Solutions of Neutral Differential Inclusions

Year 2022, , 74 - 92, 31.03.2022

Sana Hadj Amor , Ameni Remadı

https://doi.org/10.31197/atnaa.983573

Cited By: 1

Abstract

Motivated by the study of neutral differential inclusions, we establish a new fixed point theorem for multivalued countably Meir-Keeler condensing mappings via an arbitrary measure of weak noncompactness which in turn include the fixed point theorems of Krasnoselskii and Dhage as special cases in non separable spaces.

Keywords

meir keeler condensing operators, measure of weak noncompactness, neutral differential inclusions

References

• [1] A. Aghajani, M. Mursaleen, A. Shole Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta. Math. Sci. 35B (3) (2015), 552-566.
• [2] D. Averna, S.A. Marano, Existence of solutions for operator inclusions, a unified approach. Rend. Sem. Mat. Univ. Padova 102, (1999).
• [3] J. Bana’s and M.-A. Taoudi, Fixed points and solutions of operator equations for the weak topology in Banach algebras, Taiwanese J. Math. 18 (2014), 871-893.
• [4] M. Belhadj, A. Ben Amar and M. Boumaiza, Some fixed point theorems for Meir-Keeler condensing operators and appli- cation to a system of integral equations, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), 223-239.
• [5] A. Ben Amar, Krasnoselskii type fixed point theorems for multi-valued mappings with weakly sequentially closed graph, Ann. Univ. Ferrara 58 (2012), 1-10.
• [6] A. Ben Amar, M. Boumaiza and D. O’Regan, Hybrid fixed point theorems for multivalued mappings in Banach algebras under a weak topology setting, J. Fixed Point Theory Appl. DOI 10.1007/s11784-016-0289-9.
• [7] A. Ben Amar, S. Chouayekh and A. Jeribi, Fixed point theory in a new class of Banach algebras and application, Afr. Mat. 24 (2013), no. 4, 725-724.
• [8] A. Ben Amar, S. Chouayekh, A. Jeribi, New fixed point theorems in Banach algebras under weak topology features and applications to nonlinear integral equations, Journal of Functional Analysis 259 (2010), 2215-2237.
• [9] A. Ben Amar, S. Derbel, D. O’Regan and T. Xiang, Fixed point theory for countably weakly condensing maps and multimaps in non-separable Banach spaces, Journal of Fixed Point Theory and Applications 21.1 (2019), 1-25.
• [10] A. Ben Amar, A. Jeribi and R. Moalla, Leray-Schauder alternatives in Banach algebras involving three operators with application, Fixed Point Theory 15 (2014), no. 2, 359-372.
• [11] A. Ben Amar, A. Sikorska-Nowak, On Some Fixed Point Theorems for 1-Set Weakly Contractive Multi-Valued Mappings with Weakly Sequentially Closed Graph, Advances in Pure Mathematics, 2011, 1, 163-169.
• [12] De Blasi, S. Francesco , On a property of the unit sphere in a Banach space, Bulletin math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de la R´ epublique Socialiste de Roumanie, 1977, 259–262, JSTOR.
• [13] G. Bonanno, S.A. Marano, Positive solutions of elliptic equations with discontinuous nonlinearities, Topol. Methods Non- linear Anal. 8, 263-273 (1996).
• [14] T. Cardinali, F. Papalini, Fixed point theorems for multifunctions in topological vector spaces, J. Math. Anal. Appl. 186 (1994),769-777.
• [15] T. Cardinali, F. Rubbioni, Multivalued fixed point theorems in terms of weak topology and measure of weak noncompact- ness, J. Math. Anal. Appl. 405, 409-415 (2013).
• [16] S. Chandrasekhar, Radiative Transfer, Dover Publications Inc.: New York, NY, USA, 1960.
• [17] B.C. Dhage, Existence results for neutral functional diffrential inclusions in Banach algebras, Nonlinear Analysis 64 (2006) 1290-1306.
• [18] B. C. Dhage, Multivalued operators and fixed point theorems in Banach algebras, I. Taiwanese J. Math. 10 (2006), 1025- 1045.
• [19] B.C. Dhage, On some variants of Schauders fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci. 25 (1988) 603-611.
• [20] R. E. Edwards, Functional Analysis, Theory and Appli-cations, Reinhart and Winston, New York, 1965.
• [21] H. George, J.R. Pimbley, Positive solutions of a quadratic integral equation, Arch. Ration. Mech. Anal. 1967,24, 107-127.
• [22] M. Ghiocel, A. Petrusel and G. Petrusel, Topics in Nonlinear Analysis and Application to Mathematical Economics, Cluj-Napoca, 2007.
• [23] G. Gripenberg, On some epidemic models, Q. Appl. Math. 1981, 39, 317-327.
• [24] J.K. Hale, Theory of Functional Differential Equations, Springer, New York 1977.
• [25] N. Hussain, M.A. Taoudi, Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations, Fixed Point Theory Appl. 2013, 196(2013).
• [26] L. Kurz, P. Nowosad, B.R. Saltzberg, On the solution of a quadratic integral equation arising in signal design, J. Franklin Inst. 1966, 281, 437-454.
• [27] T. C. Lim, On characterizations of Meir-Keeler contractive maps, Nonlinear Anal. 46 (2001), 113-120.
• [28] K. Musial, Pettis integral, In: Handbook of Measure Theory, Vol. I, II, NorthHolland, Amsterdam, 2002, 531-58.
• [29] ] S.K. Ntouyas, Initial and boundary value problems for functional differential equations via topological transversality method : A Survey, Bull. Greek Math. Soc. 40 (1998), 3-41.
• [30] D. O’Regan, Fixed point theorems for weakly sequentially closed maps, Archivum Mathematicum (Brno) Tomus. 36, 61-70 (2000).
• [31] D. O’Regan, M.A.Taoudi, Fixed point theorems for the sum of two weakly sequentially continuous mappings, Nonlinear Anal. Nonlinear Anal. 73(2), 283-289 (2010).
• [32] W. Rudin, Functional Analysis, 2nd ed.; McGraw-Hill: New York, NY, USA, 1991.
• [33] T. Suzuki, Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces, Nonlinear Anal. 64 (5) (2006), 971-978.