Therefore, from the Proposition 3. Lemma 3. Therefore, S" is a projection operator. Now, take x e ker S". Therefore, we obtain. From Lemma 3. Beside this, from Lemma 3. Let the operator a : Coker L x [0,1] — ker L be the operator given in Definition 3.
Coincidence Degree, and Nonlinear Differential Equations
Using Proposition 3. So, in order to show that the set M q x [0,1] is relatively compact the only delicate point is the last term. Therefore, compactness can be proven like in the proof of Proposition 3. Using the invariance of Leray-Schauder degree with respect to compact homotopy, we obtain that. Now, let us indicate how the degree d I - M, Q, 0 depends on homotopy class of a.
For this, let us prove the following lemma. For this take, x e ker L, then we have. If we substitute this result in 3. For surjectivity, take y e X. Therefore, A is an automorphism on X. Corollary 3. Under the assumptions of Proposition 3.
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Now, if the orientation on the spaces ker L and Coker L is fixed, then we can give the following beautiful and fruitful definition. If the operators L and N satisfy the conditions i - v then the coincidence degree of L and N in q defined by. In this section, we will see that the coincidence degree satisfies all the basic properties of the Leray-Schauder degree.
Therefore, the conditions iii and iv reduced to the compactness of N on q.
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Theorem 4. Assume that the conditions i to c are satisfied. Then coincidence degree satisfies the following basic properties. In particular. But in fact, we know that x e Dom L n q. Also, by Proposition 3. So the result follows from the definition of coincidence degree. Since the operator N is L-compact on q x [0,1] then it is a homotopy of compact operators on q.
Therefore, by invariance of the Leray-Schauder degree under homotopy property the result follows. We proved that the operator M is compact on q. Therefore, the result follows from the validity of Borsuck theorem in the Leray-Schauder degree. Gaines and J. Li and Y. Kuang, "Periodic solutions in periodic state-dependent delay equations and population models," Proceedings of the American Mathematical Society, vol. Sirma, C. Tunc, and S. Ozlem, "Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with finitely many deviating arguments," Nonlinear Analysis, vol.
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He, L. Share This Paper. Citations Publications citing this paper. Existence of periodic solution for fourth-order generalized neutral p-Laplacian differential equation with attractive and repulsive singularities Yun Xin , Hongmin Liu. Novel existence criteria for periodic solutions of a prescribed mean curvature Rayleigh equation Shulin Yan. Mawhin, J. Publication Timeline. Most widely held works about J Mawhin. Most widely held works by J Mawhin.
Study of a class of arbitrary order differential equations by a coincidence degree method
The first 60 years of nonlinear analysis of Jean Mawhin : April , Sevilla, Spain by J Mawhin 9 editions published in in English and held by 1, WorldCat member libraries worldwide The work of Jean Mawhin covers different aspects of the theory of differential equations and nonlinear analysis. On the occasion of his sixtieth birthday, a group of mathematicians gathered in Sevilla, Spain, in April to honor his mathematical achievements as well as his unique personality.
This book provides an extraordinary view of a number of ground-breaking ideas and methods in nonlinear analysis and differential equations. List of contributors: H. Amann, M. Delgado, J. Krasnoselskij, E. Liz, J. Mawhin, P. Quittner, B. Rynne, L. Sanchez, K.
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Schmitt, J. Ward, F. Zanolin, and others. Coincidence degree and nonlinear differential equations by Robert E Gaines Book 26 editions published between and in English and German and held by WorldCat member libraries worldwide.