A CLASS OF THIRD-ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITION AT RESONANCE

In this paper, we consider third-order boundary value problem with, Dirichlet, Neumann and integral conditions at resonance case, where the kernel’s dimension of the ordinary differential operator is equal to one and the ordinary differential equation which can be written as the abstract equation Lu = Nu, called semilinear form, where L is a linear Fredholm operator of index zero, and N is a nonlinear operator. First, we prove a priori estimates, and then we use Mawhin’s coincidence degree theory to deduce the existence of solutions. One important ingredient to be able to apply this abstract results (Mawhin’s coincidence degree theory) is proving the Fredholm property of the operator L. An example is also presented to illustrate the effectiveness of the main results with integral condition at resonance case The existence of solution is etablished via Mawhin's coincidence degree theory. The results are illustrated with an example.

Keywords:

Resonance, Coincidence degree theory, Fredholm operato, One dimensional kernels Nonlocal boundary value problem,

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[1] A. Yong, B. Sun and W. Ge; Existence of positive solutions for self-adjoint boundary valueproblems with integral boundary condition at resonance, Electron. J. Differential Equations.(2011), 11, 1-8.
[2] A. Guezane-Lakoud, A. Frioui; Third-order boundary value problem with integral condition atresonance, Theory and Application of Mathematics and Comput Science. 3 (1)(2013), 56-64.
[3] B. Przeradzki and R. Stanczy; Solvability of multi-point boundary value problems at reso-nance, J. Math. Anal. Appl. 264 (2001), No. 2, 253261.
[4] C. P. Gupta; Solvability of a three-point nonlinear boundary value problem for a second orderordinary differential equations, J. Math. Anal. Appl. 168 (1998), 540{551.[
5] C. Xue, Z. Du and W. Ge; Solutions of M-point boundary value problems of third-orderordinary differential equations at resonance, J. Appl. Math. Comput. 17 (2005), No. 1-2,229{244.
[6] E. R. Kaufmann; A third-order nonlocal boundary value problem at resonance, Electron. J.Qual. Theory Differ. Equ. 2009 (2009), No. 16, 1-11.
[7] F. Meng and Z. Du; Solvability of a second order multi-point boundary value problems atresonance, Appl. Math. Comput 208, (2010), 23-30.
[8] G. Karakostas and P. Ch. Tsamatos; On a nonlocal boundary value problem at resonance, J.Math. Anal. Appl. 259 (2001), No. 1, 209-218.
[9] G. Karakostas and P. Ch. Tsamatos; Suffcient conditions for the exsistence of nonnegativesolutions of a nonlocal boundary value problem. Appl. Math. Letters 15(4) (2002), 401{407.
[10] H. Zhang. W. Liu and T. Chen; Exsistence of solutions for three-point boundary value prob-lem. J. Appl. Math and Informatics. 27(5-6) (2009), 35{51.
[11] H. Z. Wang, Y. Li; Neumann boundary value problems for second-order ordinary differentialequations across resonance, SIAM J. Control Optim. 33 (1995), 1312{1325.
[12] J. Mawhin; Tobological degree Methods in Nonlinear Boundary Value Problems, NSF-CBMSRegional Conference Series in Mathematics,40, Amer. Math. Soc., Providence, RI, 1979.
[13] J.R.L. Webb, M. Zima; Multiple positive solutions of resonant and non-resonant nonlocalboundary value problems, Nonlinear Anal. 71 (2009) 1369{1378.
[14] N. Kosmatov; Multi-point boundary value problems on an unbounded domain at resonance,Nonlinear Anal. 68 (2008), 2158{2171.
[15] R. Ma; Suffcient Conditions for the Existence of Nonnegative Solutions of a Nonlocal Bound-ary Value Problem, Appl. Math. Letters 15 (2002) 401{407.
[16] R. Ma; Multiplicity results for a third order boundary value problem at resonance, NonlinearAnal. 32 (2005), 493{499.
[17] W. Feng and J. R. L. Webb; Solvability of three-point boundary value problem at resonance,Nonlinear Anal. Theory, Methods and Appl 30, (1997) 3227{3238.
[18] W. H. Jiang, Y. P. Guo and J. Qiu; Solvability of 2n-order m-point boundary value problemat resonance, Appl. Math. Mech. (English Ed) 28(2007), No. 9, 1219-1226.
[19] X. Liu, J. Qiu, Y. Guo; Three positive solutions for second-order m-point boundary valueproblems, Appl. Math. Comput. 156 no. 3 (2004) 733{742
[20] X. Zhang, M. Feng and W. Ge; Existence result of second-order differential equations withintegral boundary conditions at resonance, J. Math. Appl. 353 (2009),1, 311-319.
[21] Y. Lin, Z. D and F. Meng; A note on third-order multi-point boundary value problem atresonance, Math. Nachr 284 (13) (2011), 1690-1700.
[22] Z. Du, W. Ge, X. Lin; Nonlocal boundary value problem of higher order ordinary differentialequations at resonance, Rocky Mountain J. Math. 36 No. 5 (2006), 1471{1486.

ISSN: 2667-7660
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2019
Yayıncı: Hüseyin ÇAKALLI

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