A FIXED POINT PROBLEM VIA SIMULATION FUNCTIONS IN INCOMPLETE METRIC SPACES WITH ITS APPLICATION

Yıl 2020, Cilt: 10 Sayı: 1, 220 - 231, 01.01.2020

R. Lashkaripour H. Baghani Z. Ahmadi

Öz

In this paper, rstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some xed point theorems for single-valued mappings in incomplete metric spaces. Secondly, as motivated by the recent work of A.H. Ansari et al. [J. Fixed Point Theory Appl. 2017 , 1145{1163], we obtain that an existence and uniqueness result for the following problem: nding x 2 X such that x = Tx, Ax R1 Bx and Cx R2 Dx, where X; d is an incomplete metric space equipped with the two binary relations R1 and R2, A;B;C;D : X ! X are discontinuous mappings and T : X ! X satises in a new contractive condition. This result is a real generalization of main theorem of A.H. Ansari's. Finally, we provide some examples for our results and as an application, we nd that the solutions of a dierential equation.

Anahtar Kelimeler

Fixed point, Constraint inequalities, ?-Z-contraction, SO-complete metric space, Fractional dierential equation.

Kaynakça

• Ahmad, B., Nieto, J.J., (2013), Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl., Article ID, pp. 149-659.
• Ansari, A.H., Kumam, P., Samet, B., (2017), A fixed point problem with constraint inequalities via an implicit contraction, J. Fixed Point Theory and Appl., pp. 1145-1163.
• Ahmadi, Z., Lashkaripour, R., Baghani, H., A fixed point problem with constraint inequalities via a contraction in incomplete metric spaces, Filomat, to appear.
• Babu, G.V.R., Sarma, K.K.M., Krishna, P.H., (2014), Fixed points of ψ-weak Geraghty contractions in partially ordered metric spaces, J. Adv. Res. Pure Math., 6, pp. 9-23.
• Baghani, H., Eshaghi Gordji, M., Ramezani, M., (2016), Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory and Appl., 18, pp. 465-477.
• Baghani, H., Ramezani, M, (2017), A fixed point theorem for a new class of set-valued mappings in R-complete(not necessarily complete) metric spaces, Filomat, 31, pp. 3875-3884.
• Baghani, H., Ramezani, M., (2017), Contractive gauge functions in strongly orthogonal metric spaces, Int. J. Nonlinear Anal., 2, 23-28.
• Eshaghi Gordji, M., Ramezani, M., De La Sen, M., Cho, Y.J., (2017), On orthogonal sets and Banach fixed Point theorem, Fixed Point Theory, 18, 569-578.
• Hussain, N., Kadelburg, Z., Radenovic, S., Al-Solamy, F.R., (2012), Comparison functions and fixed point results in partial metric spaces, Abstract and Applied Analysis Article ID., 15, 605-781.
• Jleli, M., Samet, B., (2016), A fixed point problem under two constraint inequalities, Fixed Point Theory and Appl., doi:10.1186/s13663-016-0504-9.
• Kawasaki, T., Toyoda, M., (2015), Fixed point theorem and fractional differential equations with multiple delays related with chaos neuron models, Appl. Math., 6, 2192-2198.
• Khojasteh, F., Shukla, S., Radenovi, S., (2015), A new approach to the study of fixed point theorems via simulation functions, Filomat, 29, 1189-1194.
• Ntouyas, S.K., Tariboon, J., (2017), Fractional boundary value problems with multiple orders of fractional with multiple orders of fractional derivatives and integrals, Electronic Journal of Diferential Equations, 100, 1-18.
• Roldan-Lopez-de-Hierro, AF., Karapinar, E, Roldan-Lopez-de-Hierro, C., Martinez-Moreno, J., (2015), Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275, 345-355.
• Wang, W.X., Zhang, L., Liang, Zh., (2006),Initial value problems for nonlinear impulsive integro- differential equations in Banach space, J. Math. Anal. Appl., 320, 510-527.
• Yang, Sh., Zhang, Sh., (2016), Impulsive boundary value problem for a fractional differential equation, Boundary Value Problems, doi:10.1186/s13661-016-0711-7.