Abstract
In this paper, we study the existence and multiplicity of solutions of the quasilinear problems with minimum and maximum(ϕ(u′(t)))′=(Fu)(t),a.e. t∈(0,T),min{u(t)∣t∈[0,T]}=A,max{u(t)∣t∈[0,T]}=B,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& \\bigl(\\phi \\bigl(u'(t)\\bigr)\\bigr)'=(Fu) (t),\\quad \\mbox{a.e. }t\\in (0,T), \\\\& \\min \\bigl\\{ u(t) \\mid t\\in [0,T]\\bigr\\} =A, \\qquad \\max \\bigl\\{ u(t) \\mid t\\in [0,T]\\bigr\\} =B, \\end{aligned}$$ \\end{document} where phi :(-a,a)rightarrow mathbb{R} (0< a<infty ) is an odd increasing homeomorphism, F:C^{1}[0,T]rightarrow L^{1}[0,T] is an unbounded operator, T>1 is a constant and A, Bin mathbb{R} satisfy B>A. By using the Leray–Schauder degree theory and the Brosuk theorem, we prove that the above problem has at least two different solutions.
Highlights
In this paper we study the following quasilinear problem φ u (t) = (Fu)(t), a.e. t ∈ (0, T), (1.1)subjected to nonlinear boundary conditions min u(t) | t ∈ [0, T] = A, max u(t) | t ∈ [0, T] = B, (1.2)where φ : (–a, a) → R is an increasing homeomorphism, φ(0) = 0, a is a positive constant, F : C1[0, T] → L1[0, T] is an unbounded operator, T > 1 is a constant and A, B ∈ R satisfy B > A
We study the existence and multiplicity of solutions of the quasilinear problems with minimum and maximum (φ(u (t))) = (Fu)(t), a.e. t ∈ (0, T), min u(t) | t ∈ [0, T] = A, max u(t) | t ∈ [0, T] = B, where φ : (–a, a) → R (0 < a < ∞) is an odd increasing homeomorphism, F : C1[0, T] → L1[0, T] is an unbounded operator, T > 1 is a constant and A, B ∈ R satisfy B > A
By using the Leray–Schauder degree theory and the Brosuk theorem, we prove that the above problem has at least two different solutions
Summary
The various boundary value problems above are reduced to the search of a fixed point for some operator defined on the space C1[0, T] Those operators are completely continuous, and a novel feature linked to the nature of the function φ lies in the fact that those operators map C1[0, T] into the cylinder of functions u ∈ C1[0, T] such that max[0,T] |u | < a. Note that φ : R → R is an odd increasing homeomorphism; the classical p-Laplacian cases, for which φ(s) = |s|p–2s, the existence and multiplicity results of p-Laplacian problem with functional boundary conditions have been studied in [18, 20] and [22]; for the other cases, see [21]. Lemma 2.8 Suppose that u is a solution of (2.12) for any λ ∈ [0, 1] and satisfies the boundary condition (1.2) with A = 0.
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