Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral
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Authors
Xuexiao You
- School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China.
- College of Computer and Information, Hohai University, Nanjing, Jiangsu 210098, P. R. China.
Wei Liu
- College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China.
Guoju Ye
- College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China.
Dafang Zhao
- School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China.
- College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China.
Abstract
In this paper, we are concerned with the existence of solutions for a second-order three-point nonlinear boundary value
problems involving the distributional Henstock-Kurzweil integral. By using the Leray-Schauder nonlinear alternative, we achieve
some results which are the generalizations of the previous results in the literatures.
Share and Cite
ISRP Style
Xuexiao You, Wei Liu, Guoju Ye, Dafang Zhao, Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 1820--1829
AMA Style
You Xuexiao, Liu Wei, Ye Guoju, Zhao Dafang, Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral. J. Nonlinear Sci. Appl. (2017); 10(4):1820--1829
Chicago/Turabian Style
You, Xuexiao, Liu, Wei, Ye, Guoju, Zhao, Dafang. "Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 1820--1829
Keywords
- Distributional Henstock-Kurzweil integral
- nonlinear boundary value problems
- distributional derivative
- Leray-Schauder nonlinear alternative.
MSC
References
-
[1]
B. Ahmad, A. Alsaedi, D. Garout, Monotone iteration scheme for impulsive three-point nonlinear boundary value problems with quadratic convergence, J. Korean Math. Soc., 45 (2008), 1275–1295.
-
[2]
D. D. Ang, K. Schmitt, L. K. Vy, A multidimensional analogue of the Denjoy-Perron-Henstock-Kurzweil integral, Bull. Belg. Math. Soc. Simon Stevin, 4 (1997), 355–371.
-
[3]
N. A. Asif, P. W. Eloe, R. A. Khan, Positive solutions for a system of singular second order nonlocal boundary value problems, J. Korean Math. Soc., 47 (2010), 985–1000.
-
[4]
D.-L. Bai, H.-Y. Feng, Eigenvalue for a singular second order three-point boundary value problem, J. Appl. Math. Comput., 38 (2012), 443–452.
-
[5]
P. S. Bullen, D. N. Sarkhel, On the solution of \((dy/dx)ap = f(x, y)\), J. Math. Anal. Appl., 127 (1987), 365–376.
-
[6]
T. S. Chew, F. Flordeliza, On \(\acute{x}= f(t, x)\) and Henstock-Kurzweil integrals, Differential Integral Equations, 4 (1991), 861–868.
-
[7]
K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin (1985)
-
[8]
P. W. Eloe, Y. Gao, The method of quasilinearization and a three-point boundary value problem, J. Korean Math. Soc., 39 (2002), 319–330.
-
[9]
G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301–325.
-
[10]
S.-M. Im, Y.-J. Kim, D.-I. Rim, A uniform convergence theorem for approximate Henstock-Stieltjes integral, Bull. Korean Math. Soc., 41 (2004), 257–267.
-
[11]
R. A. Khan, Higher order nonlocal nonlinear boundary value problems for fractional differential equations, Bull. Korean Math. Soc., 51 (2014), 329–338.
-
[12]
R. A. Khan, J. R. L. Webb, Existence of at least three solutions of a second-order three-point boundary value problem, Nonlinear Anal., 64 (2006), 1356–1366.
-
[13]
M. K. Kwong, J. S. W. Wong, Solvability of second-order nonlinear three-point boundary value problems, Nonlinear Anal., 73 (2010), 2343–2352.
-
[14]
P. Y. Lee, Lanzhou lectures on Henstock integration, Series in Real Analysis, World Scientific Publishing Co., Inc., Teaneck, NJ (1989)
-
[15]
S.-Q. Liang, L. Mu, Multiplicity of positive solutions for singular three-point boundary value problems at resonance, Nonlinear Anal., 71 (2009), 2497–2505.
-
[16]
H.-S. Lü, C.-K. Zhong, Existence theorems for three-point boundary value problem, Ann. Differential Equations, 17 (2001), 162–172.
-
[17]
R.-Y. Ma, Existence theorems for a second order three-point boundary value problem, J. Math. Anal. Appl., 212 (1997), 430–442.
-
[18]
R.-Y. Ma, Positive solutions for second-order three-point boundary value problems, Appl. Math. Lett., 14 (2001), 1–5.
-
[19]
J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee, The ap-Denjoy and ap-Henstock integrals, Czechoslovak Math. J., 57(132) (2007), 689–696.
-
[20]
S. Schwabik, G.-J. Ye, Topics in Banach space integration, Series in Real Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2005)
-
[21]
Y.-P. Sun, L.-S. Liu, Solvability for a nonlinear second-order three-point boundary value problem, J. Math. Anal. Appl., 296 (2004), 265–275.
-
[22]
Y.-P. Sun, M. Zhao, Existence of positive pseudo-symmetric solution for second-order three-point boundary value problems, J. Appl. Math. Comput., 47 (2015), 211–224.
-
[23]
E. Talvila, The distributional Denjoy integral, Real Anal. Exchange, 33 (2008), 51–82.
-
[24]
G.-J. Ye, W. Liu, The distributional Henstock-Kurzweil integral and applications, Monatsh. Math., 181 (2016), 975–989.
-
[25]
J.-F. Zhao, W.-G. Ge, A necessary and sufficient condition for the existence of positive solutions to a kind of singular three-point boundary value problem, Nonlinear Anal., 71 (2009), 3973–3980.
