Solvability of functional differential equations with multi-point boundary value problems at resonance. (English) Zbl 1142.34357

Du, Zengji

Comput. Math. Appl. 55, No. 11, 2653-2661 (2008).

Summary: We discuss the following third order functional differential equations \(x^{\prime\prime\prime(t)}=f(t,x(t),(Fx)(t),x'(t),(Gx')(t),x^{\prime\prime} (t), (Hx^{\prime\prime})(t))\), \(t\in [0,1]\), subject to the boundary conditions \(x(0)=0\), \(x^{\prime\prime}(0)=0\), \(x'(1)=\sum_{i=1}^{m-2} \alpha_i x' (\eta_i)\), where \(f:[0,1] \times\mathbb{R}^6\to \mathbb{R}\), \(F,G,H\) are three operators, \(\alpha_i\) \((i=1,\dots ,m-2) \geq 0\), \(0<\eta_1<\eta_2<\dots <\eta_{m-2} <1\). Under some appropriate conditions, some existence and multiplicity results are given for the problem at resonance by using a priori estimates and the topological degree theory of Mawhin.

MSC:

34K10	Boundary value problems for functional-differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations

Keywords:

third-order functional differential equations; m-point boundary value problem; topological degree; Carathéodory conditions; resonance; multiplicity

References:

[1]	Gupta, C.P., A second-order \(m\)-point boundary value problems at resonance, Nonlinear anal., 24, 1483-1489, (1995) · Zbl 0824.34023
[2]	Feng, W.; Webb, J.R.L., Solvability of three-point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019
[3]	Ma, R.Y., Multiplicity results for a third order boundary value problem at resonance, Nonlinear anal., 32, 493-499, (1998) · Zbl 0932.34014
[4]	Nagle, R.K.; Pothoven, K.L., On a third-order nonlinear boundary value problems at resonance, J. math. anal. appl., 195, 148-159, (1995) · Zbl 0847.34026
[5]	Gupta, C.P., On a third-order boundary value problem at resonance, Differential intergral equations, 2, 1-12, (1989) · Zbl 0722.34014
[6]	Du, Zengji; Lin, Xiaojie; Ge, Weigao, On a third order multi-point boundary value problem at resonance, J. math. anal. appl., 302, 1, 217-229, (2005) · Zbl 1072.34012
[7]	Du, Zengji; Lin, Xiaojie; Ge, Weigao, Some higher order multi-point boundary value problem at resonance, J. comput. appl. math., 177, 1, 55-65, (2005) · Zbl 1059.34010
[8]	Kuo, C.C., Solvability of a nonlinear two-point boundary value problems at resonance, J. differential equations, 140, 1-9, (1997) · Zbl 0887.34016
[9]	Krasnosel’skii, A.M.; Mawhin, J., On some higher order boundary value problems at resonance, Nonlinear anal., 24, 1141-1148, (1995) · Zbl 0829.34016
[10]	Rachånkové, I.; Staněk, S., Topological degree method in functional boundary value problems at resonance, Nonlinear anal., 27, 271-285, (1996) · Zbl 0853.34062
[11]	Tsamatos, P.Ch., Third order boundary value problems for differential equations with deviating arguments, (), 277-287 · Zbl 0844.34065
[12]	Liu, Bing, Note on third order boundary value problems for differential equations with deviating arguments, Appl. math. comput., 15, 371-379, (2002) · Zbl 1026.34074
[13]	Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025