On the Existence of Three Solutions for Some Classes of Two-Point Semi-linear and Quasi-linear Differential Equations
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On the Existence of Three Solutions for Some Classes of Two-Point Semi-linear and Quasi-linear Differential Equations Somayeh Saiedinezhad1 Received: 10 August 2019 / Revised: 25 September 2019 / Accepted: 29 October 2019 © Iranian Mathematical Society 2019
Abstract A general theorem concerning the three critical points for some classes of coercive functionals depending on a real parameter is established, which may derive existence’s results of three solutions with various sufficient conditions for some classes of two-point semi-linear boundary value problems. Moreover, by applying known three existence theorems, we derive multiple existence results for a class of quasi-linear differential equation. Keywords Critical points · Three solutions · Two-point boundary value problem · Eigenvalue problem Mathematics Subject Classification 34B09 · 58E05
1 Introduction and Preliminary Results Ricceri in [18] established an interesting method which leads to the existence of three critical points for some classes of coercive functionals depending on a real parameter λ ∈ . Ricceri’s result has wide applications to derive the existence of at least three weak solutions for some differential equations. Many authors by assembling a class of appropriate assumptions for various differential equations make a suitable geometry of corresponding energy functional to applying Ricceri’s theorems. (for example, see [1,2,4–7,9–16])
Communicated by Asadollah Aghajani.
B 1
Somayeh Saiedinezhad [email protected] School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
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Bulletin of the Iranian Mathematical Society
Bonanno in [8] established some remarks on a strict minimax inequality, which play a fundamental role in Ricceri’s three critical points theorem. The Bonanno’s results in [8] gives an estimate of where can be located. Theorem 1.1 [8, Theorem 2.1] Let X be a separable and reflexive real Banach space, and let φ, J : X → R be two continuously Gateaux differentiable functionals. Assume that there exists u 0 ∈ X such that φ(u 0 ) = J (u 0 ) = 0, φ(u) ≥ 0 for every u ∈ X and there exists u 1 ∈ X , r > 0 such that (I) r < φ(u 1 ); J (u 1 ) . (II) supφ(u) inf u∈X φ(u) put α(r ) := inf φ(u) 0, 0 ≤ a1 < a < b < b1 ≤ 1, A(x) ∈ C 1 ([a1 , a]) and B(x) ∈ C 1 ([b, b1 ]) where A(a1 ) = B(b1 ) = 0 and A(a) = B(b) = d. Moreover, there is c ∈ R, which 2c2 ≤ η :=
1 2
a
A2 (x)dx +
a1
b1
B 2 (x)dx ,
(2.1)
b
and sup|ξ |
3 2 1 2
ξ 8; 3 ξ2 −
13 2 ;
we have lim|ξ |→∞
0 ≤ ξ ≤ 1, ξ > 1. g(ξ ) |ξ |s = 0, 1 < 200 g(1)
the growth condition (S1 ) is
satisfied. Letting c := = = of Corollary 2.3 are satisfied and the proof is completed. we have g( 21 )
1 28
1 200
and so the assumptions
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Bulletin of the Iranian Mathematical Society
Corollary 2.5 Let f be a non-negative function, which satisfies (S1 ). Suppose that there exits c < 3.8d, where g(c) g(d) ≤ 0.08 2 . 2 c d Then there exists an open interval ⊆ (0, +∞) and a positive number ρ such that for every λ ∈ , problem (Pλ )
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