Citation: Hemant Kumar Nashine, Rabha W. Ibrahim. Solvability of a fractional Cauchy problem based on modified fixed point results of non-compactness measures[J]. AIMS Mathematics, 2019, 4(3): 847-859. doi: 10.3934/math.2019.3.847
[1] | R. P. Agarwal, M. Benchohra and D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math., 55 (2009), 221-230. |
[2] | R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001. |
[3] | A. Aghajani, J. Banas and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin., 20(2013), 345-358. |
[4] | A. Aghajani, J. Banas and Y. Jalilian, Existence of solution for a class of nonlinear Volterra singular integral equations, Comp. Math. Appl., 62(2011), 1215-1227. |
[5] | A. Aghajani and N. Sabzali, A coupled fixed point theorem for condensing operators with application to system of integral equations, J. Nonlinear Convex Anal., 15 (2014), 941-952. |
[6] | A. Aghajani, R. Allahyari and M. Mursaleen, A generalization of Darbo's theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math., 260 (2014), 68-77. |
[7] | R. Arab, 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(2015), 125-139. |
[8] | R. Arab, The existence of fixed points via the measure of noncompactness and its application to functional-integral equations, Mediterr. J. Math., 13 (2016), 759-773. |
[9] | J. Banas, Measures of noncompactness in the space of continuous tempered functions, Demonstr. Math., 14 (1981), 127-133. |
[10] | J. Banas and K. Goebel, Measures of noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 1980. |
[11] | L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505. |
[12] | G. Darbo, Punti uniti i transformazion a condominio non compatto, Rend. Sem. Math. Univ. Padova, 4 (1995), 84-92. |
[13] | M. M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput., 149 (2004), 823-831. |
[14] | M. A. A. El-Sayeed, Fractional order diffusion wave equation, Int. J. Theor. Phys., 35 (1996), 311-322. |
[15] | D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Dordrecht, The Netherlands : Kluwer Academic Publishers, 1996. |
[16] | R. W.Ibrahim andH. A. Jalab, Existence of entropy solutions for nonsymmetric fractional systems, Entropy, 16 (2014), 4911-4922. |
[17] | R. W.Ibrahim andH. A. Jalab, Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy, 17 (2015), 3172-3181. |
[18] | O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problem, Nonlinear Anal., 69 (2008), 3153-3159. |
[19] | A. Kilbas, and S. Marzan, Cauchy problem for differential equation with Caputo derivative, Fract. Calc. Appl. Anal., 7 (2004), 297-321. |
[20] | M. Mursaleen and S. A.Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $l_p$ spaces, Nonlinear Anal. Theory, MethodsAppl., 75 (2012), 2111-2115. |
[21] | M. Mursaleen and A. Alotaibi, Infinite system of differential equations in some BK spaces, Abstr. Appl. Anal., 2012 (2012), Article ID 863483. |
[22] | C. Tsallis, Generalized entropy-based criterion for consistent testing, Phys. Rev. E, 58 (1998), 1442. |