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Boundary value problems for differential equations in Hilbert spaces involving reflection of the argument. (English) Zbl 0658.34053
Under the monotonicity type conditions, by using Leray-Schauder degree argument, some natural results concerning the existence and uniqueness of solutions of the equation $-x''(t)+\alpha x'(t)+f(t,x(t),x(-t))=e(t)$ satisfying the boundary value condition $x(-1)=x(1)=0$ (resp. $x(- 1)=hx'(-1)$, $x(1)=-hx'(1))$ are given. Here f: [-1,1]$\times H\times H\to H$ is completely continuous, H is a Hilbert space, $e\in L\sp 1([- 1,1],H)$ and h,k$\ge 0$, $h+k>0:$ This paper generalizes a paper of the author [Nonlinear Analysis and Appl., Proc. 7th Int. Conf. Arlington/Tex. 1986, Lect. Notes Pure Appl. Math. 109, 223-228 (1987; Zbl 0636.34013)], and originally derives from {\it J. Mawhin} [Tohoku Math. J. 32, 225-233 (1980; Zbl 0436.34057)].
Reviewer: S.Myjak

34G10Linear ODE in abstract spaces
34B10Nonlocal and multipoint boundary value problems for ODE
47J05Equations involving nonlinear operators (general)
Full Text: DOI
[1] Hardy, G. H.; Littlewood, J. E.; Polya, G.: Inequalities. (1952) · Zbl 0047.05302
[2] Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces. (1976) · Zbl 0327.47022
[3] C. P. Gupta, Existence and uniqueness theorems for boundary value problems involving reflection of the argument, Nonlinear Anal. Theory Meth. Applications, to appear.
[4] Mawhin, J.: Topological degree methods in nonlinear boundary value problems. Regional conference series in math. 40 (1979) · Zbl 0414.34025
[5] Mawhin, J.: Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces. Tohoku math. J. 32, 225-233 (1980) · Zbl 0436.34057
[6] Weiner, J.; Aftabizadeh, A. R.: Boundary value problems for differential equations with reflection of argument. Internat. J. Math. math. Sci. 8, 151-163 (1985) · Zbl 0583.34055