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- W2351738862 abstract "A further treatment of the Non-Linear integral equations of Hammerstein type in Lp leads to the resolution of Linear operators of Lq to Lp. The question is: If Linear operator A transforms Lq to Lp, Can the relation A=HH~* hold true, here H being Linear and Continuous, which transforms L_2 to Lp? In addition to this, we also want to study the Complete Continuity of H. A Systematic treatment on this had already been given by only not So Complete. But among the results not accomplished by Several have been worked out by together with the author. This paper is written as the result of a further investigation of the resolution of the operators. It is based upon the author's latest result, which says: If A transform Lq to Lp Linearly and Continuously, L_1 to L_2 Self-conjugately and positively, the A=HH~*, where H transforms L_2 to Lp linearly and Continuously. In Section 2 by making use of a method different from those used in [2], [1]. The author has proved that: 1) If A transforms Lq to Lp Complete Continuously, then H transforms L_2 to Lp Complete Continuously (theorem) 2) If A transforms L_2 to L_2 Complete Continously, then H transforms L_2 to L_p' (p'p) Complete Continuously theorem). In Section 3, an example is given to illustrate how the theorem due to. can given no better result for when p'=p, the theorem may not hold true. In Sections 4 and 5. The author has considered the resolution of the operators of a different type suggested by, The fact is that if we only Suppose A~γ (γ1, any integer), but not A transforms Lq to Lp. We want to know whether the relation A=HH~* still hold true. Concerning this problem, this paper has not given a satisfying result up to now the author has only proved the existence of H, its Continuity and its Complete Continuity, under the resriction that A is strictly positive. In Section 6, the author has applied the results obtained above to the problem Concerning the existence of the Solution of the Non-Linear integral equations of Hammerstein type and the problem Concerning eigen-values, thus improve Certain results given by [1]." @default.
- W2351738862 created "2016-06-24" @default.
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- W2351738862 date "1958-01-01" @default.
- W2351738862 modified "2023-09-28" @default.
- W2351738862 title "ON SOME PROBLEMS OF THE DECOMPOSITION OF LINEAR OPERATORS" @default.
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