A Quadruple Fixed Point Theorem for Contractive Type Condition by Using ICS Mapping and Application to Integral Equation

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1 Mathematica Moravica Vol. - () 3 A Quadruple Fixed Point Theorem for Contractive Type Condition by Using ICS Mapping and Application to Integral Equation K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi Abstract. In this paper we obtain a Quadruple fixed point theorem for ψ φ contractive condition in partially ordered partial metric spaces by using ICS mapping. We are also given an example and an application to integral equation which supports our main theorem.. Introduction The notion of partial metric space was introduced by Matthews [] as a part of the study of denotational semantics of data flow networks. In fact it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation and domain theory in computer science. First we recall some basic definitions and lemmas which play crucial role in the theory of partial metric spaces. Definition.. (See [ 9]) A partial metric on a nonempty set X is a function p : X X R + such that for all x y z X: (p ) x = y p(x x) = p(x y) = p(y y) (p ) p(x x) p(x y) p(y y) p(x y) (p 3 ) p(x y) = p(y x) (p ) p(x y) p(x z) + p(z y) p(z z). The pair (X p) is called a partial metric space (PMS). Clearly p(x y) = implies x = y and x y implies p(x y) >. If p is a partial metric on X then the function d p : X X R + given by d p (x y) = p(x y) p(x x) p(y y) is a metric on X. Example. (See e.g. [9]). Consider X = [ ) with p(x y) = maxx y. Then (X p) is a partial metric space. It is clear that p is not a (usual) metric. Note that in this case d p (x y) = x y. Mathematics Subject Classification. Primary: 7H 5H5. Key words and phrases. Partial metric space Quadruple fixed point ICS mapping mixed monotone property. c Mathematica Moravica

2 A Quadruple Fixed Point Theorem for Contractive Type Condition... Each partial metric p on X generates a T topology τ p on X which has as a base the family of open p-balls B p (x ε) x X ε > where B p (x ε) = y X : p(x y) < p(x x) + ε for all x X and ε >. We now state some basic topological notions (such as convergence completeness) on partial metric spaces (see e.g. [ 9].) Definition.. () A sequence x n in the PMS (X p) converges to the limit x if and only if p(x x) = lim p(x x n). () A sequence x n in the PMS (X p) is called a Cauchy sequence if lim p(x n x m ) exists and is finite. nm (3) A PMS (X p) is called complete if every Cauchy sequence x n in X converges with respect to τ p to a point x X such that p(x x) = lim p(x n x m ). nm We need the following lemmas in PMS([ 9]). Lemma.. () A sequence x n is a Cauchy sequence in the PMS (X p) if and only if it is a Cauchy sequence in the metric space (X d p ). () A PMS (X p) is complete if and only if the metric space (X d p ) is complete. Moreover lim d p(x x n ) = p(x x) = lim p(x x n) = lim p(x n x m ). nm Lemma.. Assume x n z as n in a PMS (X p) such that p(z z) =. Then lim p(x n y) = p(z y) for every y X. In 9 K.P. Chi [3] introduced the concept of ICS mapping as follows. Definition.3 ([37]). Let (X d) be a metric space. A mapping T : X X is said to be ICS if T is injective continuous and has the property: for every sequence x n in X if T x n is convergent then x n is also convergent. Now we introduce the notion of Quadruple fixed point as follows. Definition.. An element (x y z w) X is called a Quadruple fixed point of F : X X if F (x y z w) = x F (y z w x) = y F (z w x y) = z and F (w x y z) = w. Definition.5 ([56]). Let (X ) be a partial ordered set and F : X X. We say that F has the mixed monotone property if F (x y z w) is monotone non decreasing in x and z and is monotone non - increasing in y and w

3 K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi 3 that is for any x y z w X. x x X x x F (x y z w) F (x y z w) y y X y y F (x y z w) F (x y z w) z z X z z F (x y z w) F (x y z w) w w X w w F (x y z w ) F (x y z w ).. Main Results Let Ψ denote the set of all continuous and monotonically increasing functions ψ : [ ) [ ) with ψ() =. Let Φ denote the set of all lower semi continuous functions φ : [ ) [ ) such that φ(t) > for t > and φ() =. Let (X ) be a partial ordered set. We consider the following partial order on the product space X = X X X X. (x y z w) (u v r t) iff x u y v z r and w t where (x y z w) and (u v r t) X. Theorem.. Let (X p ) be a complete ordered partial metric space. Suppose T : X X is an ICS mapping and F : X X is such that F has the mixed monotone property. Assume that there exist ψ Ψ and φ Φ such that (..) ψ (p(t F (x y z w) T F (u v r t))) p(t x T u) p(t y T v) p(t z T r) p(t w T t) ( p(t x T u) p(t y T v) p(t z T r) p(t w T t) ) for all x y z w u v r t X for which x u y v z r and w t. Suppose X has the following property (A): I. If non - decreasing sequence x n x then x n x for all n II. if non - increasing sequence y n y then y n y for all n. Suppose there exist x y z w X such that x F (x y z w ) y F (y z w x ) z F (z w x y ) w F (w x y z ). Then there exist x y z w X such that F (x y z w) = x F (y z w x) = y F (z w x y) = z F (w x y z) = w that is F has a quadruple fixed point.

4 A Quadruple Fixed Point Theorem for Contractive Type Condition... Proof. Let x y z w X be such that Set x F (x y z w ) y F (y z w x ) z F (z w x y ) w F (w x y z ). x = F (x y z w ) x y = F (y z w x ) y z = F (z w x y ) zo w = F (w x y z ) w and by the mixed monotone property of F for n inductively we get x n = F (x n y n z n w n ) x n x y n = F (y n z n w n x n ) y n y z n = F (z n w n x n y n ) z n z w n = F (w n x n y n z n ) w n w. Assume for some n N x n+ = x n y n+ = y n z n+ = z n and w n+ = w n. Then (x n y n z n w n ) is a quadruple fixed point of F. Hence the theorem. Now assume that x n+ x n or y n+ y n or z n+ z n or w n+ w n for any n N. Since T is injective we have a n = maxp(t x n+ T x n ) p(t y n+ T y n ) p(t z n+ T z n ) p(t w n+ T w n ) >. ψ (p(t x n+ T x n )) = ψ (p(t F (x n y n z n w n ) T F (x n y n z n w n ))) p(t xn T x n ) p(t y n T y n ) p(t z n T z n ) p(t w n T w n ) p(t xn T x n ) p(t y n T y n ) p(t z n T z n ) p(t w n T w n ) Similarly = ψ (a n ) φ (a n ). ψ (p(t y n+ T y n )) ψ (a n ) φ (a n ) ψ (p(t z n+ T z n )) ψ (a n ) φ (a n ) ψ (p(t w n+ T w n )) ψ (a n ) φ (a n ).

5 K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi 5 Hence Thus () p(t xn+ T x ψ (a n ) = ψ max n ) p(t y n+ T y n ) p(t z n+ T z n ) p(t w n+ T w n ) ψ (p(t xn+ T x = max n )) ψ (p(t y n+ T y n )) ψ (p(t z n+ T z n )) ψ (p(t w n+ T w n )) ψ (a n ) φ (a n ). Since ψ is increasing we have ψ (a n ) ψ (a n ) φ (a n ) < ψ (a n ). a n < a n n = 3 Thus a n is a positive decreasing sequence of real numbers. Hence there exists r such that lim a n = r. Suppose r >. Letting n in () we obtain that a contradiction. Hence r =. Thus () lim max From (p ) we have ψ(r) ψ(r) φ(r) < ψ(r). p(t xn+ T x n ) p(t y n+ T y n ) p(t z n+ T z n ) p(t w n+ T w n ) =. (3) lim maxp(t x n T x n) p(t y n T y n ) p(t z n T z n ) p(t w n T w n ) =. From definition of d p and from () and (3) it follows that () lim max dp (T x n+ T x n ) d p (T y n+ T y n ) =. d p (T z n+ T z n ) d p (T w n+ T w n ) Now we shall prove that T x n T y n T z n and T w n are Cauchy sequences in the metric space (X d p ). Assume on the contrary that T x n or T y n or T z n or T w n is not a Cauchy sequence in (X d p ). Then there exists ɛ > for which we can find subsequences of integers m k and n k with n k > m k > k such that dp (T x (5) max mk T x nk ) d p (T y mk T y nk ) ɛ d p (T z mk T z nk ) d p (T w mk T w nk ) Further corresponding to m k we may choose n k such that it is the smallest integer satisfying (5) and n k > m k. Then dp (T x (6) max mk T x nk ) d p (T y mk T y nk ) < ɛ. d p (T z mk T z nk ) d p (T w mk T w nk )

6 6 A Quadruple Fixed Point Theorem for Contractive Type Condition... We have (7) () d p (T x mk T x nk ) d p (T x mk T x mk ) + d p (T x mk T x nk ) + d p (T x nk T x nk ) d p (T x mk T x mk ) + d p (T x mk T x mk ) d p (T x mk T x nk ) + d p (T x nk T x nk ) < d p (T x mk T x mk ) + ɛ + d p (T x nk T x nk ) from (6). Letting k in (7) and () and using () we get (9) lim k d p(t x mk T x nk ) lim k d p(t x mk T x nk ) ɛ. Similarly () () () lim d p(t y mk T y nk ) lim d p(t y mk T y nk ) ɛ k k lim d p(t z mk T z nk ) lim d p(t z mk T z nk ) ɛ k k lim d p(t w mk T w nk ) lim d p(t w mk T w nk ) ɛ k k (3) Using (5) and (9) - () we have max dp (T x mk T x nk ) d p (T y mk T y nk ) k d p (T z mk T z nk ) d p (T w mk T w nk ) ɛ lim lim k max ɛ. Now using (3) and (3) we obtain ɛ = lim k max dp (T x mk T x nk ) d p (T y mk T y nk ) d p (T z mk T z nk ) d p (T w mk T w nk ) p(t x mk T x nk ) p(t x mk T x mk ) p(t x nk T x nk ) p(t y mk T y nk ) p(t y mk T y mk ) p(t y nk T y nk ) p(t z mk T z nk ) p(t z mk T z mk ) p(t z nk T z nk ) p(t w mk T w nk ) p(t w mk T w mk ) p(t w nk T w nk ) () ɛ p(t = lim max xmk T x nk ) p(t y mk T y nk ) k p(t z mk T z nk ) p(t w mk T w nk ). (5) Similarly from (3) and () we obtain ɛ p(t = lim max xmk T x nk ) p(t y mk T y nk ) k p(t z mk T z nk ) p(t w mk T w nk ).

7 K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi 7 Now using (..) we have ψ (p(t x mk T x nk )) = ψ (p(f (x mk y mk z mk w mk ) F (x nk y nk z nk w nk ))) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) T x nk ) p(t y mk T y nk ). p(t z mk T z nk ) p(t w mk T w nk ) and Similarly ψ (p(t y mk T y nk )) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) ψ (p(t z mk T z nk )) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) ψ (p(t w mk T w nk )) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) T x nk ) p(t y mk T y nk ). p(t z mk T z nk ) p(t w mk T w nk ) Thus ψ(max p(t x mk T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) ) = max ψ (p(t x mk T x nk )) ψ (p(t y mk T y nk )) ψ (p(t z mk T z nk )) ψ (p(t w mk T w nk )) T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) ( T x nk ) p(t y mk T y nk ) p(t z mk T z nk ) p(t w mk T w nk ) ).

8 A Quadruple Fixed Point Theorem for Contractive Type Condition... Letting k and using () and (5) we obtain ( ɛ ( ɛ ( ɛ ( ɛ ψ ψ φ < ψ ) ) ) ) a contradiction. Thus T x n T y n T z n and T w n are Cauchy sequences in the metric space (X d p ). That is lim d p(t x n T x m ) = m lim d p(t z n T z m ) = m lim d p(t y n T y m ) = m lim d p(t w n T w m ) =. m From the definition of d p and from (3) we have lim p(t x n T x m ) = lim p(t y n T y m ) = m m (6) lim p(t z n T z m ) = lim p(t w n T w m ) =. m m Thus T x n T y n T z n and T w n are Cauchy sequences in (X p). Since (X p) is complete the Cauchy sequences T x n T y n T z n and T w n are convergent. Since T is an ICS mapping there exist x y z w X such that lim p(x n x) = p(x x) lim p(z n z) = p(z z) lim p(y n y) = p(y y) lim p(w n w) = p(w w). Since T is continuous we have lim p(t x n T x) = p(t x T x) lim p(t z n T z) = p(t z T z) lim p(t y n T y) = p(t y T y) lim p(t w n T w) = p(t w T w). These implies that T x n T y n T z n and T w n are convergent to T x T y T z and T w respectively. Using Lemma. () and from (6) it follows that lim p(t x n T x) = lim p(t y n T y) = (7) lim p(t z n T z) = and lim p(t w n T w) =. Suppose X has the property (A). Since x n z n are non - decreasing with x n x z n z and also y n w n are non - increasing with y n y w n w then by the property (A) we have x n x y n y z n z and w n w for all n.

9 K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi 9 Consider now ψ (p(t x n+ T F (x y z w))) = ψ (p(t F (x n y n z n w n ) T F (x y z w))) p(t xn T x) p(t y n T y) p(t z n T z) p(t w n T w) p(t xn T x) p(t y n T y). p(t z n T z) p(t w n T w) Taking n and using (7) we get p(t x T F (x y z w)) = so that T x = T F (x y z w).since T is injective we obtain x = F (x y z w). Similarly we can show that y = F (y z w x) z = F (z w x y) w = F (w x y z). Thus (x y z w) is a quadruple fixed point of F. Theorem.. Let (X p) be a complete partial metric space. Suppose T : X X is an ICS mapping and F : X X. Assume that there exist ψ Ψ and φ Φ such that (..) ψ (p(t F (x y z w) T F (u v r t))) p(t x T u) p(t y T v) p(t z T r) p(t w T t) p(t x T u) p(t y T v) p(t z T r) p(t w T t) for all x y z w u v r t X. Then F has a quadruple fixed point of the form (x x x x) where x X. Proof. By proceeding the proof of Theorem. we get for some x y z w X. Now from (..) we have F (x y z w) = x F (y z w x) = y F (z w x y) = z F (w x y z) = w ψ(p(t x T y)) = ψ (p(t F (x y z w) T F (y z w x))) p(t x T y) p(t y T z) p(t z T w) p(t w T x) p(t x T y) p(t y T z) p(t z T w) p(t w T x)

10 3 A Quadruple Fixed Point Theorem for Contractive Type Condition... Similarly we have p(t x T y) p(t y T z) ψ(p(t y T z)) p(t z T w) p(t w T x) p(t x T y) p(t y T z) p(t z T w) p(t w T x) p(t x T y) p(t y T z) ψ(p(t z T w)) p(t z T w) p(t w T x) p(t x T y) p(t y T z) p(t z T w) p(t w T x) and Now p(t x T y) p(t y T z) ψ(p(t w T x)) p(t z T w) p(t w T x) p(t x T y) p(t y T z) p(t z T w) p(t w T x) p(t x T y) p(t y T z) ψ max p(t z T w) p(t w T x) ψ(p(t x T y)) ψ(p(t y T z)) = max ψ(p(t z T w)) ψ(p(t w T x)) p(t x T y) p(t y T z) p(t z T w) p(t w T x) ( p(t x T y) p(t y T z) p(t z T w) p(t w T x) ) Hence p(t x T y) p(t y T z) max p(t z T w) p(t w T x) = since φ(t) > for t >. T x = T y T y = T z T z = T w T w = T x. Since T is injuctive we have x = y = z = w. Thus F has a quadruple fixed point of the form (x x x x). The following example illustrates our Theorem.

11 K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi 3 Example.. Let X = [ ] p(x y) = maxx y and T : X X be defined by T (x) = x. Let F : X X X X X by F (x y z w) = x +y +z +w and ψ φ : [ ) [ ) by ψ(t) = t φ(t) = t clearly all conditions of Theorem. are satisfied and ( F (x y z w) ψ(p(t F (x y z w) T F (u v r t))) = p x + y + z + w = max u + v + r + t 6 6 = 6 max x + y + z + w u + v + r + t = 6 ) F (u v r t) [max x u + max y v + max z r + max w t ] max max x u max y v max z r max w t maxx max u maxy v maxz r maxw t maxt max x T u maxt y T v maxt z T r maxt w T t = ψ (max p(t x T u) p(t y T v) p(t z T r) p(t w T t) ) ( p(t x T u) p(t y T v) p(t z T r) p(t w T t) ). Clearly ( ) is quadruple fixed point of F. 3. Application In this section we study the existence of a unique solution to an initial value problem as an application to Theorem.. Consider the initial value problem () x (t ) = l(t x(t ) x(t ) x(t ) x(t )) t I = [ ] x() = x where l : I [ x ) [ x ) [ x ) [ x ) [ x ) and x R +. Theorem 3.. Consider the initial value problem with l C ( I [ x ) [x ) [x ) [x ) )

12 3 A Quadruple Fixed Point Theorem for Contractive Type Condition... and t l(s x(s) y(s) z(s) w(s))ds max t t t t l(s x(s) x(s) x(s) x(s))ds x l(s y(s) y(s) y(s) y(s))ds x. l(s z(s) z(s) z(s) z(s))ds x l(s w(s) w(s) w(s) w(s))ds x Then there exists unique solution in C(I [ x )) for the initial value problem. Proof. The integral equation corresponding to initial value problem is t (9) x(t) = x + l(s x(s) x(s) x(s) x(s))ds. Let X = C(I [ x )) and p(x y) = maxx x y x for x y X. Define ICS mapping T : X X by T x = x ψ φ : [ ) [ ) by ψ(t) = t φ(t) = t and F : X X X X by Now F (x y z w)(t ) = x + t l(s x(s) y(s) z(s) w(s))ds. ψ(p(t F (x y z w)(t ) F (u v r t)(t ))) F (x y z w) = max x F (u v r t) x t x = + l(s x(s) y(s) z(s) w(s))ds max t x l(s u(s) v(s) r(s) t(s))ds +

13 K.P.R. Rao G.N.V. Kishore and K.V. Siva Parvathi 33 max = max = max x + max x + max max x(t ) max t t t t t t l(s x(s) x(s) x(s) x(s))ds x l(s y(s) y(s) y(s) y(s))ds x l(s z(s) z(s) z(s) z(s))ds x l(s w(s) w(s) w(s) w(s))ds x t t l(s u(s) u(s) u(s) u(s))ds x l(s v(s) v(s) v(s) v(s))ds x l(s r(s) r(s) r(s) r(s))ds x l(s t(s) t(s) t(s) t(s))ds x x y(t ) x z(t ) x w(t ) x x t(t ) x u(t ) x v(t ) x r(t ) max T x(t ) x T u(t ) x max T y(t ) x T v(t ) x max T z(t ) x T r(t ) x max T w(t ) x T t(t ) x = max p(t x T u) p(t y T v) p(t z T r) p(t w T t) p(t x T u) p(t y T v) = ψ max p(t z T r) p(t w T t) p(t x T u) p(t y T v) p(t z T r) p(t w T t) Thus F satisfies the condition (..) of Theorem.. From Theorem. we conclude that F has a quadruple fixed point. In particular x(t) is the unique solution of the integral equation (9). References [] T. Abdeljawad E. Karapınar K. TasExistence and uniqueness of a common fixed point on partial metric spaces Appl. Math. Lett. () () [] I. Altun F. Sola and H. Simsek Generalized contractions on partial metric spaces Topology and its Applications 57() ()

14 3 A Quadruple Fixed Point Theorem for Contractive Type Condition... [3] K.P. Chi On a fixed point theorem for certain class of maps satisfying a contractive condition depended on an another function Lobachevskii J. Math. 3() (9) 9 9. [] E. Karapınar I.M. Erhan Fixed point theorems for operators on partial metric spaces Applied Mathematics Letters () () 9 9.6/j.aml [5] E. Karapínar Quadruple fixed point theorems for weak φ - contractions ISRN Mathematical Analysis Article ID 993 () 5 pages. [6] E. Karapínar and V. Berinde Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces Banach Journal of Mathematical Analysis 6() () 7 9. [7] N.V. Luong N.X. Thuan and T.T. Hai Coupled fixed point theorems in partially ordered metric spaces depended on an another function Bulletin of mathematical analysis and applications 3(3) () 9. [] S.G. Matthews. Partial metric topology Research Report Dept. of Computer Science University of Warwick 99. [9] S.G. Matthews Partial metric topology in Proceedings of the th Summer Conference on General Topology and Applications Annals of the New York Academy of Sciences 7 (99) K.P.R. Rao Department of Mathematics Acharya Nagarjuna University Nagarjuna Nagar Guntur Andhra Pradesh India address: kprrao@yahoo.com G.N.V. Kishore Department of Mathematics Kl University Vaddeswaram Guntur Andhra Pradesh India address: kishore.apr@gmail.com gnvkishore@kluniversity.in K.V. Siva Parvathi Department of Applied Mathematics Krishna University M.R. Appa Row P.G.Center Nuzvid-5 Andhra Pradesh India address: kvsp979@yahoo.com