Generalizations of Darbo’s fixed point theorem for new condensing operators with application to a functional integral equation

• Autorzy: Rehman Habib ur , Gopal Dhananjay , Kumam Poom

Identyfikatory

DOI

10.1515/dema-2019-0012

• Języki publikacji: EN

Abstrakty

EN

In this paper, we provide some generalizations of the Darbo’s fixed point theorem associated with the measure of noncompactness and present some results on the existence of the coupled fixed point theorems for a special class of operators in a Banach space. To acquire this result, we define α-ψ and β-ψ con-densing operators and using them we propose new fixed point results. Our results generalize and extendsome comparable results from the literature. Additionally, as an application, we apply the obtained fixedpoint theorems to study the nonlinear functional integral equations.

Słowa kluczowe

EN

Darbo fixed point theorem; measure of noncompactness; fixed point; coupled fixed point; functional integral equations

PL

twierdzenie Darbo o punkcie stałym; miara niezwartości; punkt stały; równania całkowe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 166--182
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Rehman Habib ur

• Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod,Thung Khru, Bangkok 10140, Thailand

autor

Gopal Dhananjay

• National Institute of Technology, Surat- 395007 Gujarat, India

autor

Kumam Poom

• Department of Mathematics, Faculty of Science, King Mongkut’sUniversity of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod,Thung Khru, Bangkok 10140, Thailand

Bibliografia

• [1] Brouwer L. E. J., Über abbildung von mannigfaltigkeiten, Math. Ann., 1911, 71(1), 97-115
• [2] Banach S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 1922, 3(1), 133-181
• [3] Schauder J., Der Fixpunktsatz in Funktional-raumen, Studia Math., 1930, 2(1), 171-180
• [4] Banaś J., On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 1980, 21(1), 131-143
• [5] Banaś J., Applications of measures of weak noncompactness and some classes of operators in the theory of functional equations in the Lebesgue space, Nonlinear Anal., 1997, 30(6), 3283-3293
• [6] Belluce L., Kirk W., Fixed-point theorems for families of contraction mappings, Pacific J. Math., 1966, 18(2), 213-217
• [7] Belluce L. P., Kirk W. A., Nonexpansive mappings and fixed-points in Banach spaces, Illinois J. Math., 1967, 11(3), 474-479
• [8] Edelstein M., An extension of Banach’s contraction principle, Proc. Amer. Math. Soc., 1961, 12(1), 7-10
• [9] Samet B., Vetro C., Vetro P., Fixed point theorems for α–ψ-contractive type mappings, Nonlinear Anal., 2012, 75(4), 2154-2165
• [10] Bhaskar T. G., Lakshmikantham V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 2006, 65(7), 1379-1393
• [11] Kuratowski C., Sur les espaces complets, Fund. Math., 1930, 1(15), 301-309
• [12] Heinz H. P., On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 1983, 7(12), 1351-1371
• [13] Appell J., Measures of noncompactness, condensing operators and fixed points, an application-oriented survey, Fixed PointTheory, 2005, 6(2), 157-229
• [14] Aghajani A., Banaś J., Sabzali N., Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin, 2013, 20(2), 345-358
• [15] Aghajani A., Haghighi A. S., Existence of solutions for a system of integral equations via measure of noncompactness, Novi Sad J. Math., 2014, 44(1), 59-73
• [16] Aghajani A., Mursaleen M., Haghighi A. S., Fixed point theorems for Meir-Keeler condensing operators via measure of non-compactness, Acta Math. Sci., 2015, 35(3), 552-566
• [17] Arab R., Some generalizations of Darbo fixed point theorem and its application, Miskolc Math. Notes, 2017, 18(2), 595-610
• [18] Cai L., Liang J., New generalizations of Darbo’s fixed point theorem, Fixed Point Theory Appl., 2015, 156
• [19] Aghajani A., Jalilian Y., Existence and global attractivity of solutions of a nonlinear functional integral equation, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(11), 3306-3312
• [20] Aghajani A., Jalilian Y., Existence of nondecreasing positive solutions for a system of singular integral equations, Mediterr. J. Math., 2011, 8(4), 563-576
• [21] Banaś J., Rzepka B., An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett., 2003, 16(1), 1-6
• [22] Darwish M. A., Henderson J., O’Regan D., Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc., 2011, 48(3), 539-553
• [23] Dhage B., Bellale S. S., Local asymptotic stability for nonlinear quadratic functional integral equations, Electron. J. Qual. Theory Differ. Equ., 2008, 10, 1-13
• [24] Mursaleen M., Arab R., On existence of solution of a class of quadratic-integral equations using contraction defined bysimulation functions and measure of noncompactness, Carpathian J. Math., 2018, 34(3), 371-378
• [25] Banaś J., Mursaleen M., Sequence spaces and measures of noncompactness with applications to differential and integral equations, 1st ed, Springer, New Delhi, 2014
• [26] Das A., Hazarika B., Arab R., Mursaleen M., Solvability of the infinite system of integral equations in two variables in thesequence spaces c0 and l1, J. Comput. Appl. Math., 2017, 326, 183-192
• [27] Zhang J., Sun J., Exponential synchronization of complex networks with continuous dynamics and Boolean mechanism, Neurocomputing, 2018, 307, 146-152
• [28] Kreyszig E., Introductory functional analysis with applications, Wiley, New York, 1978
• [29] Darbo G., Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova., 1955, 24, 84-92
• [30] Akhmerov R. R., Kamenskii M. I., Patapov A. S., Rodkina A. E., Sadovskii B. N., Measures of noncompactness and condensing operators, 1st ed, Birkhäuser Basel, Springer Basel AG, 1992
• [31] Appell J., Banaś J., Merentes N., Measures of noncompactness in the study of asymptotically stable and ultimately nondecreasing solutions of integral equations, Z. Anal. Anwend., 2010, 29(3), 251-273
• [32] Toledano J. M., Benavides T. D., Acedo G. L., Measures of noncompactness in metric fixed point theory, 1st ed, Birkhäuser Basel, Springer Basel AG, 1997

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).