Lower and upper solutions for a discrete first-order nonlocal problems at resonance
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Authors
Faxing Wang
- TongDa College of Nanjing University of Posts and Telecommunications, 225127 Yangzhou, China.
Ying Zheng
- College of Science, Nanjing University of Posts and Telecommunications, 210046 Nanjing, China.
Abstract
We discuss the existence of solutions for the discrete first-order nonlocal problem
\[
\begin{cases}
\Delta u(t - 1) = f(t, u(t)),\quad t \in \{1, 2, ... , T\},\\
u(0) +\Sigma_{i=1}^m \alpha_iu(\xi_i) = 0,
\end{cases}
\]
where \(f : \{1,..., T\} \times \mathbb{R}\rightarrow \mathbb{R}\) is continuous, \(T > 1\) is a fixed natural number, \(\alpha_i \in (-\infty; 0],\, \xi_i \in \{1,...,T\}(i = 1,..., m; 1 \leq m \leq T; m \in \mathbb{N})\) are given constants such that
\(\Sigma_{i=1}^m \alpha_i+ 1 = 0\). We develop the
methods of lower and upper solutions by the connectivity properties of the solution set of parameterized
families of compact vector fields.
Share and Cite
ISRP Style
Faxing Wang, Ying Zheng, Lower and upper solutions for a discrete first-order nonlocal problems at resonance, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 3, 174--183
AMA Style
Wang Faxing, Zheng Ying, Lower and upper solutions for a discrete first-order nonlocal problems at resonance. J. Nonlinear Sci. Appl. (2015); 8(3):174--183
Chicago/Turabian Style
Wang, Faxing, Zheng, Ying. "Lower and upper solutions for a discrete first-order nonlocal problems at resonance." Journal of Nonlinear Sciences and Applications, 8, no. 3 (2015): 174--183
Keywords
- Coincidence point
- first-order discrete nonlocal problem
- contraction
- lower and upper solutions
- connected sets.
MSC
References
-
[1]
R. D. Anderson, Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions, J. Math. Anal. Appl., 408 (2013), 318-323.
-
[2]
D. Y. Bai, Y. T. Xu, Nontrivial solutions of boundary value problems of second-order difference equations, J. Math. Anal. Appl., 326 (2007), 297-302.
-
[3]
E. Carlini, F. J. A. Silva , Fully Discrete Semi-Lagrangian Scheme for a First Order Mean Field Game Problem Carlinilly Discrete Semi-Lagrangian Scheme for a First Order Mean Field Game Problem, J. Numerical Anal., 52 (2014), 45-47.
-
[4]
S. C. Goodrich, On a first-order semipositone discrete fractional boundary value problem, Arch. Math. (Basel), 99 (2012), 509-518.
-
[5]
J. Henderson, Positive solutions for nonlinear difference equations, Nonlinear Stud., 4 (1997), 29-36.
-
[6]
J. Henderson, H. B. Hompson, Thompson. Existence of multiple solutions for second-order discrete boundary value problems, Comput. Math. Appl., 43 (2002), 1239-1248.
-
[7]
J. Mawhin, P. M. Fitzpatric, Topological degree and boundary value problem for nonlinear differential equations, Lecture Notes in Math., 1537 (1991), 74-172.
-
[8]
L. Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Archiv. der. Mathematik, 99 (2012), 509-518.
-
[9]
R. Y. Ma, Multiplicity results for a three-point boundary value problem at resonance, Nonlinear Anal., 53 (2003), 777-789.
-
[10]
R. Y. Ma, Existence and uniqueness of solutions to first-order three-point boundary value problems, Appl. Math. Letters, 15 (2002), 211-216.
-
[11]
R. Y. Ma, Multiplicity results for an m-point boundary value problem at resonance, Indian J. Math., 47 (2005), 15-31.
-
[12]
E. Mahmudov, Optimal control of Cauchy problem for first-order discrete and partial differential inclusions, J. Dyn. Control Syst., 15 (2009), 587-610.
-
[13]
H. H. Pang, H. Y. Feng, W. G. Ge, Multiple positive solutions of quasi-linear boundary value problems for finite difference equations , Appl. Math. Comput., 197 (2008), 451-456.
-
[14]
P. J. Y. Wong, R. P. Agarwal , Fixed-sign solutions of a system of higher order difference equations , J. Comput. Appl. Math., 113 (2000), 167-181.
-
[15]
R. P. Agarwal, D. O'Regan , Nontrivial solutions of boundary value problems of second-order difference equations, Nonlinear Anal., 39 (2000), 207-215.
-
[16]
J. P. Sun, Positive solution for first-order discrete periodic boundary value problem, Appl. Math. Letters, 19 (2006), 1244-1248.
-
[17]
J. P. Sun, W. T. Li , Existence of positive solutions of boundary value problem for a discrete difference system, Appl. Math. Comput., 156 (2004), 857-870.
-
[18]
P. X. Weng, Z. H. Guo, Existence of Positive Solutions for BVP of Nonlinear Functional Difference Equation with p-Laplacian Operator, Acta Math. Sinica, 1 (2006), 187-194.