A novel approach to construct the adjoint problem for a first-order functional integro-differential equation with general nonlocal condition

作者：Ozen Kemal^*; Orucoglu Kamil

来源：Lithuanian Mathematical Journal, 2014, 54(4): 482-502.

DOI：10.1007/s10986-014-9259-3

• Green%26apos
• s function
• functional integro-differential equation
• nonsmooth coefficient
• nonlocal condition
• adjoint problem

In this work, with the aim of determining Green%26apos;s solution or generalized Green%26apos;s solution, we propose a novel constructive approach by which a linear or specific nonlinear problem involving general linear nonlocal condition for a first-order functional ordinary integro-differential equation with general nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness is reduced to an integral equation. A system of two integro-algebraic equations, called %26quot;the adjoint system,%26quot; is constructed for this problem. Green%26apos;s functional for the problem with trivial kernel and generalized Green%26apos;s functional for the problem with nontrivial kernel are the unique solutions to the specific cases of this adjoint system. Green%26apos;s functional and generalized Green%26apos;s functional have two components. Their first components correspond to Green%26apos;s function and generalized Green%26apos;s function for the problem, respectively. Some illustrative applications are provided with known and unknown results.

• 出版日期2014-10