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A new existence theorem for right focal boundary value problems on a measure chain. (English) Zbl 1074.34017
Summary: We consider the following differential equation on a measure chain \(\mathbb{T}\) \[ u^{\Delta\Delta}(t)+ f(u(\sigma(t)))= 0,\quad t\in [a,b]\cap \mathbb{T}, \] satisfying the right focal boundary value conditions \(u(a)= 0= u^\Delta(\sigma(b))\).
An existence result is obtained by using a fixed-point theorem due to Krasnosels’kii and Zabreiko. Our conditions imposed on \(f\) are very easy to verify and our result is even new for special cases of differential equations and difference equations, as well as in the general time scale setting.

34B15 Nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
Full Text: DOI
[1] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[2] Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003
[3] Aulbach, B.; Hilger, S., Linear dynamic processes with inhomogeneous time scale, () · Zbl 0719.34088
[4] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differ. equations dyn. syst., 1, 223-246, (1993) · Zbl 0868.39007
[5] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. modelling, 32, 5-6, 571-585, (2000) · Zbl 0963.34020
[6] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic press San Diego · Zbl 0661.47045
[7] Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen
[8] Agarwal, R.P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear anal., 44, 527-535, (2001) · Zbl 0995.34016
[9] Agarwal, R.P.; O’Regan, D., A coupled system of difference equations, Appl. math. comput., 114, 39-49, (2000) · Zbl 1023.39001
[10] Krasnoselskii, M.A.; Zabreiko, P.P., Geometrical methods of nonlinear analysis, (1984), Spriger-Verlag New York
[11] Kaymakcalan, B.; Lakshmikantham, V.; Sivasundaram, S., Dynamic systems on measure chains, (1996), Kluwer Academic Publishers Boston · Zbl 0869.34039
[12] Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential equation on measure chains, Dyn. contin. discrete impuls. syst., 6, 121-137, (1999) · Zbl 0938.34027
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