Abstract
The aim of this paper is to explore the method of lower and upper solutions in order to give some existence results for equations of the form \t\t\ty(4)(x)+(k1+k2)y″(x)+k1k2y(x)=f(x,y(x)),x∈(0,1),\\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}$$y^{(4)}(x)+(k_{1}+k_{2}) y''(x)+k_{1}k_{2} y(x)=f\\bigl(x,y(x)\\bigr), \\quad x\\in(0,1), $$\\end{document} with the Navier condition \t\t\ty(0)=y(1)=y″(0)=y″(1)=0\\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}$$y(0) = y(1) = y''(0) = y''(1) = 0 $$\\end{document} under the condition k_{1}<0<k_{2}<pi^{2}. The main tool is the Schauder fixed point theorem.
Highlights
The aim of this paper is to explore the method of lower and upper solutions in order to give some existence results for equations of the form y(4)(x) + (k1 + k2)y (x) + k1k2y(x) = f (x, y(x)), x ∈ (0, 1), with the Navier condition y(0) = y(1) = y (0) = y (1) = 0 under the condition k1 < 0 < k2 < π 2
1 Introduction The aim of this paper is to explore the method of lower and upper solutions in order to give some existence of solutions for equations of the form y( )(x) + (k + k )y (x) + k k y(x) = f x, y(x), x ∈ (, ), ( . )
The use of lower and upper solutions in boundary value problems of the fourth order, even for the simple boundary conditions ( . ), is heavily dependent on the positiveness properties for the corresponding linear operators, see the counterexample in [, Remark . ]. It is the purpose of this paper to establish the method of lower and upper solutions for fourth order problem ( . ), ( . ) under condition (H )
Summary
The aim of this paper is to explore the method of lower and upper solutions in order to give some existence of solutions for equations of the form y( )(x) + (k + k )y (x) + k k y(x) = f x, y(x) , x ∈ ( , ),. With the Navier condition y( ) = y( ) = y ( ) = y ( ) =. Such boundary value problems appear, as it is well known [ – ], in the theory of hinged beams. ) under the assumption (H ) k and k are two constants with k < k < Vrabel [ ] studied problem ( . ), ( . ) under the assumption (H ) k and k are two constants with k < k
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