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The Qualitative Sturm–Liouville Theory on Spatial Networks
Journal of Mathematical Sciences, 2004The authors consider boundary value problems generated by the differential expression \[ -(p(x)y')'+q(x)y=f(x) \] on the edges of a graph domain. At the pendant vertices Dirichlet boundary conditions and at the vertices of degree greater 1 the continuity conditions together with Kirchhoff-type conditions are imposed.
Pokornyi, Yu. V., Pryadiev, V. L.
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SOME PROBLEMS IN THE THEORY OF A STURM-LIOUVILLE EQUATION
Russian Mathematical Surveys, 1960CONTENTSIntroductionChapter One. The solution of the Cauchy problem for the one dimensional wave equation § 1. The application of the method of successive approximations § 2. Reduction to the Goursat problem § 3. The solution of the mixed problem on the half line § 4. The solution of a mixed problem on a finite intervalChapter Two.
Levitan, B. M., Sargsjan, I. S.
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1990
1. In a series of articles dating from 1836–1837, Sturm and Liouville created a whole new subject in mathematical analysis. The theory, later known as Sturm-Liouville theory, deals with the general linear second-order differential equation $$(k(x)V'(x))' + (g(x)r - l(x))V(x) = 0\quad for\;x \in (\alpha ,\beta ),$$ (1) with the imposed ...
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1. In a series of articles dating from 1836–1837, Sturm and Liouville created a whole new subject in mathematical analysis. The theory, later known as Sturm-Liouville theory, deals with the general linear second-order differential equation $$(k(x)V'(x))' + (g(x)r - l(x))V(x) = 0\quad for\;x \in (\alpha ,\beta ),$$ (1) with the imposed ...
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1997
In diesem Kapitel werden wir Anfangs- und Randwertprobleme fur eine lineare gewohnliche Differentialgleichung 2. Ordnung mit variablen Koeffizienten in eine Integralgleichung umformen, sodas wir die Ergebnisse (insbesondere uber Eigenwerte) aus Kapitel 2 und 4 benutzen konnen. Auf einem Intervall [a, b] betrachten wir die Gleichung $$\bar{p}(s){x}''
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In diesem Kapitel werden wir Anfangs- und Randwertprobleme fur eine lineare gewohnliche Differentialgleichung 2. Ordnung mit variablen Koeffizienten in eine Integralgleichung umformen, sodas wir die Ergebnisse (insbesondere uber Eigenwerte) aus Kapitel 2 und 4 benutzen konnen. Auf einem Intervall [a, b] betrachten wir die Gleichung $$\bar{p}(s){x}''
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Sturm Liouville eigenvalue theory
2014In this chapter we deal with Dirichlet boundary value problems as $$\left\{ {\begin{array}{*{20}{r}} {\begin{array}{*{20}{c}} {x''\left( t \right) + A\left( t \right)x'\left( t \right) + B\left( t \right) + x\left( t \right) + \lambda C\left( t \right)x\left( t \right)}&{ = 0} \end{array}}\\ {\begin{array}{*{20}{c}} {x\left( a \right)}&{ = 0} \end ...
Shair Ahmad, Antonio Ambrosetti
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Sturm-Liouville theory for the radial $\Delta_p$ -operator
Mathematische Zeitschrift, 1998Let \(s^{(p)}=| s| ^{p-1}s\) (\(s\) real). The differential operator \[ L_p^\alpha=r^{-\alpha}\bigl(r^\alpha{u'}^{p-1}\bigr)' \] is considered, where \(s\) is the independent variable, \(\alpha\geq 0\), and \(p>1\). For \(\alpha=n-1\) and \(r=| x| \), this is the radial \(\Delta_p\)-operator in \(\mathbb{R}^n\).
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Nonlinear Sturm–Liouville Theory
2014In the next chapter, we’ll apply the Krasnoselskii-Rabinowitz bifurcation theorem in a very specific way: to the Euler buckling problem. The buckling problem belongs to an important class of problems in ordinary differential equations called nonlinear Sturm-Liouville problems.
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