• Title, Summary, Keyword: Fredholm operator

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WEYL TYPE-THEOREMS FOR DIRECT SUMS

  • Berkani, Mohammed;Zariouh, Hassan
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.1027-1040
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    • 2012
  • The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $S{\oplus}T$, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property ($gb$) and if ${\Pi}(T){\subset}{\sigma}_a(S)$, ${\PI}(S){\subset}{\sigma}_a(T)$, then $S{\oplus}T$ possesses property ($gb$) if and only if ${\sigma}_{SBF^-_+}(S{\oplus}T)={\sigma}_{SBF^-_+}(S){\cup}{\sigma}_{SBF^-_+}(T)$. Moreover, we prove that if T and S both satisfies generalized Browder's theorem, then $S{\oplus}T$ satis es generalized Browder's theorem if and only if ${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$.

EXISTENCE AND UNIQUENESS THEOREMS OF SECOND-ORDER EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

  • Bougoffa, Lazhar;Khanfer, Ammar
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.899-911
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    • 2018
  • In this paper, we consider the second-order nonlinear differential equation with the nonlocal boundary conditions. We first reformulate this boundary value problem as a fixed point problem for a Fredholm integral equation operator, and then present a result on the existence and uniqueness of the solution by using the contraction mapping theorem. Furthermore, we establish a sufficient condition on the functions ${\mu}$ and $h_i$, i = 1, 2 that guarantee a unique solution for this nonlocal problem in a Hilbert space. Also, accurate analytic solutions in series forms for this boundary value problems are obtained by the Adomian decomposition method (ADM).

Inverse problem for semilinear control systems

  • Park, Jong-Yeoul;Jeong, Jin-Mun;Kwun, Young-Chel
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.603-611
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    • 1996
  • Let consider the following problem: find an element u(t) in a Banach space U from the equation $$ x'(t) = Ax(t) + f(t,x(t)) + \Phi_0 u(t), 0 \leq t \leq T $$ with initial and terminal conditions $$ x(0) = 0, x(T) = \phi $$ in a Banach space X where $\phi \in D(A)$. This problem is a kind of control engineering inverse problem and contains nonlinear term, so that it is difficult and interesting. Thee proof main result in this paper is based on the Fredholm property of [1] in section 3. Similar considerations of linear system have been dealt with in many references. Among these literatures, Suzuki[5] introduced this problem for heat equation with unknown spatially-varing conductivity. Nakagiri and Yamamoto[2] considered the identifiability problem, which A is a unknown operator to be identified, where the system is described by a linear retarded functional differential equation. We can also apply to determining the magnitude of the control set for approximate controllability if X is a reflexive space, i.e., we can consider whether a dense subset of X is covered by reachable set in section 4.

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