Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 05, pp. 1–33. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu FRACTIONAL-POWER APPROACH FOR SOLVING COMPLETE ELLIPTIC ABSTRACT DIFFERENTIAL EQUATIONS WITH VARIABLE-OPERATOR COEFFICIENTS FATIHA BOUTAOUS, RABAH LABBAS, BOUBAKER-KHALED SADALLAH Abstract. This work is devoted to the study of a complete abstract second- order differential equation of elliptic type with variable operators as coeffi- cients. A similar equation was studied by Favini et al [6] using Green’s kernels and Dunford functional calculus. Our approach is based on the semigroup theory, the fractional powers of linear operators, and the Dunford’s functional calculus. We will prove the main result on the existence and uniqueness of a strict solutions using combining assumptions from Yagi [16], Da Prato-Grisvard [3], and Acquistapace-Terreni [1]. 1. Introduction In a complex Banach space X, we consider the complete abstract second-order differential equation with variable operators as coefficients u ′′ (x)+ B(x)u ′ (x)+ A(x)u(x) − λu(x)= f (x),x ∈ (0, 1) (1.1) under the Dirichlet boundary conditions u(0) = ϕ, u(1) = ψ. (1.2) Here λ is a positive real number, f ∈ C θ ([0, 1]; X), 0 <θ< 1, ϕ and ψ are given elements in X,(B(x)) x∈[0,1] is a family of bounded linear operators, and (A(x)) x∈[0,1] is a family of closed linear operators whose domains D(A(x)) are not necessarily dense in X. Set Q(x)= A(x) − λI, λ> 0, and consider Problem (1.1)-(1.2) in an elliptic setting. We assume that the family of closed linear operators (Q(x)) x∈[0,1] with domains D(Q(x)) satisfies the condition: There exists C> 0 such that for all x ∈ [0, 1] and all z ≥ 0, exists (Q(x) − zI ) −1 in L(X) and ‖(Q(x) − zI ) −1 ‖ L(X) ≤ C/(1 + z), (1.3) 2000 Mathematics Subject Classification. 34G10, 34K10, 34K30,35J25, 44A45, 47D03. Key words and phrases. Fractional powers of linear operators; analytic semigroup; strict solution; Dunford’s functional calculus. c 2012 Texas State University - San Marcos. Submitted June 14, 2011. Published January 9, 2012. Supported by grant 08 MDU 735 from EGIDE under the CMEP Program. 1