Abstract
This paper aims to establish the existence of best proximity points for Sadovskii type map**s. The key concept used is measure of noncompactness which allows us to select a class of map**s which is general than that of compact map**s. The main results in this article are extension of a Sadovskii type fixed point theorem. As an application of the main result, the existence of optimum solutions for a system of fractional differential equations of arbitrary order featuring initial conditions at integer derivatives.
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References
Aghajani, A., Banaś, J., Sabzali, N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin 20, 345–358 (2013)
Ambrosetti, A.: Un teorema di esistenza per Ie equazioni differenziali negli spazi di Banach. Rend. Sem. Mat. Padova 39, 349–361 (1967)
Toledano, M.A., Benavides, T.D., Acedo, G.L.: Measures of Noncompactness in Metric Fixed Point Theory. Operator Theory: Advances and Applications, vol. 99 Birkhauser, Basel (1997)
Banaś, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent II. Fract. Calc. Appl. Anal. 11, 4–14 (2008). (reprinted from Geophys. J.R. Astr. Soc. 13 (1967), 529–539)
Chen, G.S.: Mean value theorems for Local fractional integrals on fractal space. Adv. Mech. Eng. Appl. 1, 5–8 (2012)
Darbo, G.: Punti uniti in transformazioni a condomino non compatto. Rend. Sem. Mat. Uni. Padova 24, 84–92 (1955)
Dunford, N., Schwartz, J.T.: Linear Operators, 3, Interscience (1971)
Eldred, A.A., Kirk, W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive map**s. Stud. Math. 171, 283–293 (2005)
Gabeleh, M.: A characterization of proximal normal structure via proximal diametral sequences. J. Fixed Point Theory Appl. 19, 2909–2925 (2017)
Gabeleh, M., Markin, J.: Optimum solutions for a system of differential equations via measure of noncompactness. Indagationes Math. 29, 895–906 (2018)
Gohberg, I., Gol’denshtein, L.S., Markus, A.S.: Investigation of some properties of bounded linear operators in connection with their q-norms. Ucen. Zap. Kishinevsk. Un-ta 29, 29–36 (1957)
Malkowsky, E., Rakočević, V.: On some results using measors of noncompactness, pp. 127-180, in J. Banas et al. (editors). Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer (2017). ISBN 978-981-10-3721-4
Malkowsky, E., Rakočević, V.: Advanced Functional Analysis. CRC Press, Taylor and Francis Group, Boca Raton, London, New York (2019)13 ISBN 978-1-1383-3715-2
Schauder, J.: Der Fixpunktsatz in Funktionalriiumen. Stud. Math. 2, 171–180 (1930)
Tisdell, C.C.: On the application of sequential and fixed point methods to fractional differential equations of arbitrary order. J. Integral Equ. Appl. 24, 283–319 (2012)
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Patle, P.R., Gabeleh, M. & Rakočević, V. Sadovskii Type Best Proximity Point (Pair) Theorems with an Application to Fractional Differential Equations. Mediterr. J. Math. 19, 141 (2022). https://doi.org/10.1007/s00009-022-02058-7
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DOI: https://doi.org/10.1007/s00009-022-02058-7