International Journal of Nonlinear Analysis and Applications

Some Wardowski-Mizogochi-Takahashi-Type generalizations of the multi-valued version of Darbo's fixed point theorem with applications

Document Type : Research Paper

Authors

1 Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran

2 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E- Gharb, Iran

3 Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran

Abstract

In this paper, we extend the multi-valued version of Darbo's fixed point theorem using generalized Mizogochi-Takahashi mappings of the Wardowski type. The technique of measure of noncompactness is the main tool in carrying out our proofs. As an application, we investigate the existence of solutions for an integral inclusion on the space $BC(\mathbb{R}_+,E)$.

Keywords

[1] R. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2004.
[2] A. Aghajani, M. Mursaleen and A. Shole Haghighi, A generalization of Darbo’s theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math. 260 (2014), 68–77.
[3] R. Arab, M. Mursaleen and S.M.H. Rizvi, Positive Solution of a quadratic integral equation using generalization of Darbo’s fixed point theorem, Numer. Funct. Anal. Optim. 40 (2019), no. 10, 1150–1168.
[4] S.J. Bana and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, New York, 1980.
[5] S. J. Bana, M. Jleli, M. Mursaleen and B. Samet, Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer, Singapore, 2017.
[6] Sh. Banaei, Solvability of a system of integral equations of Volterra type in the Frechet space Lploc(R+) via measure of noncompactness, Filomat 32 (2018), 5255–5263.
[7] Sh. Banaei, An extension of Darbo’s theorem and its application to existence of solution for a system of integral equations, Cogent Math. Statist. 6 (2019), no. 1, 1614319.
[8] Sh. Banaei and M. Ghaemi, A generalization of the Meir-Keeler condensing operators and its application to solvability of a system of nonlinear functional integral equations of Volterra type, Sahand Commun. Math. Anal. 15 (2019), 19–35.
[9] Sh. Banaei, M. Ghaemi and R. Saadati, An extension of Darbo’s theorem and its application to system of neutral differential equations with deviating argument, Miskolc Math. Notes 18 (2017), 83–94.
[10] Sh. Banaei, M. Mursaleen and V. Parvaneh, Some fixed point theorems via measure of noncompactness with applications to differential equations, Comput. Appl. Math. 39 (2020), no. 139.
[11] Sh. Banaei, V. Parvaneh and M. Mursaleen, Measures of noncompactness and infinite systems of integral equations of Urysohn type in L∞(G), Carpathian J. Math. 37 (2021), no. 3, 407–416.
[12] H.F. Bohnenblust, S. Karlin and A.W. Tucker, On a theorem of Games, Princeton Univ. Press, Princeton, 1950, pp. 155–160.
[13] L. Cai, J. Liang and J. Zhang, Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems, J. Fixed Point Theory Appl. 2018 (2018).
[14] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova. 24 (1955), 84–92.
[15] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
[16] B. Dhage, Some generalizations of multi-valued version of Schauder’s fixed point theorem with applications, CUBO 12 (2010), 139–151.
[17] M. Erturk and V. Karakaya, n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces, J. Inequal. Appl. 139 (2013).
[18] Z. Goodarzi and A. Razani, A periodic solution of the generalized forced Lienard equation, Abstr. Appl. Anal. 2014 (2014).
[19] M. Haddadi, H. Alaeidizaj and V. Parvaneh, A new version of the Hahn Banach theorem in b-Banach spaces, Math. Anal. Contemp. Appl. 3 (2021), no. 3, 27–32.
[20] S. Hong and L. Wang, Existence of solutions for integral inclusions, J. Math. Anal. Appl. 317 (2006), 429–441.
[21] H. Hosseinzadeh, H. Isık, H. Hadi Bonab and R. George, Coupled measure of noncompactness and functional integral equations, Open Math. 20 (2022), no. 1,38–39.
[22] K. Javed, F. Uddin, F. Adeel, M. Arshad, H. Alaeidizaji and V. Parvaneh, Fixed point results for generalized contractions in S-metric spaces, Math. Anal. Contemp. Appl. 3 (2021), no. 2, 27–39.
[23] S. Kakutani, A generalization of Brower’s fixed point theorem, Duke Math. J. 8 (1941), 457–459.
[24] K. Kuratowski, Sur les espaces, Fund. Math. 15 (1930), 301–309.
[25] N. Papageorgiou, Boundary value problems for evolution, Comment. Math. Univ. Carolin 29 (2019), 355–363.
[26] V. Parvaneh, Sh. Banaei, J.R. Roshan and M. Mursaleen, On tripled fixed point theorems via measure of noncompactness with applications to a system of fractional integral equations, Filomat 35 (2021), no. 14, 4897–4915.
[27] V. Parvaneh, M. Khorshid, M.D.L. Sen, H. Isik and M. Mursaleen, Measure of noncompactness and a generalized Darbo’s fixed point theorem and its applications to a system of integral equations, Adv. Differ. Equ. 243 (2020).
[28] A. Razani, An existence theorem for ordinary differential equation in Menger probabilistic metric space, Miskolc Math. Notes 15 (2014), no. 2, 711–716.
[29] A. Razani, Fixed points for total asymptotically nonexpansive mappings in a new version of bead space, Int. J. Ind. Math. 6 (2014), no. 4.
[30] A. Samadi, Applications of measure of noncompactness to coupled fixed points and systems of integral equations, Miskolc Math. Notes 19 (2018), no. 1, 537–553.
[31] D. Sekman, N.E.H. Bouzara and V. Karakaya, n-Tuplet fixed points of multivalued mappings via measure of noncompactness, Commun. Optim. Theory 2017 (2017).
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 11
November 2023
Pages 115-125
  • Receive Date: 04 April 2022
  • Revise Date: 05 January 2023
  • Accept Date: 07 January 2023