The influences of his job in Dusseldorf, as a conductor of the Municipal Orchestra and Choir, and the friendship with his violinist colleague, Joseph von Wasielewski might have something to do with the writing of the first violin sonata. This violinist was connected to the Gewandhaus-orchestra en was much younger than Schumann himself, 28 years. Schumann did write the Marchenbilder Op.113 for viola for Wasielewski who also knew how to play the viola.
For the second sonata, which he as well wrote in 1851, was dedicated to Ferdinand David. Some sources say that the piece was written with him in mind, others do not mention this. Both works are written soon after each other and it is hard to say which violinist Schumann had in mind if one at all.
The job, as a conductor, which Schumann took was not always easy for him. He composed less than he wanted and he was at times unappreciated according to Clara and himself. It is known that they wanted to leave Dusseldorf because the public wanted to hear different things than he programmed with the orchestra. He was a big promoter of new music and the old Italian; French or even Mozart were almost never performed.
Due to the pressure and the stress that came from the expensive life in Dusseldorf, the 'amateur' choir' as he called it and the lack of rest to compose was of a great mental strain to Robert Schumann. He was quickly exhausted and especially in 1852/1853 his health deteriorated fast. It went so far that he had attacks:
'that resembled mild strokes, he had no strength or Stamina, and worst of all, he suffered from auditory hallucinations in which certain tones sounded incessantly' (Nancy B. Reich "Clara Schumann, the artist and the woman" P.138)
The violin sonatas stand out from the works that were written until then for violin and piano works. They are less brilliant but more dramatic than Mendelssohn's work which really wrote in the style-Brilliante. The pieces are full of gorgeous melodies in the slow movements, remembering us of the songs, and the passionate dramatic style in the faster movements of which his works, especially for solo piano, are so well known. One can definitely imagine from the music all the struggles that he had during his time in Düsseldorf.
Regarding Schumann's interest in the violin we can look at his fascination for Nicolo Paganini that started at the beginning of his composing career. In his popular piano-work, Carnaval Op.9 did he include one piece that he called 'Paganini'. His op.3 for solo piano, as well, are 'Etudes after caprices of Paganini' and he wrote a piano-accompaniment for all the Caprices as well. He kept interested and inspired by the composer until the end of his life. There is a letter to Clara in which he asks for the score of the Paganini caprices and empty paper to write his musical ideas for accompaniments. When Clara (due to the worries of the stress that composing might take upon his mental state) does not respond to this letter, Schumann writes a Letter to Johannes Brahms asking to remind her to bring the score.