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On Two-Parameter Regularized Semigroups and the Cauchy Problem

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  • Mohammad Janfada

Abstract

Suppose that ð ‘‹ is a Banach space and ð ¶ is an injective operator in ð µ ( ð ‘‹ ) , the space of all bounded linear operators on ð ‘‹ . In this note, a two-parameter ð ¶ -semigroup (regularized semigroup) of operators is introduced, and some of its properties are discussed. As an application we show that the existence and uniqueness of solution of the 2-abstract Cauchy problem ( 𠜕 / ( 𠜕 ð ‘¡ ð ‘– ) ) ð ‘¢ ( ð ‘¡ 1 , ð ‘¡ 2 ) = ð » ð ‘– ð ‘¢ ( ð ‘¡ 1 , ð ‘¡ 2 ) , ð ‘– = 1 , 2 , ð ‘¡ ð ‘– > 0 , ð ‘¢ ( 0 , 0 ) = ð ‘¥ , ð ‘¥ ∈ ð ¶ ( ð · ( ð » 1 ) ∩ ð · ( ð » 2 ) ) is closely related to the two-parameter ð ¶ -semigroups of operators.

Suggested Citation

  • Mohammad Janfada, 2009. "On Two-Parameter Regularized Semigroups and the Cauchy Problem," Abstract and Applied Analysis, Hindawi, vol. 2009, pages 1-15, August.
    • Handle: RePEc:hin:jnlaaa:415847
    DOI: 10.1155/2009/415847
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