# Article

Full entry | PDF   (0.3 MB)
Keywords:
Fredholm operator; coincidence degree; left focal problem; nontrivial solution; resonance
Summary:
This paper deals with the generalized nonlinear third-order left focal problem at resonance $$\begin {cases} (p(t)u''(t))'-q(t)u(t)=f(t, u(t), u'(t), u''(t)), \quad t\in \mathopen ]t_0, T[, m(u(t_0), u''(t_0))=0, n(u(T), u'(T))=0, l(u(\xi ), u'(\xi ), u''(\xi ))=0, \end {cases}$$ where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the topological degree technique as well as some recent generalizations of this technique. Our results are generalizations and extensions of the results of several authors. An application is included to illustrate the results obtained.
References:
[1] Agarwal, R. P.: Focal Boundary Value Problems for Differential and Difference Equations. Mathematics and its Applications 436 Kluwer Academic Publishers, Dordrecht (1998). MR 1619877 | Zbl 0914.34001
[2] Anderson, D. R.: Discrete third-order three-point right-focal boundary value problems. Comput. Math. Appl. 45 (2003), 861-871. DOI 10.1016/S0898-1221(03)80157-8 | MR 2000563 | Zbl 1054.39010
[3] Anderson, D. R., Davis, J. M.: Multiple solutions and eigenvalues for third-order right focal boundary value problems. J. Math. Anal. Appl. 267 (2002), 135-157. DOI 10.1006/jmaa.2001.7756 | MR 1886821 | Zbl 1003.34021
[4] Gupta, C. P.: Solvability of multi-point boundary value problem at resonance. Result. Math. 28 (1995), 270-276. DOI 10.1007/BF03322257 | MR 1356893
[5] Kosmatov, N.: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal., Theory Methods Appl. 68 (2008), 2158-2171. DOI 10.1016/j.na.2007.01.038 | MR 2398639 | Zbl 1138.34006
[6] Liao, S. J.: A second-order approximate analytical solution of a simple pendulum by the process analysis method. J. Appl. Mech. 59 (1992), 970-975. DOI 10.1115/1.2894068 | Zbl 0769.70017
[7] Liu, B., Yu, J. S.: Solvability of multi-point boundary value problem at resonance III. Appl. Math. Comput. 129 (2002), 119-143. DOI 10.1016/S0096-3003(01)00036-4 | MR 1897323 | Zbl 1054.34033
[8] Liu, Z., Debnath, L., Kang, S.: Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem. Comput. Math. Appl. 55 (2008), 356-367. DOI 10.1016/j.camwa.2007.03.021 | MR 2384152 | Zbl 1155.34312
[9] Mawhin, J.: Topological Degree Methods in Nonlinear Boundary Value Problems. Regional Conference Series in Mathematics 40. AMS, Providence, RI (1979). MR 0525202 | Zbl 0414.34025
[10] Mawhin, J., Ruiz, D.: A strongly nonlinear Neumann problem at resonance with restrictions on the nonlinearity just in one direction. Topol. Methods Nonlinear Anal. 20 (2002), 1-14. DOI 10.12775/TMNA.2002.021 | MR 1940526 | Zbl 1018.34016
[11] Nasr, H., Hassanien, I. A., El-Hawary, H. M.: Chebyshev solution of laminar boundary layer flow. Int. J. Comput. Math. 33 (1990), 127-132. DOI 10.1080/00207169008803843 | Zbl 0756.76058
[12] Nieto, Juan J.: Existence of solutions in a cone for nonlinear alternative problems. Proc. Am. Math. Soc. 94 (1985), 433-436. DOI 10.1090/S0002-9939-1985-0787888-1 | MR 0787888 | Zbl 0585.47050
[13] Rachůnková, I., Staněk, S.: Topological degree method in functional boundary value problems at resonance. Nonlinear Anal., Theory Methods Appl. 27 (1996), 271-285. DOI 10.1016/0362-546X(95)00060-9 | MR 1391437 | Zbl 0853.34062
[14] Santanilla, J.: Some coincidence theorem in wedges, cones, and convex sets. J. Math. Anal. Appl. 105 (1985), 357-371. DOI 10.1016/0022-247X(85)90053-8 | MR 0778471
[15] Wong, P. J. Y.: Multiple fixed-sign solutions for a system of generalized right focal problems with deviating arguments. J. Math. Anal. Appl. 323 (2006), 100-118. DOI 10.1016/j.jmaa.2005.10.016 | MR 2261154 | Zbl 1107.34014

Partner of