FIXED POINT AND COINCIDENCE POINT THEOREMS FOR A PAIR OF SINGLE-VALUED AND MULTI-VALUED MAPS ON A METRIC SPACE

Bogin [1] proved a fixed point theorem for a nonexpansive type self-map on a complete metric space. While Rhoades obtained a generalization of it (see [8, Theorem 1]) by replacing the constant coefficients in the governing inequality of the map with nonnegative real-valued functions of the independent variables, Ćirić [3] obtained a generalization by further weakening the governing inequality without allowing the coefficients to vary. Chandra et al. obtained a coincidence point theorem (see [2, Theorem 2.1]) for a pair of self-maps on a metric space unifying the results of Rhoades and Ćirić. They also obtained a corresponding version for multimaps (see [2, Theorem 2.2]). In this paper, we obtain proper generalizations of Theorems 2.1 and 2.2 of Chandra et al. [2]. Throughout, unless otherwise stated, (X,d) is a metric space, K(X) is the collection of all nonempty compact subsets of X, CL(X) is the collection of all nonempty closed subsets of X, H is the extended Hausdorff metric on CL(X), F is a mapping from X into CL(X), f , S are self-maps on X, I is the identity map on X, for any self-map h on X, (h) = {hx : x ∈ X}, R+ is the set of all nonnegative real numbers, N is the set of all positive integers, Ω : (R+)5 → R+ is monotonically increasing in each coordinate variable, for any t1, t2, t3, t4, t5 ∈ R+, Ω(t+ 1 , t+ 2 , t+ 3 , t+ 4 , t+ 5 ) = inf{Ω(s1,s2,s3,s4,s5) : sj ∈ (tj,+∞) for all j = 1,2,3,4,5}, Ω(t1, t+ 2 , t+ 3 , t+ 4 , t+ 5 ) = inf{Ω(t1,s2,s3,s4,s5) : sj ∈ (tj,+∞) for all j = 2,3,4,5}, σj : R+ → R+ (j = 1,2) and ζ :R+ →R+ are defined as σ1(t)=Ω ( 0+,0+, t+, t+, t+ ) , σ2(t)=Ω ( t,0+,0+, t+,0+ ) , ζ(t)=max{σ1(t),σ2(t)}, (1)

Bogin [1] proved a fixed point theorem for a nonexpansive type self-map on a complete metric space.While Rhoades obtained a generalization of it (see [8, Theorem 1]) by replacing the constant coefficients in the governing inequality of the map with nonnegative real-valued functions of the independent variables, Ćirić [3] obtained a generalization by further weakening the governing inequality without allowing the coefficients to vary.Chandra et al. obtained a coincidence point theorem (see [2, Theorem 2.1]) for a pair of self-maps on a metric space unifying the results of Rhoades and Ćirić.They also obtained a corresponding version for multimaps (see [2, Theorem 2.2]).In this paper, we obtain proper generalizations of Theorems 2.1 and 2.2 of Chandra et al. [2].
Proof.If possible, suppose that {y n } is not Cauchy.Then there exists a positive real number ε with the following property: given N ∈ N there exists m, n ∈ N m > n ≥ N and d(y n ,y m ) ≥ ε.Hence there exist strictly increasing sequences {n k } ∞ k=1 and {m k } ∞ k=1 in N such that k < n k < m k , d(y n k ,y m k ) ≥ ε, and d(y n k ,y m k −1 ) < ε for all k ∈ N. Since {d(y n ,y n+1 )} converges to zero, it follows that {d(y n k ,y m k )} ∞ k=1 converges to ε and that for any fixed r ,s in {−1, 0, 1}, the sequence {d(y n k +r ,y m k +s )} ∞ k=1 also converges to ε.We have α( )) for all k ∈ N. We note that the limit superior of {α(x n k ,x m k +1 )} ∞ k=1 is less than or equal to Ω(0 for all k ∈ N. By taking limit superiors on both sides of (4) as k → +∞ we obtain ε ≤ σ 1 (ε).This is a contradiction since σ 1 (t) < t for all t ∈ (0, ∞) and ε > 0. Hence {y n } is Cauchy.
Definition 3. We say that Ω has property A if Ω(t,s,t,0,t +s) < s for all s, t ∈ R + with t < s.Definition 4. We say that Ω has property B if there exist (i) a monotonically increasing function ϕ : R + → R + with ϕ(t + ) < t for all t ∈ (0, ∞), and (ii) for each t ∈ R + a nonempty index set I t and nonnegative real numbers Lemma 5. Suppose that (f , S) has property A, Ω has properties A and B, and that inequality (3) is true for all x, y in X.Then {d(y n ,y n+1 )} ∞ n=0 converges to zero.Hence in view of the hypothesis that 1 > sup{γ i : i ∈ I r } (= µ, say), we have Definition 6 (see [5]).A pair (f 1 ,f 2 ) of self-maps on (X, d) is said to be compatible converges to zero whenever {x n } is a sequence in X such that {f 1 x n } and {f 2 x n } are convergent in X and have the same limit.Definition 7 (see [4]).A pair (f 1 ,f 2 ) of self-maps on an arbitrary set E is said to be weakly compatible (w.co. Definition 9 (see [7]).A pair Lemma 11.Suppose that (f , S) has property A, {y n } converges to an element z of X and that ( 3) is true for all x, y in X.Then the following statements are true: (i) If σ 1 (t) < t for all t ∈ (0, ∞) and f p = Sp for some p ∈ X, then f p = z.In particular, f and S cannot have a common fixed point or coincidence value other than Proof.(i) Suppose that σ 1 (t) < t for all t ∈ (0, ∞) and f p = Sp for some p ∈ X.We have for all n ∈ N. We note that the limit superior of the sequence {α(p, x n+1 )} is less than or equal to Ω(0, 0 + ,d(f p,z) + ,d(f p,z) + ,d(f p,z) + ) which in turn is less than or equal to σ 1 (d(f p, z)).From (3) we have for all n ∈ N. By taking limit superiors on both sides of (17) as n → +∞ we obtain Hence f and S cannot have a common fixed point other than z.If p, q ∈ X are such that f p = Sp and f q = Sq, then we have f p = z = f q.Hence f and S cannot have a coincidence value other than z.
(ii) Suppose that σ 2 (t) < t for all t ∈ (0, ∞) and z ∈ (S).Then there exists w ∈ X Sw = z.We have d y n ,y n+1 ,d z, y n ,d f w,y n ,d z, y n+1 (18) for all n ∈ N. We note that the limit superior of the sequence {α(w, x n+1 )} is less than or equal to Ω(d(f w, z), 0 for all n ∈ N. By taking limit superiors on both sides of (19) as n → +∞ we obtain (iii) Suppose that ζ(t) < t for all t ∈ (0, ∞), z ∈ (S) and (f , S) is w.co.From statement (ii) it follows that there exists w ∈ X f w = Sw = z.Hence from the weak compatibility of (f , S) we have f z = f Sw = Sf w = Sz.We have for all n ∈ N. We note that the limit superior of the sequence {α(z, x n+1 )} is less than or equal to Ω(0, 0 for all n ∈ N. By taking limit superiors on both sides of (21) as n → +∞ we obtain (iv) Suppose that σ 1 (t) < t for all t ∈ (0, ∞), S is continuous at z and (f , S) is co.Since {y n } converges to z and S is continuous at z, {Sy n } converges to Sz. Hence the sequences {SSx n } and {Sf x n } converge to Sz.Since (f , S) is co., and {f x n } and {Sx n } are convergent sequences having the same limit z, it follows that {d(Sf x n ,f Sx n )} converges to zero.Since {Sf x n } converges to Sz, it follows that {f Sx n } also converges to Sz.We have for all n ∈ N. We note that the limit superior of the sequence {α(Sx n ,x n+1 )} is less than or equal to Ω(0 for all n ∈ N. By taking limit superiors on both sides of (23) as n → +∞ we obtain d(Sz, z) ≤ σ 1 (d(Sz, z)).Hence d(Sz, z) = 0. Hence Sz = z.
(v) Suppose that σ 1 (t) < t for all t ∈ (0, ∞), f is continuous at z and (f , S) is co.Since {y n } converges to z and f is continuous at z, {f y n } converges to f z.Hence the sequences {f f x n } and {f Sx n } converge to f z.Since (f , S) is co., and {f x n } and {Sx n } are convergent sequences having the same limit z, it follows that {d(Sf x n ,f Sx n )} converges to zero.Since {f Sx n } converges to f z, it follows that {Sf x n } also converges to f z.We have for all n ∈ N. We note that the limit superior of the sequence {α(f x n ,x n+1 )} is less than or equal to Ω(0 for all n ∈ N. By taking limit superiors on both sides of (25) as n → +∞ we obtain (vi) Suppose that σ 1 (t) < t for all t ∈ (0, ∞), and (f , S) is co.and reciprocally continuous at z. Since {y n } converges to z, the sequences {f x n } and {Sx n } are convergent and have the same limit z.Hence from the reciprocal continuity of (f , S) at z it follows that {Sf x n } converges to Sz and {f Sx n } converges to f z and from the compatibility of (f , S) it follows that {d(Sf x n ,f Sx n )} converges to zero.Hence f z = Sz.We have for all n ∈ N. We note that the limit superior of the sequence {α(z, x n+1 )} is less than or equal to Ω(0, 0 + , d(f z, z) + , d(f z, z) + , d(f z, z) + ) which in turn is less than or equal to σ 1 (d(f z, z)).From (3) we have for all n ∈ N. By taking limit superiors on both sides of the above inequality as n → +∞ we obtain Theorem 12. Suppose that (f , S) has property A, Ω has properties A and B, σ 1 (t) < t for all t ∈ (0, ∞) and that inequality (3) is true for all x, y in X.Then {y n } is Cauchy.Suppose that it converges to an element z of X.Then the following statements are true: (i) (v) If σ 2 (t) < t for all t ∈ (0, ∞) and z ∈ (S), then there exists w ∈ X such that f w = Sw = z.
Corollary 13.Suppose that (f , S) has property A and that there is a monotonically decreasing function δ : R + → (0, 1/3] such that for all x, y in X, where Then {y n } is Cauchy.Suppose that it converges to an element z of X.Then the following statements are true: (i) f and S cannot have a common fixed point or coincidence value other than z.
Corollary 14 (see [2, Theorem 2.1]).Suppose that f (X) ⊆ S(X) and that there are nonnegative real-valued functions a, b, c on X × X such that d(f x, f y) ≤ a(x, y)d(Sx, Sy) + b(x, y) max d(Sx,f x),d(Sy,f y) for all x, y in X, inf{b(u, v) : u, v ∈ X} > 0, inf{c(u, v) : u, v ∈ X} > 0, and sup{a(u, v) Suppose also that either (a) X is complete and S is surjective; or (b) X is complete, S is continuous and (f , S) is co.; or (c) S(X) is complete: or (d) f (X) is complete.Then f and S have a coincidence point in X.Further, the coincidence value is unique.
Proof.The proof follows from Corollary 13 by taking δ as a constant function on R + with its value in (0, min{1/3,δ 1 ,δ 2 }], where Corollary 15.Suppose that (f , S) has property A and that there are a positive integer N, nonnegative constants a 1 ,...,a N , and positive constants b 1 ,...,b N ,c 1 ,...,c N such that a i + b i + 2c i ≤ 1 for all i = 1, 2,...,N and for all x, y in X.Then {y n } is Cauchy.Suppose that it converges to an element z of X.
Then statements (i) to (vi) of Corollary 13 are true here also.
Corollary 16 (see uniqueness part of [2, Theorem 2.1]).Suppose that (X, d) is complete and d(x, f x), d(y, f y) + c d(f x, y) + d(x, f y for all x, y in X and for some constants a, b, c with a ≥ 0, b > 0, c > 0 and a+b+2c = 1.
Then f has a unique fixed point in X.
Proof.The proof follows from Corollary 15 by taking Corollary 17.Let ϕ : R + → R + be a monotonically increasing map such that ϕ(t + ) < t for all t ∈ (0, ∞).Suppose that (f , S) has property A and for all x, y in X.Then {y n } is Cauchy.Suppose that it converges to an element z of X.
Then statements (i) to (vi) of Corollary 13 are true here also.
Definition 20.We say that Ω has property C if there exist (i) a monotonically increasing function ϕ : R + → R + with ∞ n=1 ϕ n (t) < +∞ for all t ∈ R + , and (ii) for each t ∈ R + a nonempty index set I t and nonnegative real numbers Definition 21.We say that (F , S) has property A if there is a sequence {u n } ∞ n=0 in X such that Su n+1 ∈ Fu n and d(Su n ,Fu n ) = d(Su n ,Su n+1 ) for all n = 0, 1, 2 .... (Let v n stand for Su n+1 .)Lemma 22. Suppose that (F , S) has property A, Ω has properties A and C, and for all x, y in X.Then {v n } ∞ n=1 is Cauchy.
)) for all n ∈ N since Ω is increasing in each coordinate variable.From (40) we have for all n ∈ N. Now proceeding as in the proof of Lemma 5 it can be seen that since the sequence {d(v k−1 ,v k )} is monotonically decreasing and Ω is increasing in each coordinate variable.Since Ω has property C, there exist (i) a monotonically increasing function ϕ : R + → R + with ∞ n=1 ϕ n (t) < +∞ for all t ∈ R + and (ii) for each t ∈ R + a nonempty index set I t and nonnegative real numbers β i , γ i (i ∈ I t ) such that sup{γ i : i ∈ I t } < 1, Ω(t, t, 2t, t, t + s) ≤ sup{(1 + β i )t + γ i s : i ∈ I t } for all s ∈ [t, 2t], and Ω(t,t,t,0,λ t t) ≤ ϕ(t), where λ t = sup{(1 Hence from (42), (43), and (44) we have ).Now proceeding as in the proof of Lemma 5, it can be shown that for all n ∈ N. Hence we have for all n ∈ N. Hence from (40) we have that is, for all n ∈ N. Since ϕ is monotonically increasing on R + , by repeatedly using (49) we obtain for all n ∈ N. Since ∞ n=1 ϕ n (t) < +∞ for all t ∈ R + , from (50) it follows that Theorem 23.Suppose that (F , S) has property A, Ω has properties A and C and that ( 40) is true for all x, y in X.Then {v n } is Cauchy.Suppose that it converges to an element z of S(X) and σ 2 (t) < t for all t ∈ (0, ∞).Then Sw ∈ Fw for any w ∈ X Sw = z.
Proof.The proof that {v n } is Cauchy follows from Lemma 22. Suppose that it converges to an element z of S(X).Let w ∈ X be such that z = Sw.We have for all n ∈ N. We note that the limit superior of the sequence {D(w, u n+1 )} is less than or equal to Ω(d(z, F w), 0 + , 0 + ,d(z,Fw) + , 0 + ) which in turn is less than or equal to σ 2 (d(z, F w)). From (40) we have for all n ∈ N. By taking limit superiors on both sides of (52) as n → +∞ we obtain d(z, F w) ≤ σ 2 (d(z, F w)). Since σ 2 (t) < t for all t ∈ (0, ∞), we have d(z, F w) = 0. Since Fw is closed, z ∈ Fw.
Corollary 24 (see [2, Theorem 2.2]).Suppose that F(x) ∈ K(X) for all x ∈ X, F(x) ⊆ S(X) for all x ∈ X and that there are nonnegative real-valued functions a, b, c on X × X such that H(Fx, Fy) ≤ a(x, y)d(Sx, Sy) + b(x, y) max d(Sx,Fx),d(Sy,Fy) orbitally complete and S is surjective; or (b) S(X) is (F , S) orbitally complete: or (c) F(X) is (F , S) orbitally complete.Then F and S have a coincidence point in X.
As in the proof of Corollary 13 it can be seen that Ω has properties A and C and that σ 2 (t) < t for all t ∈ (0, ∞).Evidently (40) is true for all x, y in X.Since the values of F are compact and Fx ⊆ S(X) for all x in X, (F , S) has property A. Now the corollary is evident from Theorem 23.
Remark 28.Following the proof of [6, Theorem 1] it can be shown that Corollary 25 remains valid if the condition (F , S) has property A is replaced with the condition Fx ⊆ S(X) for all x in X provided ϕ is subjected to the additional condition ϕ(t + ) < t for all t ∈ (0, ∞) .With this modification Corollary 25 is a generalization of [2,Theorem 2.3].Example 27 shows that the generalization is proper.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: and S cannot have a common fixed point or coincidence value other than z.(ii) If S is continuous at z and (f , S) is co., then Sz = z.(iii) If f is continuous at z and (f , S) is co., then f z = z.(iv) If (f , S) is co.and reciprocally continuous at z, then f z = Sz = z.
The corollary follows from Theorem 23 by defining Ω as in the proof of Corollary 17 and noting that σ 2 (t) = ϕ(t) for all t ∈ (0, ∞).