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Discrete approaches to continuous boundary value problems: existence and convergence of solutions. (English) Zbl 1470.39010

Summary: We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs.

MSC:

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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[1] Falb, P. L.; de Jong, J. L., Some Successive Approximation Methods in Control and Oscillation Theory. Some Successive Approximation Methods in Control and Oscillation Theory, Mathematics in Science and Engineering, 59, (1969), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0202.09603
[2] Agarwal, R. P.; Cabada, A.; Otero-Espinar, V.; Dontha, S., Existence and uniqueness of solutions for anti-periodic difference equations, Archives of Inequalities and Applications, 2, 4, 397-412, (2004) · Zbl 1087.39001
[3] Atici, F. M.; Cabada, A.; Ferreiro, J. B., Existence and comparison results for first order periodic implicit difference equations with maxima, Journal of Difference Equations and Applications, 8, 4, 357-369, (2002) · Zbl 1005.39012
[4] Cabada, A., The method of lower and upper solutions for periodic and anti-periodic difference equations, Electronic Transactions on Numerical Analysis, 27, 13-25, (2007) · Zbl 1171.39301
[5] Tisdell, C. C., On first-order discrete boundary value problems, Journal of Difference Equations and Applications, 12, 12, 1213-1223, (2006) · Zbl 1115.39022
[6] Mohamed, M.; Thompson, H. B.; Jusoh, M. S.; Jusoff, K., Discrete first-order three point boundary value problem, Journal of Mathematics Research, 2, 2, 207-215, (2009) · Zbl 1258.39002
[7] Gaines, R., Difference equations associated with boundary value problems for second order nonlinear ordinary differential equations, SIAM Journal on Numerical Analysis, 11, 411-434, (1974) · Zbl 0279.65068
[8] Rachůnková, I.; Tisdell, C. C., Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions, Nonlinear Analysis: Theory, Methods and Applications, 67, 4, 1236-1245, (2007) · Zbl 1130.39017
[9] Thompson, H. B.; Tisdell, C., Systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations, Journal of Mathematical Analysis and Applications, 248, 2, 333-347, (2000) · Zbl 0963.65081
[10] Thompson, H. B.; Tisdell, C., Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations, Applied Mathematics Letters, 15, 6, 761-766, (2002) · Zbl 1003.39012
[11] Thompson, H. B.; Tisdell, C. C., The nonexistence of spurious solutions to discrete, two-point boundary value problems, Applied Mathematics Letters, 16, 1, 79-84, (2003) · Zbl 1018.39009
[12] Reid, W. T., Ordinary Differential Equations, (1971), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0212.10901
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