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Zbl 1152.34322
Wang, Da-Bin; Sun, Jian-Ping
Existence of a solution and a positive solution of a boundary value problem for a nonlinear fourth-order dynamic equation.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 5-6, A, 1817-1823 (2008). ISSN 0362-546X

Summary: By using the Schauder fixed point theorem, we offer some existence criteria for a solution and a positive solution to the following fourth-order two-point boundary value problem on time scale $\Bbb T$: $$u^{\Delta\Delta\Delta\Delta}(t)= f(t,u(t), u^{\Delta\Delta}(t))= 0, \quad t\in [a,\rho^2(b)],$$ $$u(a)=A, \quad u(\sigma^2(b))=B, \quad u^{\Delta\Delta}(a)= C, \quad u^{\Delta\Delta}(b)=D,$$ where $a,b\in\Bbb T$ and $a<b$.
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE
34B18 Positive solutions of nonlinear boundary value problems
39A10 Difference equations
47N20 Appl. of operator theory to differential and integral equations

Keywords: time scale; dynamic equation; boundary value problem; solution and positive solution; fixed point

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