Existence of solutions for generalized 2d fractional integral equations via Petryshyn’s fixed point theorem. (English) Zbl 07915545
Summary: We establish the existence results of the solutions for fractional Volterra-type integral equations of two variables. We use the method of measure of noncompactness and Petryshyn’s fixed point theorem to obtain these results. Our results contain many previously obtained existence results with more relaxed conditions. Finally, we give an example to illustrate our obtained results.
MSC:
| 45D05 | Volterra integral equations |
| 26A33 | Fractional derivatives and integrals |
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
| 47H10 | Fixed-point theorems |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
fixed-point theorems; fractional integral equation; measures of noncompactness; Volterra integral equationsReferences:
| [1] | J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York, 1980. · Zbl 0441.47056 |
| [2] | M. A. Darwish and S. K. Ntouyas, “On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument”, Nonlinear Anal. 74:11 (2011), 3510-3517. Digital Object Identifier: 10.1016/j.na.2011.02.035 Google Scholar: Lookup Link · Zbl 1228.45006 · doi:10.1016/j.na.2011.02.035 |
| [3] | A. Das, B. Hazarika, and P. Kumam, “Some new generalization of Darbo’s fixed point theorem and its application on integral equations”, Mathematics 7:3 (2019), art. id. 214. Digital Object Identifier: 10.3390/math7030214 Google Scholar: Lookup Link · doi:10.3390/math7030214 |
| [4] | A. Deep, Deepmala, and B. Hazarika, “An existence result for Hadamard type two dimensional fractional functional integral equations via measure of noncompactness”, Chaos Solitons Fractals 147 (2021), art. id. 110874. Digital Object Identifier: 10.1016/j.chaos.2021.110874 Google Scholar: Lookup Link · Zbl 1486.45006 · doi:10.1016/j.chaos.2021.110874 |
| [5] | A. Deep, Deepmala, and J. R. Roshan, “Solvability for generalized nonlinear functional integral equations in Banach spaces with applications”, J. Integral Equations Appl. 33:1 (2021), 19-30. Digital Object Identifier: 10.1216/jie.2021.33.19 Google Scholar: Lookup Link · Zbl 1476.45011 · doi:10.1216/jie.2021.33.19 |
| [6] | M. Kazemi, A. Deep, and J. Nieto, “An existence result with numerical solution of nonlinear fractional integral equations”, Math. Methods Appl. Sci. 46:9 (2023), 10384-10399. Digital Object Identifier: 10.1002/mma.9128 Google Scholar: Lookup Link · Zbl 1530.45005 · doi:10.1002/mma.9128 |
| [7] | M. Kazemi, A. Deep, and A. Yaghoobnia, “Application of fixed point theorem on the study of the existence of solutions in some fractional stochastic functional integral equations”, Math. Sci. (Springer) 18:2 (2024), 125-136. Digital Object Identifier: 10.1007/s40096-022-00489-7 Google Scholar: Lookup Link · Zbl 07886701 · doi:10.1007/s40096-022-00489-7 |
| [8] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006. · Zbl 1092.45003 |
| [9] | C. Kuratowski, “Sur les espaces complets”, Fundam. Math. 15:1 (1934), 301-309. · JFM 56.1124.04 |
| [10] | R. D. Nussbaum, The fixed point index and fixed point theorems for \(k\)-set-contractions, Ph.D. thesis, University of Chicago, Ann Arbor, MI, 1969, available at http://search.proquest.com/docview/302487975. |
| [11] | B. G. Pachpatte, Multidimensional integral equations and inequalities, Atlantis Studies in Mathematics for Engineering and Science 9, Atlantis Press, Paris, 2011. Digital Object Identifier: 10.2991/978-94-91216-17-6 Google Scholar: Lookup Link · Zbl 1232.45001 · doi:10.2991/978-94-91216-17-6 |
| [12] | W. V. Petryshyn, “Structure of the fixed points sets of \(k\)-set-contractions”, Arch. Rational Mech. Anal. 40 (1970/71), 312-328. Digital Object Identifier: 10.1007/BF00252680 Google Scholar: Lookup Link · Zbl 0218.47028 · doi:10.1007/BF00252680 |
| [13] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering 198, Academic Press, San Diego, CA, 1999. · Zbl 0924.34008 |
| [14] | P. Saini, U. Çakan, and A. Deep, “Existence of solutions for 2D nonlinear fractional Volterra integral equations in Banach space”, Rocky Mountain J. Math. 53:6 (2023), 1965-1981. Digital Object Identifier: 10.1216/rmj.2023.53.1965 Google Scholar: Lookup Link · Zbl 1541.45002 · doi:10.1216/rmj.2023.53.1965 |
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