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Multiple solutions of a third order boundary value problem. (English) Zbl 0755.34017
The aim of the authors is to give a lower estimation for the number of solutions to the two-point boundary-value problem associated with a third order nonlinear differential equation of the form (1) \(x'''=f(t,x,x',x'')\), \(t\in[a,b]\), (2) \(x(a)=A\), \(x'(a)=A_ 1\), \(x(b)=B\). Here \(f\) as well as its first derivatives \(f_ x\), \(f_{x'}\) and \(f_{x''}\) are supposed to be continuous on \([a,b]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\). The result is stated through the solution of a certain linear differential equation which is established by using any one of the solutions to the above-mentioned boundary-value problem, say \(\xi(t)\), namely: (3) \(y'''=f_{x''}(t,\xi,\xi',\xi'')y''+f_{x'}(t,\xi,\xi',\xi'')y'+f_ x(t,\xi,\xi',\xi'')y\), (4) \(y(a)=y'(a)=0\), \(y''(a)=1\). They prove that under certain conditions (1)–(2) has at least \(m+1\) solutions if the unique solution to (3)–(4) has \(m\) zeros in the interval \((a,b)\).

34B15 Nonlinear boundary value problems for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] SADYRBAEV F.: The number of solutions of a two-points boundary value problem. (Russian), Latv. Mat. Ezhegodnik. 32 (1988), 37-41. · Zbl 0702.34026
[2] GERA M.: Bedingungen der Nichtoszillations fähigkeit für die lineare Differentialgleichung dritter Ordnung y”’ + 1p(x)y” + p2(x)y’ + p3(x)y = 0. Acta Fac. Rer. Nat. Univ. Comenian. Math. XXIII (1969), 13-34. · Zbl 0216.11303
[3] FILIPPOV A. F.: Differential Equations with Discontinuous Right-hand Side. (Russian), Mir, Moscow, 1985. · Zbl 0571.34001
[4] GREGUŠ M., ŠVEC M., ŠEDA V.: Ordinary Differential Equations. (Slovak), Alfa, Bratislava, 1985.
[5] BESMERTNYCH G. A., LEVIN A. JU.: On some estimates of differentiable functions of one variable. (Russian), Soviet. Math. Dokl. 144 (3) (1962), 471-474.
[6] DUGUNDJI J.: An extension of Tietze’s theorem. Pacific J. Math. 1 (1951), 353-367. · Zbl 0043.38105
[7] CODDINGTON A. E., LEVINSON N.: Theory of Ordinary Differential Equations. New York-Toronto-London, 1955. · Zbl 0064.33002
[8] JERUGIN N. P.: Reading for General Course on Differential Equations. (Russian), Minsk, 1970.
[9] PETERSON A.: Existence-uniqueness for ordinary differential equations. J. Math. Anal. Appl. 64 (1978), 166-172. · Zbl 0386.34022
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