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Monotone solutions of iterative fractional equations found by modified Darbo-type fixed-point theorems. (English) Zbl 1376.34011
Summary: There are many forms of solutions that occur obviously, and in various situations one cannot just switch to one of them and establish solutions in that sense. It is valuable to study solutions with strong differentiability properties, but it might be difficult to confirm their existence. Therefore, one essentially considers solutions in a weaker sense. In this note, we establish the existence of monotone solutions of the dynamic system of the bacterial growth of iterative fractional differential equations, found by a new coupled and triple fixed point theorem in the sense of Darbo processing. We propose these theorems (Darbo type) for a new contraction by using a measure of non-compactness in Banach spaces. The new outcomes show some special well-known recent results.
##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 47H10 Fixed-point theorems
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##### References:
 [1] Aghajani, A; Banas, J; Sabzali, N, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin., 20, 345-358, (2013) · Zbl 1290.47053 [2] Aghajani, A; Sabzali, N, A coupled fixed point theorem for condensing operators with application to system of integral equations, J. Nonlinear Convex Anal., 15, 941-952, (2014) · Zbl 1418.47003 [3] Appell, J, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl., 83, 251-263, (1981) · Zbl 0495.45007 [4] Appell, J, Measure of noncompactness, condensing operators and fixed points: an application oriented survey, Fixed Point Theory, 6, 157-229, (2005) · Zbl 1102.47041 [5] Arab, R, Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations, J. Korean Math. Soc., 52, 125-139, (2015) · Zbl 1305.47038 [6] Arab, R, The existence of fixed points via the measure of noncompactness and its application to functional-integral equations, Mediterr. J. Math., 13, 759-773, (2016) · Zbl 1348.47041 [7] Arab, R.: Some generalizations of Darbo fixed point theorem and its application. Miskolc Math. Notes (in press) · Zbl 1399.54082 [8] Banas, J, Measures of noncompactness in the space of continuous tempered functions, Demonstr. Math., 14, 127-133, (1981) · Zbl 0462.47035 [9] Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics. Dekker, New York (1980) · Zbl 0441.47056 [10] Berinde, V; Borcut, M, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. Theory Methods Appl., 74, 4889-4897, (2011) · Zbl 1225.54014 [11] Burton, TA, Krasnoselskii’s inversion principle and fixed points, Nonlinear Anal., 30, 3975-3986, (1997) · Zbl 0894.47041 [12] Burton, TA; Kirk, C, A fixed point theorem of Krasnoselskii-Schaefer type, Mathematische Nachrichten, 189, 23-31, (1998) · Zbl 0896.47042 [13] Darbo, G, Punti uniti i transformazion a condominio non compatto, Rend. Sem. Math. Univ. Padova, 4, 84-92, (1995) [14] Eder, E, The functional differential equation $$x^{′ }(t) = x(x(t)),$$, J. Differ. Equ., 54, 390-400, (1984) · Zbl 0497.34050 [15] Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Volume 373 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1996) · Zbl 0866.45004 [16] Ibrahim, RW, Existence of deviating fractional differential equation, CUBO A Math. J., 14, 127-140, (2012) [17] Ibrahim, RW, Existence of iterative Cauchy fractional differential equation, J. Math., 2013, 1-8, (2013) · Zbl 1268.34017 [18] Ibrahim, R.W., Darus, M.: Infective disease processes based on fractional differential equation. In: Proceedings of the 3rd International Conference on Mathematical Sciences, vol. 1602, pp. 696-703 (2014) · Zbl 1385.47021 [19] Ibrahim, RW; Kilicman, A; Damag, FH, Existence and uniqueness for a class of iterative fractional differential equations, Adv. Differ. Equ., 2015.1, 1-13, (2015) · Zbl 1351.34006 [20] Ibrahim, R.W.: Fractional Calculus of Multi-objective Functions and Multi-agent Systems. Lambert Academic Publishing, Saarbrucken (2017) [21] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland, Mathematics Studies. Elsevier, Amsterdam (2006) [22] Kuratowski, K, Sur LES espaces completes, Fund. Math., 15, 301-309, (1930) · JFM 56.1124.04 [23] Mursaleen, M; Mohiuddine, SA, Applications of measures of noncompactness to the infinite system of differential equations in $$l_p$$ spaces, Nonlinear Anal. Theory Methods Appl., 75, 2111-2115, (2012) · Zbl 1256.47060 [24] Mursaleen, M; Rizvi, SMH, Solvability of infinite systems of second order differential equations in $$c_0$$ and $$ℓ _1$$ by Meir-Keeler condensing operators, Proc. Am. Math. Soc., 144, 4279-4289, (2016) · Zbl 1385.47021 [25] Reich, S, Fixed points of condensing functions, J. Math. Anal. Appl., 41, 460-467, (1973) · Zbl 0252.47062
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