SciELO - Scientific Electronic Library Online

 
vol.40 issue2The forcing total monophonic number of a graph author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.40 no.2 Antofagasta Apr. 2021

http://dx.doi.org/10.22199/issn.0717-6279-2021-02-0032 

Artículos

Applications of measure of noncompactness for the solvability of an infinite system of second order differential equations in some integrated sequence spaces

1Pandu College, Dept. of Mathematics, Guwahati, AS, India. E-mail: ri2p.das@gmail.com

1Sikkim Manipal University, Dept. of Mathematics, Gangtok, SK, India. E-mail: niraj.sapkota13@gmail.com

Abstract

The aim of this paper is to study the infinite system of second order differential equations along with the given boundary conditions for its solvability in some integrated sequence spaces. The result is achieved with the analytical tool namely the measure of noncompactness along with the idea of Meir-Keeler condensing operator and provides the realization of the sufficient conditions for the existence results in these Banach Sequence spaces. We also illustrate the results with examples.

Keywords: Sequence spaces; Measures of noncompactness; Infinite system of differential equations; Fixed point theory; Meir-Keeler condensing operators

Texto completo disponible sólo en PDF

Full text available only in PDF format.

References

[1] K. Kuratowski, “Sur les espaces complets”, Fundamenta mathematicae, vol.15, no. 1, pp. 301-309, 1930. [On line]. Available: https://bit.ly/3cxtQr5Links ]

[2] G. Darbo, “Punti uniti in transformazioni a condominio non compatto”, Rendiconti del Seminario Matematico della Università di Padova, vol. 24, pp. 84-92, 1955. [On line]. Available: https://bit.ly/3ctf8kMLinks ]

[3] L. S. Goldenstein and A. S. Markus, “On a measure of noncompactness of bounded sets and linear operators”, Studies in algebra and mathematical analysis, vol. 29, pp. 45-54, 1965. [ Links ]

[4] A. Meir and E. Keeler, “A theorem on contraction mappings”, Journal of mathematics analysis and applications, vol. 28, no. 1, 1969, pp. 326-329, doi: 10.1016/0022-247X(69)90031-6 [ Links ]

[5] G. Goes and S. Goes, “Sequences of bounded variation and sequences of Fourier coefficients. I”, Mathematische zeitschrift, vol. 118, no. 2, pp. 93-102, 1970, doi: 10.1007/BF01110177 [ Links ]

[6] V. Istrăţescu, “On a measure of noncompactness”, Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série, vol. 16, no. 2, pp. 195-197, 1972. [On line]. Available: https://bit.ly/3vlMcDQLinks ]

[7] J. Banaś and K. Goebel , Measures of noncompactness in Banach spaces. New York, NY: Marcel Dekker, 1980. [ Links ]

[8] K. P. Persidskii, “Countable systems of differential equations and stability of their solutions”, Izvestiya Akademii Nauk Kazakh SSR, vol. 7, pp. 52-71, 1959. [ Links ]

[9] K. P. Persidskii, “Countable systems of differential equations and stability of their solutions III: fundamental theorems on stability of countable many differential equations”, Izvestiya Akademii Nauk Kazakh SSR , vol. 9, pp. 11-34, 1961. [ Links ]

[10] K. P. Persidskii, “Infinite system of differential equations”, Izvestiya Akademii Nauk Kazakh SSR Alma Ata Differential Equations. Nonlinear Spaces, 1976. [ Links ]

[11] J. Banaś, “Measures of noncompactness in the space of continuous tempered functions”, Demonstratio mathematica, vol. 14, no. 1, pp. 127-133, 1981, doi: 10.1515/dema-1981-0110 [ Links ]

[12] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measures of noncompactness and condensing operators, operator theory: advances and applications. Basel: Birkhäauser, 1992, doi: 10.1007/978-3-0348-5727-7 [ Links ]

[13] J. Banaś and A. Martinon, “Some properties of the Hausdorff distance in metric spaces”, Bulletin of the Australian Mathematical Society, vol. 42, no. 3, pp. 511-516, 1990, doi: 10.1017/S0004972700028677 [ Links ]

[14] J. Banaś and A. Martinon, “Measures of noncompactness in Banach sequence spaces”, Mathematica Slovaca, vol. 42, no. 4, pp. 497-493, 1992. [On line]. Available: https://bit.ly/3qZ6PTbLinks ]

[15] J. Banaś and M. Lecko, “Solvability of infinite systems of differential equations in Banach sequence spaces”, Journal of computational and applied mathematics, vol. 137, no. 2, pp. 363-375, 2001, doi: 10.1016/S0377-0427(00)00708-1 [ Links ]

[16] D. G. Duffy, Greens functions with applications. Boca Raton, FL: Chapman & Hall/CRC, 2001. [ Links ]

[17] R. P. Agarwal, M. Benchohra, and D. Seba, “On the application of measure of noncompactness to the existence of solutions for fractional differential equations”, Results in mathematics, vol. 55, no. 3-4, pp. 221-230, 2009, doi: 10.1007/s00025-009-0434-5 [ Links ]

[18] M. Mursaleen and A. K. Noman, “Compactness by the Hausdorff measure of noncompactness”, Nonlinear analysis: theory, methods & applications, vol. 73, no. 8, pp. 2541-2557, 2010, doi: 10.1016/j.na.2010.06.030 [ Links ]

[19] M. Mursaleen and S. A. Mohiuddine, “Applications of measures of noncom-pactness to the infinite system of differential equations in ℓp spaces”, Nonlinear analysis: theory, methods & applications , vol. 75, no. 4, pp. 2111-2115, 2012, doi: 10.1016/j.na.2011.10.011 [ Links ]

[20] J. Banaś, and M. Mursaleen , Sequence spaces and measures of non-compactness with applications to differential and integral equations. New Delhi: Springer, 2014, doi: 10.1007/978-81-322-1886-9 [ Links ]

[21] A. Aghajani, R. Allahyari, andM. Mursaleen , “A generalization of Darbos theorem with application to the solvability of systems of integral equations”, Journal of computational and applied mathematics , vol. 260, pp. 68-77, 2014, doi: 10.1016/j.cam.2013.09.039 [ Links ]

[22] A. Aghajani , M. Mursaleen , and A. S. Haghighi, “Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness”, Acta mathematica scientia, vol. 35, no. 3, pp. 552-566, 2015, doi: 10.1016/S0252-9602(15)30003-5 [ Links ]

[23] A. Alotaibi, M. Mursaleen , andS. A. Mohiuddine , “Application of measure of noncompactness to infinite system of linear equations in sequence spaces”, Bulletin of the Iranian Mathematical Society, vol. 41, no. 2, pp. 519-527, 2015. [On line]. Available: https://bit.ly/30MVB9FLinks ]

[24] M. Kirişci, “Integrated and differentiated sequence spaces”, Journal of nonlinear analysis and application, vol. 2015, no. 1, pp. 2-16, 2015, doi: 10.5899/2015/jnaa-00266 [ Links ]

[25] M. Mursaleen and S. M. H. Rizvi, “Solvability of infinite systems of second order differential equations in c0 and ℓ1 by Meir-Keeler condensing operators”, Proceeding of the American Mathematical Society, vol. 144, no. 10, pp. 4279-4289, 2016, doi: 10.1090/proc/13048 [ Links ]

[26] H. M. Srivastava, A. Das, B. Hazarika, andS. A. Mohiuddine , “Existence of solutions of infinite systems of differential equations of general order with boundary conditions in the spaces c0 and ℓ1 via the measure of noncompactness”, Mathematical methods in the applied sciences, vol. 41, no. 10, pp. 3557-3569, 2018, doi: 10.1002/mma.4845 [ Links ]

[27] A. Das , B. Hazarika, andM. Mursaleen , “Application of measure of noncom-pactness for solvability of the infinite system of integral equations in two variables in ℓp (1 <p< ∞)”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 1, pp. 31-40, 2019, doi: 10.1007/s13398-017-0452-1 [ Links ]

[28] R. Saadati, E. Pourhadi, andM. Mursaleen , “Solvability of infinite systems of third-order differential equations in c0 by Meir-Keeler condensing operators”, Journal of fixed point theory and applications, vol. 21, Art. ID. 64, 2019, doi: 10.1007/s11784-019-0696-9 [ Links ]

[29] B. Hazarika, R. Arab, andM. Mursaleen , “Application measure of noncompactness and operator type contraction for solvability of an infinite system of differential equations in ℓp -space”, Filomat, vol. 33, no. 7, pp. 2181-2189, 2019, doi: 10.2298/FIL1907181H [ Links ]

Received: January 31, 2020; Accepted: November 30, 2020

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License