(1) He had to rely on whatever books were lying around.
(2) He knew nothing whatever about it.
(3) There is no scientific evidence whatever to support such a view.

以上三句都正确。

句（1）whatever 是融合型关系限定词，引导名词性关系分句。whatever 没有疑问含义。分句相当于一个名词词组。

whatever books were lying around = any books that were lying around （任何在周围的书籍）

句（2）和（3）whatever 是用来强调否定的副词，意为“任何；丝毫”。用于名词性词组（包括不定代词）后，强调否定陈述。跟 at all 的用法类似，可以省去。

(2)He knew nothing about it.
(3)There is no scientific evidence to support such a view.

【翻译】

（1）他不得不依靠阅读身边周围的书籍。

（2）他对此一无所知

（3）没有任何证据表明，广播业的竞争使成本有所降低。