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Zbl 0658.34053
Gupta, Chaitan P.
Boundary value problems for differential equations in Hilbert spaces involving reflection of the argument. (English)
[J] J. Math. Anal. Appl. 128, 375-388 (1987). ISSN 0022-247X

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)].
[S.Myjak]

MSC 2000:: *34G10 Linear ODE in abstract spaces
34B10 Multipoint boundary value problems
47J05 Equations involving nonlinear operators (general)

Keywords: monotonicity type conditions

Citations: Zbl 0636.34013; Zbl 0436.34057

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