) has a positive solution for λ ∈ (a, b). In this paper, we first extend this result for the inclusion x ∈ λA(x), where A : K → 2 K \{∅} is a given multivalued operator. This extension has not been considered yet to the best of our knowledge. Next, using obtained result we prove the existence of positive solutions for the problem
u
(t) + q(t) f (t, u(t)) = 0, t ∈ [0, 1], q(t) ∈ F(t, u(t)) a.e on [0, 1]
(1.1)
with one of the boundary conditions m m (BC1): u (0) = 0, u(1) = i=1 αi u(ηi ), 0 < ηi < 1, αi ≥ 0, i=1 αi < 1 , m m αi u(ηi ), 0 < ηi < 1, αi ≥ 0, i=1 αi ηi < 1. (BC2): u(0) = 0, u(1) = i=1 Here we suppose 0 < η1 < · · · < ηm < 1, where m > 1. In what follows, if A is a subset of a Banach space, we denote by CC(A) (K C(A), respectively) the class of all nonempty closed (compact, respectively)
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