Örtüşen Derece Teorisi Üzerine Bazı Düzeltme ve İzahlar

Gaines-Mawhin örtüşen derece teorisi olarak da bilinen örtüşen derece teorisi, özellikle doğrusal olmayan denklemlerdeki çözümün varlığı probleminde güçlü bir tekniktir. Özellikle doğrusal olmayan differensiyel denklemlerin periyodik çözümlerinin varlığının gösterilmesinde çok geniş bir uygulaması olduğundan pek çok araştırması çalışmalarında bu metodu kullanmışlardır. Örtüşen derece teorisinde, bir Banach uzayındaki açık ve sınırlı kümesinde tanımlı formundaki bir operatör denkleminin çözümlerinin varlığı araştırılır. Burada bir doğrusal operatör ve doğrusal olmayan bir operatör olmak üzere ve bazı özel koşulları sağlayan operatörlerdir. Bu çalışmada esas olarak Gaines ve Mawhin’in çalışmaları takip edilmiş, örtüşen teorinin sürdürülebilirlik teoreminin ifadesindeki ikinci sonuç düzeltilmiş ve gerekçesi belirtilmiş. Her ne kadar uygulamalarda birinci sonuç kullanılsa da bu ikincisinin düzeltilmesi de önemli bir çalışmadır. Bu sürdürülebilirlik teoreminin farklı bir şekilde ifadesi verilmiş. Gaines ve Mawhin’in çok az izahla verdiği ispat ilerideki çalışmalara bir yardımcı olmak amacıyla yeterince detaylı bir şekilde izah edilmeye çalışılmıştır.

Anahtar Kelimeler:

Coincidence Degree Theory, L-compact operator, Homotopi Teorisi

On Coincidence Degree Theory Some Corrections and Explanations

Coincidence degree theory, also known Mawhin’s coincidence theory is very powerful technique especially in existence of solutions problems in nonlinear equations. It has especially so broad applications in the existence of periodic solutions of nonlinear differential equations so that many researchers have used it for their investigations. In coincidence degree, mainly existence of solutions of the operator equation in the form in an open and bounded set in some Banach space was researched. Here, is a linear operator and is a nonlinear operator satisfying some special properties. In this study mainly the studies of Gaines and Mahwin are followed, the statement of continuation theory in a coincidence degree theory was corrected and the reason is expressed. A continuation theorem was expressed in different manner. In order to help the researchers with their studies on this subject, the proof that was provided by Gaines and Mawhin has now been presented with more detailed explanation.

Keywords:

Coincidence Degree Theory, -compact operator, Homotopy Theory,

PDF

___

[1] Gaines, R. E., and Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, 1977.
[2] Sırma, A., and Şevgin S., A Note on Coincidence Degree Theory, Abstract and Applied Analysis, (2012) 370946; 1-18. Doi:10.1155/2012/370946.
[3] Mawhin, J. L., Reduction and Continuation Theorems for Brouwer Degree and Applications to Nonlinear Difference Equations, Opuscula Math., (2008) 28(4); 541-560.
[4] Feltrin G., and Zanolin F., Existence of Positive Solutions in Superlinear Case via Coincidence Degree: The Neumann and the Periodic Boundary Value Problems, Adv. Differential Equations, 20(9-10), (2015) 937-982.
[5] Dong, X., Bai, Z. and Zhang, S., Positive Solutions to Boundary Value Problems of p-Laplacian with Fractional Derivative, Boundary Value Problems, 2017(5), (2017). Doi:10.1186/s13661-016-0735-Z
[6] Tian-Wei-Tian, Z., Multiplicity of Positive Almost Periodic Solutions in a Delayed Hassell-Varley-Type Predator-Prey Model with Harvesting on Prey, Math. Meth. Appl. Sci., 37(5), (2014) 686-697. Doi:10.1002/mma.2826
[7] Liu Z., Dynamics of Positive Solutions to SIR and SEIR Epidemic Models with Saturated Incidence Rates, Nonlinear Analysis: Real Word Applications, 14 (2013) 1286-1295. Doi:10/1016/j.nonrwa.2012.09.2016
[8] Muhammadhaji, A., and Teng, Z., Global Attractivity of a Periodic Delayed N-Species Model of Facultative Mutualism, Discrete Dynamics in Nature and Society, 2013, 580185, (2013) 1-11. Doi:10.1155/2013/580185
[9] Li Y., and Ye Y., Multiple Positive Almost Periodic Solutions to an Impulsive Non-autonomous Lotka-Violtera Predator-prey System with Harvesting Terms, Commun Nonlinear Sci Numer Simulat, 18(2013) 3190-3201. Doi:10.1016/j.cnsns.2013.03.014
[10] Hou X., Duan L., and Huang Z., Permanence and Periodic Solutions for a Class of Delay Nicholson’s Blowfies Models, Applied Mathematical Modelling, 37(2013) 1537-1544. Doi:10.1016/j.apm.2012.04.2018
[11] Moussaoui A., Bassaid S., and Ait Dads El H., The Impact of Water Level Fluctuations on a Delayed Prey-Predator Model, Nonlinear Analysis: Real World Applications, 21(2015) 170-184. Doi:10.1016/j.nonrwa.2014.07.011
[12] Niyaz T., and Muhammadhaji A., Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls, International Journal of Differential Equations, (2013) 502963 1-9. Doi:10.1155/2013/502963
[13] Zhang A., Qiu J., and She J., Existence and Global Stability of Periodic Solution for High-order Discrete-time BAM Neural Networks, Neural Networks, 50(2014) 98-109. Doi:10/1016/j.neuret.2013.11.005
[14] Barcaggin A., Feltrin G., and Zanolin F., Positive Solutions for Super-sublinear Indefinite Problems: High Multiplicity Results via Coincidence Degree, Trans. Amer. Math. Soc., 370(2018) 791-845. Doi:10.1090/tran/6992
[15] Xie D., and Jiang Y., Global Exponential Stability of Periodic Solution for Delayed Complex-valued Networks with Impulses, Neurocomputing 207(2016) 528-538. Doi:10.1016/j.neucom.2016.04.054
[16] Feltrin G., and Zanolin F., An Application of Coincidence Degree Theory to Cyclic Feedback Type Systems Associated with Nonlinear Differential Operators, Topol. Methods Nonlinear Anal., 50(2), (2017) 683-726. Doi:10.12775/TMNA.2017.038
[17] Lv W., Solvability for a Discrete Fractional Three-Point Bondary Value Problem at Resonance, Abstract and Applied Analysis, (2014) 601092, 1-7. Doi:10.1155/2014/601092
[18] Li Y., and Qin J., Existence and Global Stability of Periodic Solutions for Quaternion-Valued Cellular Neural Networks with Time-Varying Delays, Neurocomputing, 292(2018) 91-103. Doi:10.1016/j.neucom2018.02.077
[19] Mohammed M.J., Ibrahim R.W., and Ahmad M.Z., Periodicity Computation of Genralized Mathematical Biology Problems Involving Delay Differential Equations, Saudi Journal of Biological Sciences, 24(2017) 737-740. Doi:10.1016/j.sjbs.2017.01.050
[20] Chen F., Coincidence Degree and Fractional Boundary Value Problems with Impulses, Computers and Mathematics with Applications, 64(2012) 3444-3455. Doi:10.1016/j.camwa.2012.02.022