We study solutions pxnq nPN of nonhomogeneous nonlinear second available online: MMM DD, 2014. 2010 Mathematics Subject Classification. 39A22, 65Q10, 65Q30. Key words and phrases. nonhomogeneous nonlinear second order difference equations, Shohat-Freud-type exponential weight functions, Painlevé's discrete equation #1, existence of solutions, unicity of solutions, asymptotic behavior.
In this paper, we introduce a new concept of probabilistic metric space, which is a generalization of the Menger probabilistic metric space, and we investigate some topological properties of this space and related examples. Also, we prove some fixed point theorems, which are the probabilistic versions of Banach's contraction principle. Finally, we present an example to illustrate the main theorems. MSC: 54E70; 47H25
Using the notion of compatible mappings in the setting of a partially ordered metric space, we prove the existence and uniqueness of coupled coincidence points involving a (j, ψ)-contractive condition for a mappings having the mixed g-monotone property. We illustrate our results with the help of an example.
The existence of fixed points, coupled fixed points, and coupled coincidence points without the assumption of compatibility is established. The results presented in this paper extend, improve, and generalize some well-known results in the literature. Also, an example is given to show that our results are real generalizations of known ones in coupled coincidence fixed point theory. MSC: 54H25; 47H10
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