This paper aims to use the Petryshyn's fixed point theorem associated with the measure of non-compactness to prove the existence of solutions of two-dimensional functional integral equations in the Banach algebra of continuous functions on the interval C([0, a] × [0, â], R), a, â > 0. Our existence results contains many functional integral equations as special case that arise in nonlinear analysis. Finally, we present some examples which show that our result is useful for various class of equations.

No abstract

In this article, we provide the existence result for functional integral equations by using Petryshyn’s fixed point theorem connecting measure of non-compactness in Banach space. The results enlarge the corresponding results of several authors. The consequences enlarge the corresponding results of several authors. We present fascinating examples of equations.

In this study the coarticulatory effects on the formant frequencies of sound /r/, one of the most context dependent sounds of Hindi, have been investigated, by taking combinations of the cluster sounds of the forms C1r, rC2 and C3C4r. In these cluster combinations C1 is one of the consonants from (/k/,/p/,/g/,/t/,/d/,/m/,/n/,/h/,/bh/,/w/,/b/,/ph/), C2 is one of the consonants from (/y/,/ñ/,/th/,/t/,/∫/,/m/,/dh/,/k/,/w/,/n/,/g/,/j/,/bh/,/t∫/, /dz/,/ph/,/d/,/s/) and C3C4 is one of the combination from (/∫t/,/st/,/nd/,/n:r/, /ndh/). A database of 438 cluster words spoken by three male speakers has been prepared. The sound /r/ has been perceptually segmented from these utterances and the first three formants and their corresponding bandwidths have been measured. For C1r combination, the deviations of the formant frequencies become larger with the order of the formants. The deviations in the formants for rC2 case are also large and it is nasalized if C2 is some nasal sound like /ñ/ and /m/. For the third case (C1C2r), the formant frequencies do not deviate much as in the earlier two cases but the effect of the frication of the combination /∫t/ makes it fricative. These results are being used in improving the quality of Hindi Klatt Synthesizer.