Third-order generalized discontinuous impulsive problems on the half-line
Abstract
In this paper, we improve the existing results in the literature by presenting weaker sufficient conditions for the solvability of a third-order impulsive problem on the half-line, having generalized impulse effects. More precisely, our nonlinearities do not need to be positive nor sublinear and the monotone assumptions are local ones. Our method makes use of some truncation and perturbed techniques and on the equiconvergence at infinity and the impulsive points. The last section contains an application to a boundary layer flow problem over a stretching sheet with and without heat transfer.
Keyword : impulsive problems, upper and lower solutions, equiconvergence, boundary layer flow
How to Cite
Minhós, F., & Carapinha, R. (2021). Third-order generalized discontinuous impulsive problems on the half-line. Mathematical Modelling and Analysis, 26(4), 548-565. https://doi.org/10.3846/mma.2021.12557
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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R.P. Agarwal and D. O’Regan. Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publisher, Glasgow, 2001. https://doi.org/10.1007/978-94-010-0718-4
R.P. Agarwal and D. O’Regan. Non-linear boundary value problems on the semi-infinite interval: an upper and lower solution approach. Mathematika, 49(12):129–140, 2002. https://doi.org/10.1112/S0025579300016120
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F. Bernis and L.A. Peletier. Two problems from draining flows involving thirdorder ordinary differential equations. SIAM J. Math. Anal., 27:515–527, 1996. https://doi.org/10.1137/S0036141093260847
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T. Chiam. Stagnation-point flow towards a stretching plate. J. Phys. Soc. Jpn., 63:2443–2444, 1994. https://doi.org/10.1143/JPSJ.63.2443
C. Corduneanu. Integral Equations and Stability of Feedback Systems. Academic Press, NewYork, 1973.
J. Graef, L. Kong and F. Minhós. Higher order boundary value problems with ΦLaplacian and functional boundary conditions. Computers and Mathematics with Applications, 61:236–249, 2011. https://doi.org/10.1016/j.camwa.2010.10.044
J. Graef, L. Kong, F. Minhós and J. Fialho. On the lower and upper solutions method for higher order functional boundary value problems. Applicable Analysis and Discrete Mathematics, 5(1):133–146, 2011. https://doi.org/10.2298/AADM110221010G
H. Lian, P. Wang and W. Ge. Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. Nonlinear Anal., 70:2627–2633, 2009. https://doi.org/10.1016/j.na.2008.03.049
Y. Liu. Existence of solutions of boundary value problems for coupled singular differential equations on whole lines with impulses. Mediterr. J. Math., 12:697– 716, 2015. https://doi.org/10.1007/s00009-014-0422-1
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R. Ma, B. Yang and Z. Wang. Positive periodic solutions of first-order delay differential equations with impulses. Appl. Math. Comput., 219:6074–6083, 2013. https://doi.org/10.1016/j.amc.2012.12.020
T.R. Mahapatra and A.S. Gupta. Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transfer, 38:517–521, 2002. https://doi.org/10.1007/s002310100215
F. Minhós and R. Carapinha. Half-linear impulsive problems for classical and singular ϕ-Laplacian with generalized impulsive conditions. Journal of Fixed Point Theory and Applications, 20(3):117, 2018. https://doi.org/10.1007/s11784-0180598-2
F. Minhós and H. Carrasco. High order boundary value problems on unbounded domains: Types of solutions, functional problems and applications: Trends in abstract and applied analysis. World Scientific, 5, 2017. https://doi.org/10.1007/s002310100215
F. Minhós and H. Carrasco. Lidstone-type problems on the whole real line and homoclinic solutions applied to infinite beams. Neural Computing & Application, 2020. https://doi.org/10.1007/s00521-020-04732-x
F. Minhós and R. de Sousa. Solvability of coupled systems of generalized Hammerstein-type integral equations in the real line. Mathematics, 8(111), 2020. https://doi.org/10.3390/math8010111
R. Nazar, N. Amin, D. Filip and I. Pop. Stagnation point flow of a micropolar fluid towards a stretching sheet. Int. J. Non-Linear Mech., 39:1227–1235, 2004. https://doi.org/10.1016/j.ijnonlinmec.2003.08.007
J. Nieto and R. López. Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl., 55:2715–2731, 2008. https://doi.org/10.1016/j.camwa.2007.10.019
J. Paullet and P. Weidmaz. Analysis of stagnation point flow toward a stretching sheet. International Journal of Non-Linear Mechanic., 42(9):1084–1091, 2007. https://doi.org/10.1016/j.ijnonlinmec.2007.06.003
M. Reza and A.S. Gupta. Steady two-dimensional oblique stagnation-point flow towards a stretching surface. Fluid Dyn. Res., 37:334–340, 2005. https://doi.org/10.1016/j.fluiddyn.2005.07.001
A.M. Samoilenko and N.A. Perestyuk. Impulsive Differential Equations. World Scientific, Singapore, 1955. https://doi.org/10.1142/2892
Y. Tian and W. Ge. Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications, 72:277–287, 2010. https://doi.org/10.1016/j.na.2009.06.051
W.C. Troya. Solutions of third-order differential equations relevant to draining and coating flows. SIAM J. Math. Anal., 24:155–171, 1993. https://doi.org/10.1137/0524010
E.O. Tuck and L.W. Schwartz. A boundary value problem from draining and coating flows involving a third-order differential equation relevant to draining and coating flows. SIAM Rev., 32:453–469, 1990. https://doi.org/10.1137/1032079
J. Xiao, J. Nieto and Z. Luo. Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Communications in Nonlinear Science and Numerical Simulation, 17:426–432, 2012. https://doi.org/10.1016/j.cnsns.2011.05.015
Y. Xu and H. Zhang. Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Applied Mathematics and Computation, 218:5806–5818, 2012. https://doi.org/10.1016/j.amc.2011.11.100
B. Yan, D. O’Regan and R.P. Agarwal. Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity. J.Comput.Appl.Math., 197:365–386, 2006. https://doi.org/10.1016/j.cam.2005.11.010
R.P. Agarwal and D. O’Regan. Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publisher, Glasgow, 2001. https://doi.org/10.1007/978-94-010-0718-4
R.P. Agarwal and D. O’Regan. Non-linear boundary value problems on the semi-infinite interval: an upper and lower solution approach. Mathematika, 49(12):129–140, 2002. https://doi.org/10.1112/S0025579300016120
C. Bai and C. Li. Unbounded upper and lower solution method for third-order boundary-value problems on the half-line. Electronic Journal of Differential Equations, 2009(119):1–12, 2009. https://doi.org/10.1023/A:1021167932414
F. Bernis and L.A. Peletier. Two problems from draining flows involving thirdorder ordinary differential equations. SIAM J. Math. Anal., 27:515–527, 1996. https://doi.org/10.1137/S0036141093260847
A. Cabada, F. Minhós and A.I. Santos. Solvability for a third order discontinuous fully equation with functional boundary conditions. J. Math. Anal. Appl., 322:735–748, 2006. https://doi.org/10.1016/j.jmaa.2005.09.065
T. Chiam. Stagnation-point flow towards a stretching plate. J. Phys. Soc. Jpn., 63:2443–2444, 1994. https://doi.org/10.1143/JPSJ.63.2443
C. Corduneanu. Integral Equations and Stability of Feedback Systems. Academic Press, NewYork, 1973.
J. Graef, L. Kong and F. Minhós. Higher order boundary value problems with ΦLaplacian and functional boundary conditions. Computers and Mathematics with Applications, 61:236–249, 2011. https://doi.org/10.1016/j.camwa.2010.10.044
J. Graef, L. Kong, F. Minhós and J. Fialho. On the lower and upper solutions method for higher order functional boundary value problems. Applicable Analysis and Discrete Mathematics, 5(1):133–146, 2011. https://doi.org/10.2298/AADM110221010G
H. Lian, P. Wang and W. Ge. Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. Nonlinear Anal., 70:2627–2633, 2009. https://doi.org/10.1016/j.na.2008.03.049
Y. Liu. Existence of solutions of boundary value problems for coupled singular differential equations on whole lines with impulses. Mediterr. J. Math., 12:697– 716, 2015. https://doi.org/10.1007/s00009-014-0422-1
Y. Liu and D. O’Regan. Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul., 16:1769–1775, 2011. https://doi.org/10.1016/j.cnsns.2010.09.001
R. Ma, B. Yang and Z. Wang. Positive periodic solutions of first-order delay differential equations with impulses. Appl. Math. Comput., 219:6074–6083, 2013. https://doi.org/10.1016/j.amc.2012.12.020
T.R. Mahapatra and A.S. Gupta. Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transfer, 38:517–521, 2002. https://doi.org/10.1007/s002310100215
F. Minhós and R. Carapinha. Half-linear impulsive problems for classical and singular ϕ-Laplacian with generalized impulsive conditions. Journal of Fixed Point Theory and Applications, 20(3):117, 2018. https://doi.org/10.1007/s11784-0180598-2
F. Minhós and H. Carrasco. High order boundary value problems on unbounded domains: Types of solutions, functional problems and applications: Trends in abstract and applied analysis. World Scientific, 5, 2017. https://doi.org/10.1007/s002310100215
F. Minhós and H. Carrasco. Lidstone-type problems on the whole real line and homoclinic solutions applied to infinite beams. Neural Computing & Application, 2020. https://doi.org/10.1007/s00521-020-04732-x
F. Minhós and R. de Sousa. Solvability of coupled systems of generalized Hammerstein-type integral equations in the real line. Mathematics, 8(111), 2020. https://doi.org/10.3390/math8010111
R. Nazar, N. Amin, D. Filip and I. Pop. Stagnation point flow of a micropolar fluid towards a stretching sheet. Int. J. Non-Linear Mech., 39:1227–1235, 2004. https://doi.org/10.1016/j.ijnonlinmec.2003.08.007
J. Nieto and R. López. Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl., 55:2715–2731, 2008. https://doi.org/10.1016/j.camwa.2007.10.019
J. Paullet and P. Weidmaz. Analysis of stagnation point flow toward a stretching sheet. International Journal of Non-Linear Mechanic., 42(9):1084–1091, 2007. https://doi.org/10.1016/j.ijnonlinmec.2007.06.003
M. Reza and A.S. Gupta. Steady two-dimensional oblique stagnation-point flow towards a stretching surface. Fluid Dyn. Res., 37:334–340, 2005. https://doi.org/10.1016/j.fluiddyn.2005.07.001
A.M. Samoilenko and N.A. Perestyuk. Impulsive Differential Equations. World Scientific, Singapore, 1955. https://doi.org/10.1142/2892
Y. Tian and W. Ge. Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications, 72:277–287, 2010. https://doi.org/10.1016/j.na.2009.06.051
W.C. Troya. Solutions of third-order differential equations relevant to draining and coating flows. SIAM J. Math. Anal., 24:155–171, 1993. https://doi.org/10.1137/0524010
E.O. Tuck and L.W. Schwartz. A boundary value problem from draining and coating flows involving a third-order differential equation relevant to draining and coating flows. SIAM Rev., 32:453–469, 1990. https://doi.org/10.1137/1032079
J. Xiao, J. Nieto and Z. Luo. Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Communications in Nonlinear Science and Numerical Simulation, 17:426–432, 2012. https://doi.org/10.1016/j.cnsns.2011.05.015
Y. Xu and H. Zhang. Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Applied Mathematics and Computation, 218:5806–5818, 2012. https://doi.org/10.1016/j.amc.2011.11.100
B. Yan, D. O’Regan and R.P. Agarwal. Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity. J.Comput.Appl.Math., 197:365–386, 2006. https://doi.org/10.1016/j.cam.2005.11.010