Some Common Fixed Point Theorems for Multi-Valued Mappings

Khan, M. S.; Khan, M. A.; Kubiaczyk, I.

doi:10.1515/dema-1984-0418

Cited by 7 publications

(8 citation statements)

References 3 publications

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“…Subsequently, a number of generalization of Nadler's contraction principle were obtained by Ciric [23], Khan [8], Kubiak [9], Kaneke [6,7], Sessa [14,19], Singh [10] and many others [22,24].…”

Section: Introductionmentioning

confidence: 99%

Coincidence and common fixed points of hybrid contractions

et al. 1993

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In this paper, we show the existence of solutions of functional equations f t x e Sx D Tx and x = fjX e Sx ("I Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to Kaneko concerning common fixed points of multivalued mappings weakly commuting with a single-valued mapping and satisfying a generalized contraction type. Some related results are also obtained.1991 Mathematics subject classification (Amer. Math. Soc): 54 H 25.

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Section: Introductionmentioning

confidence: 99%

Coincidence and common fixed points of hybrid contractions

et al. 1993

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“…By choosing φ suitably one can derive improved versions of a multitude of relevant known common fixed point theorems involving six mappings especially those contained in Singh and Meade [14], Husain and Sehgal [5], Khan and Imdad [10], Jungck [6],Ćirić [1], S. L. Singh and S. P. Singh [13], Fisher [3,4], Das and Naik [2], Kannan [9], Rhoades [12], and several others. Also setting p = 0 and choosing A, B, S, T , I, J, and φ suitably one can deduce the results proved in the above cited references and many others.…”

Section: Proofmentioning

confidence: 99%

Remarks on certain selected fixed point theorems

Imdad

2002

International Journal of Mathematics and Mathematical Sciences

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Fixed point theorems due to Lal et al. (1996) and Jungck (1988) are used to derive two common fixed point theorems involving six mappings in complete and compact metric spaces, respectively. for all x, y ∈ X where p ≥ 0 and φ ∈ Ψ . Then A, S, I, and J have a unique common fixed point provided one of these four functions is continuous.Remark 2. Theorem 1 was originally proved for "weakly compatible mappings of type (A)" (cf.[11]) but for a more natural setting we have adopted it for commuting mappings.In this paper, as an application of Theorem 1, we derive a common fixed point theorem for six mappings which runs as follows. Proof. We begin by observing that continuity of AB (resp., ST ) does not demand the continuities of the component maps A or B or both (resp., S or T or both). Since the pairs (A, B), (A, I), (B, I) (S, T ), (S, J), and (T , J) are commuting which force the pairs (AB, I) and (ST , J) to be commuting. After observing this we note that all the conditions of Theorem 1 for four mappings AB, ST , I, and J are satisfied, hence (in view of Theorem 1) AB, ST , I, and J have a unique common fixed point z. Theorem 3. Let A, B, S, T , I, and J be self-mappings of a complete metric space (X, d) such that the pairs (A, B), (A, I)(B, I), (S, T ), (S, J), and (T , J) are commuting and AB(X) ⊂ J(X), ST (X) ⊂ I(X) satisfying the inequalityHere one can note that z also remains the unique common fixed point of the pairs (AB, I) and (ST , J) separately. Now it remains to show that z is also a common fixed point of A, B, S, T , I, and J. For this let z be the unique common fixed point of the pair (AB, I), thenwhich shows that Az and Bz are other fixed points of the pair (AB, I) yielding therebyin view of the uniqueness of common fixed point of the pair (AB, I). Similarly, it can be shown that z is also the unique common fixed point of S, T , ST , and J. This completes the proof. Next we wish to indicate a similar result in compact metric spaces. For this purpose one can adopt a general fixed point theorem for commuting mappings in compact metric spaces due to Jungck [8], which was originally proved for compatible mappings (a notion due to Jungck [7]). Theorem 5 (see [8]). Let A, S, I, and J be self-mappings of a compact metric space (X, d) with A(X) ⊂ J(X) and S(X) ⊂ I(X). If the pairs (A, I) and (S, J) are commuting andfor all x, y ∈ X where If the pairs (A, B), (A, I)(B, I), (S, T ), (S, J), and (T , J) are commuting andfor all x, y ∈ X where Proof. The proof is essentially the same as that of Theorem 3, hence we omit the proof.Remark 7. By choosing A, B, S, T , I, and J suitably one can derive a multitude of known theorems.Acknowledgement. The author is grateful to both the learned referees for their pertinent remarks which has significantly improved the contents of an earlier version of this paper.

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“…Subsequently a number of fixed point theorems in metric space have been proved for multi-valued mapping satisfying contractive type conditions (see, for instance [8], [9], [18], [20], [32] and references therein). Later on the study of hybrid fixed point theory for nonlinear single-valued and multi-valued mappings is a new development in the domain of contractive type multi-valued theory( see, for instance [4], [5], [12], [17], [21], [24], [28], [29], [30], [31], [33], [34] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Let (X, d) be a complete metric space. Let S : X → C(X) be a multi-valued mapping and f : X → X be a single-valued mapping such that for all x, y ∈ X S(X) ⊂ f (X) (18) f (X) is closed (19) Proof. If we take S = T and f as an identity mapping in Theorem 3.1, then we can get the proof.…”

Section: Introductionmentioning

confidence: 99%