Some local fixed point results under $C$-class functions with applications to coupled elliptic systems

Journal of Linear and Topological Algebra

مقاله 2، دوره 07، شماره 03، آذر 2018، صفحه 169-182   

نوع مقاله: Special Issue on Fixed Point Theory

نویسندگان

A. Hojat Ansari¹؛ A. Benterki^{* 2}؛ M. Rouaki²

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

²LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Algeria

چکیده

The main objective of the paper is to state newly fixed point theorems for set-valued mappings in the framework of 0-complete partial metric spaces which speak about a location of a fixed point with respect to an initial value of the set-valued mapping by using some $C$-class functions. The results proved herein generalize, modify and unify some recent results of the existing literature. As an application, we provide an existence theorem for a coupled elliptic system subject to various two-point boundary conditions.

کلیدواژه‌ها

fixed point؛ set-valued mapping؛ partial metric space؛ coupled elliptic system؛ $C$-class function

مراجع

[1] I. Addou, A. Benmezaı, Boundary-value problems for the one-dimensional p-laplacian with even superlinearity, Elect. J. Differ. Equ. (1999), 1999:9.

[2] R. P. Agarwal, P. J. Wong, Existence of solutions for singular boundary problems for higher order differential equations, Milan J. of Math. 65 (1995), 249-264.

[3] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (2010), 2778-2785.

[4] A. H. Ansari, Note on φ-ψ-contractive type mappings and related fixed point, in The 2nd Regional Conference on Mathematics and Appl. 2014 (2014), 377-380.

[5] H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces, Topology Appl. 159 (2012), 3234-3242.

[6] H. Aydi, S. H. Amor, E. Karapınar, Berinde-type generalized contractions on partial metric spaces, Abstr. Appl. Anal. (2013), 2013:312479.

[7] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund.
Math. 3 (1922), 133-181.

[8] A. Benterki, A local fixed point theorem for set-valued mappings on partial metric spaces, Appl. Gen. Topol. 17 (2016), 37-49.

[9] R. M Bianchini, M. Grandolfi, Transformazioni di tipo contracttivo generalizzato in uno spazio metrico, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat, 45 (1968), 212-216.

[10] A. Castro, R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities, J. Math. Anal. Appl. 133 (1988), 509-528.

[11] S. Chandok, K. Tas, A. H. Ansari, Some fixed point results for TAC-type contractive mappings, J. Funct. Spaces. (2016), 2016:1907676.

[12] X. Cheng, C. Zhong, Existence of three nontrivial solutions for an elliptic system, J. Math. Anal. Appl. 327 (2007), 1420-1430.

[13] M. Chhetri, P. Girg, Existence and nonexistence of positive solutions for a class of superlinear semipositone systems, Nonlinear Anal. 71 (2009), 4984-4996.

[14] A. L. Dontchev, W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481-489.

[15] W.-S. Du, E. Karapınar, N. Shahzad, The study of fixed point theory for various multivalued non-self-maps, Abstr. Appl. Anal. (2013), 2013:938724.

[16] A. D. Ioffe, V. M. Tihomirov, Theory of extremal problems, Amsterdam, New York, Oxford: North-Holland Publishing Company, 1979.

[17] E. Kreyszig, Introductory functional analysis with applications, Wiley, New York, 1989.

[18] A. Latifa, H. Isik, A. H. Ansari, Fixed points and functional equation problems via cyclic admissible gener- alized contractive type mappings, J. Nonlinear Sci. Appl. 9 (2016), 1129-1142.

[19] P. S. Macansantos, A fixed point theorem for multifunctions in partial metric spaces, J. Nonlinear Anal. Appl. (2013), jnaa:00200.

[20] P. S. Macansantos, A generalized Nadler-type theorem in partial metric spaces, Int. J. Math. Anal. 7 (2013), 343-348.

[21] R. T. Marinov, D. K. Nedelcheva, Implicit mapping theorem for extended metric regularity in metric spaces, Ric. Mat. 62 (2013), 55-66.

[22] S. G. Matthews, Partial metric topology, Annals of the New York Acad. Sci, 728 (1994), 183-197.

[23] D. Motreanu, Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations, Set-Valued Var. Anal. 19 (2011), 255-269.

[24] S. B. Nadler, Multi-valued contraction mappings, Pac. J. Math. 30 (1969), 475-488.

[25] H. Rahimi, P. Vetro, G. Soleimani Rad, Coupled fixed-point results for T-contractions on cone metric spaces with applications, Math. Notes. 98 (2015), 158-167.

[26] S. Romaguera, A kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. (2009) 2009:493298.

[27] S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl. 159 (2012), 194-199.

[28] M. Rouaki, Nodal radial solutions for a superlinear problem, Nonlinear Anal. (Real World Appl). 8 (2007), 563-571.

[29] M. Rouaki, Existence and classifiction of radial solutions of a nonlinear nonautonomous Dirichlet problem, arXiv preprint arXiv:1110.4019.

[30] B. Ruf, S. Solimini, On a class of superlinear Sturm-Liouville problems with arbitrarily many solutions, SIAM J. Math. Anal. 17 (1986), 761-771.

[31] P. J. Wong, R. P. Agarwal, Eigenvalues of boundary value problems for higher order differential equations, Math. Prob. Engin. 2 (1996), 401-434.