EXISTENCE OF SYMMETRIC POSITIVE SOLUTIONS FOR LIDSTONE TYPE INTEGRAL BOUNDARY VALUE PROBLEMS

Yıl 2018, Cilt: 8 Sayı: 1.1, 295 - 305, 01.09.2018

N. Sreedhar K. R. Prasad S. Balakrishna

Öz

This paper establishes the existence of even number of symmetric positive solutions for the even order differential equation −1 n u 2n t = f t, u t , t ∈ 0, 1 , satisfying Lidstone type integral boundary conditions of the form u 2i 0 = u 2i 1 = Z 1 0 ai+1 x u 2i x dx, for 0 ≤ i ≤ n − 1, where n ≥ 1, by applying Avery–Henderson fixed point theorem.

Anahtar Kelimeler

Green’s function, integral boundary conditions, cone, positive solution, fixedpoint theorem

Kaynakça

• Agarwal, R. P., O’Regan, D. , Wong, P. J. Y., 1999, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands.
• Avery, R. I., Henderson, J., (2001), Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal., 8, no. 1, 27–36.
• Bai, Z, Ge, W., (2003), Solutions of 2nth Lidstone boundary value problems and dependence on higher order derivatives, J. Math. Anal. Appl., 279 , 442–450.
• Belarbi, A., Benchohra, M., (2005), Existence results for nonlinear boundary value problems with integral boundary conditions, Electron. J. Differ. Equ., No. 6, 1-10.
• A. K. Boucherif, (2007), Positive solutions of second order differential equations with integral boundary conditions, Discrete Cont. Dyn. Syst., 155–159.
• Davis, J. M., Henderson, J., (1999), Triple positive symmetric solutions for a Lidstone boundary value problem,Differ. Equ. Dyn. Syst. 7, 321–330.
• Davis, J. M., Henderson, J, Wong, P. J. Y., (2000) General Lidstone problems: Multiplicity and symmetry of solutions, J. Math. Anal. Appl. 251, 527–548.
• Ehme, J., Henderson, J., (2000), Existence and local uniqueness for nonlinear Lidstone boundary value problems, J. Inequalities Pure Appl. Math., No. 1, Article 8, 1–9.
• Galvis, J., Rojas, E. M., Sinitsyn, A. V., (2015), Existence of positive solutions of a nonlinear second order boundary value problem with integral boundary conditions, Electron. J. Differ. Equ., No. 236, 1–7.
• Ji, Y., Guo, Y., Yao, Y., (2015), Positive solutions for higher order differential equations with integral boundary conditions, Boundary Value Problems, No. 214, 1–11.
• Kang, P., Wei, Z, Xu, J., (2008), Positive solutions to fourth order singular boundary value problems with integral boundary conditions in abstract spaces, Appl. Math. Comput., No. 1, 245–256.
• Liu, L, Hao, X, Wu, Y., (2013), Positive solutions for singular second order differential equations with integral boundary conditions, Math. Comp. Modelling,57, No. 3–4, 836–847.
• Ma, H., (2008), Symmetric positive solutions for nonlocal boundary value problems of fourth order, Nonlinear Anal., 68, No. 3, 645–651.
• Sun, Y. P., (2011), Three Symmetric positive solutions for second order nonlocal boundary value problems, Acta Math. Appl. Sin. Engl. Ser., 27, No. 2, 233–242.
• Wang, Q., Guo, Y., Ji, Y., (2013), Positive solutions for fourth order nonlinear differential equation with integral boundary conditions, Discrete Dyn. Nature Soc., article ID: 684962, 1–10.
• Wong, P. J. Y., Agarawal, R. P., (1999), Eigenvalues of Lidstone boundary value problems, Appl. Math. Comput., 104, 15–31.
• Xu, F., (2011) Three symmetric positive solutions of fourth order nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ., No. 96, 1–11.
• Xu, F., Liu, J., (2010), Symmetric positive solutions for nonlinear singular fourth order eigenvalue problems with nonlocal boundary condition, Discrete Dyn. Nature Soc., article ID: 187827, 1–16.
• Zhang, B., Liu, X., (2003), Existence of multiple symmetric positive solutions of higher order Lidstone problems, J. Math. Anal. Appl., 284, 672–689.
• Zhang, L., Xuan, Z., (2016), Multiple Positive solutions for a second order boundary value problem with integral boundary conditions, Boundary Value Problems, No. 60, 1–8.
• Zhang, X., Feng, M., Ge, W., (2008), Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces, Nonlinear Anal., 69,No. 10, 3310–3321.
• Zhang, X., Ge, W., (2009), Positive solutions for a class of boundary value problems with integral boundary conditions, Comp. Math. Appl., 58, no. 2, 203–215.