Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

Abstract

In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation X = Q + ∑ i = 1 m A i * γ ( X ) A i and give a numerical example.

The Best Proximity Point Problem for New Types of Ciric a 􀀀b􀀀 Contraction with Application

“…Anuradha and Veeramani (6) proved the existence of a best proximity point for proximal pointwise contraction. Recently many authors have studied and generalize various concept related to the best proximity points (7)(8)(9)(10)(11)(12)(13)(14) .…”

Best Proximity Point for Generalized and S -Geraphty Contractions

“…Whereas a cyclic mapping does not necessarily have a fixed point, it is desirable to determine an element x which is somehow closest to Tx: More precisely, an element x for which the error dðx, TxÞ assumes the least possible value distðA, BÞ where distðA, BÞ = inf fdðx, yÞ: x ∈ A, y ∈ Bg, such a point is called a best proximity point of the cyclic mapping T: Since 2003, research on best proximity points of cyclic mapping became an important topic in nonlinear analysis and has been studied by many authors [2][3][4][5][6][7][8].…”

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“…We represent by the Ω set of all mappings Θ: (0, ∞) ⟶ (1, ∞) satisfying (Θ 1 )-(Θ 3 ) consistent with Samet et al [11] . For more details, we refer the following [12,[14][15][16][18][19][20] to the readers.…”