Abstract

In this paper we discuss solutions of the Abstract Cauchy Problem Bu(α) (t) = Au(t)+ f (t) u(0) = x0 that are of the form u = u1⊗δ1+u2⊗δ2 , where u and f are functions defined on [0,a] with values in the Hilbert space l2.

Highlights

  • In 2010, Ziqan, A.M., Al Horani, M. and Khalil, R. used tensor product technique to find a unique solution for the Abstract Cauchy Problem under certain conditions on the operators A and B [8], [9]

  • In this paper we studied the non-homogeneous Abstract Cauchy Problem using the tensor product technique when the non-homogeneous part of the problem is a two rank function

  • In this paper we study the non-homogeneous α−Abstract Cauchy Problem using the tensor product technique when the non-homogeneous part of the problem is a two rank function

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Summary

Introduction

One of the most important differential equations in Banach spaces, which is involved in many sectors of sciences, is the so-called the Abstract Cauchy Problem, and it has the form: Many researchers were and are interested in this Problem and studied it using a variety of methods. In 2002, the inverse form of Problem (E) was studied by Al Horani, M. In 2010, Ziqan, A.M., Al Horani, M. and Khalil, R. used tensor product technique to find a unique solution for the Abstract Cauchy Problem under certain conditions on the operators A and B [8] , [9]. In this paper we studied the non-homogeneous Abstract Cauchy Problem using the tensor product technique when the non-homogeneous part of the problem is a two rank function. We did prove the existence of a unique solution for this type of problems when u has the form u1 ⊗ δ 1 + u2 ⊗ δ 2, where u1, u2 are real-valued continuously differentiable functions on I, and with some conditions on the operators A and B. We found an atomic solution for non-degenerate problem of this type of the non-homogeneous

Basic Facts On Conformable Fractional Derivatives
Two Rank Solution

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