Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equa
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Journal of Fixed Point Theory and Applications
Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations Farzaneh Pouladi Najafabadi, Juan J. Nieto Hojjatollah Amiri Kayvanloo
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Abstract. First, we introduce a new measure of noncompactness in weighted Sobolev spaces Wwm,p (Ω), and then as an application, we study the existence of solutions for a class of the nonlinear convolution type integral equations using Darbo’s fixed point theorem associated with this new measure of noncompactness. Further, an example is presented to verify the effectiveness and applicability of our main results. Mathematics Subject Classification. Primary 47H08, Secondary 45J05, 47H10. Keywords. Measures of noncompactness, Darbo’s fixed point theorem, convolution differential equations, weighted Sobolev spaces, Carath¨eodory condition.
1. Introduction Sobolev spaces, which are the classes of functions with derivatives in Lp , play an outstanding role in the modern analysis; see [15,31]. In the last decades, there have been increasing attempts to study of these spaces. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces. They are also highlighted in the approximation theory, calculus of variation, differential geometry, spectral theory, and so on. On the other hand, integral–differential equations have a great deal of applications in some branches of sciences. They arise especially in a variety of models from applied mathematics, biological science, physics, and another phenomenon, such as the theory of electrodynamics, electromagnetic, fluid dynamics, heat and oscillating magnetic, and so on; see [6,10,13,14,23,24]. Recently, a number of interesting papers [2,5,9,11,25,27,28,32,33] on the solvability of various integral equations with the help of measures of noncompactness have been presented. 0123456789().: V,-vol
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Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. Details can be found in almost any book on partial differential equations. For degenerate partial differential equations, which are equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces; see, for example, [20–22,26]. The first such a measure was defined by Kuratowski [29]. Next, Bana´s and Goebel [8] proposed a generalization of this notion, which is more convenient in the applications. The technique of measures of noncompactness is frequently applicable in several branches of nonlinear analysis; in particular, the technique turns out to be very useful tool in the existence theory for several types of integral and integral–differential equations. Furthermore, these measures are often used in the functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory, and optimal control theory; see [1,3,7,12,16–19].
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