We investigate the existence of a fixed point of certain contractive multivalued mappings of integral type by using the admissible mapping. Our results generalize the several results on the topic in the literature involving Branciari, and Feng and Liu. We also construct some examples to illustrate our results.

1. Preliminaries and Introduction

Fixed point theory is one of the most celebrated research areas that has an application potential not only in nonlinear but also in several branches of mathematics. As a consequence of this fact, several fixed point results have been reported. It is not easy to know, manage, and use all results of this reich theory to get an application. To overcome such problems and clarify the literature, several authors have suggested a more general construction in a way that a number of existing results turn into a consequence of the proposed one. One of the examples of this trend is the investigations of fixed point of certain operator by using the α-admissible mapping introduced Samet et al. . This paper has been appreciated by several authors and this trend has been supported by reporting several interesting results; see for example .

In this paper, we define (α*,ψ)-contractive multivalued mappings of integral type and discuss the existence of a fixed point of such mappings. Our construction and hence results improve, extend, and generalize several results including Branciari  and Feng and Liu .

In what follows, we recall some basic definitions, notions, notations, and fundamental results for the sake of completeness. Let Ψ be a family of nondecreasing functions, ψ:[0,)[0,) such that n=1ψn(t)<  for each t>0, where ψn is the nth iterate of ψ. It is known that, for each ψΨ, we have ψ(t)<t for all t>0 and ψ(0)=0  for  t=0  . We denote by  Φ  the set of all Lebesgue integrable mappings,  ϕ:[0,)[0,)  which is summable on each compact subset of  [0,)  and  0ϵϕ(t)dt>0, for each  ϵ>0.

Let  (X,d)  be a metric space. We denote by  N(X)  the space of all nonempty subsets of  X, by  B(X)  the space of all nonempty bounded subsets of  X, and by  CL(X)  the space of all nonempty closed subsets of  X. For  AN(X)  and  xX, (1)d(x,A)=inf{d(x,a):aA}. For every  A,BB(X), (2)δ(A,B)=sup{d(a,b):aA,bB}. We denote  δ(A,B)  by  δ(x,B)  when  A={x}. If, for  x0X, there exists a sequence  {xn}nN  in  X  such that  xnGxn-1, then  O(G,x0)={x0,x1,x2,}  is said to be an orbit of  G:XCL(X)  at  x0. A mapping  f:XR is  G  orbitally lower semicontinuous at  x, if  {xn}  is a sequence in  O(G,x0)  and  xnx  implies  f(x)liminfnf(xn). Branciari  extended the Banach contraction principle  in the following way.

Theorem 1.

Let  (X,d)  be a complete metric space and let  G:XX  be a mapping such that (3)0d(Tx,Ty)ϕ(t)dtc0d(x,y)ϕ(t)dt for each  x,yX, where  c[0,1)  and  ϕΦ. Then  G  has a unique fixed point.

Since then many authors used integral type contractive conditions to prove fixed point theorems in different settings; see for example [12, 1622]. Feng and Liu  extended the result of Branciari  to multivalued mappings as follows.

Theorem 2 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Let  (X,d)  be a complete metric space and let  G:XCL(X)  be a mapping. Assume that for each  xX  and  yGx, there exists  zGy  such that (4)0d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt), where  ψΨ  and  ϕΦ. Then  G  has a fixed point in  X  provided  f(ξ)=d(ξ,Gξ)  is lower semicontinuous, with  ξX.

Definition 3 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let  (X,d)  be a metric space and  α:X×X[0,)  be a mapping. A mapping  G:XCL(X)  is  α*-admissible if  α(x,y)1α*(Gx,Gy)1, where  α*(Gx,Gy)=inf{α(a,b):aGx,bGy}.

Definition 4 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let  (X,d)  be a metric space. A mapping  G:XCL(X)  is called  α*-ψ-contractive if there exist two functions  α:X×X[0,)  and  ψΨ  such that (5)α*(Gx,Gy)H(Gx,Gy)ψ(d(x,y)) for all  x,yX.

Theorem 5 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let  (X,d)  be a complete metric space, let  α:X×X[0,)  be a function, let  ψΨ  be a strictly increasing map, and let  G  be a closed-valued  α*-admissible and  α*-ψ-contractive multifunction on  X. Suppose that there exist  x0X  and  x1Gx0  such that  α(x0,x1)1. Assume that if  {xn}  is a sequence in  X  such that  α(xn,xn+1)1  for all  n  and  xnx, then  α(xn,x)1  for all  n. Then  G  has a fixed point.

Definition 6 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let  (X,d)  be a metric space and let  G:XCL(X)  be a mapping. We say that  G  is a generalized  (α*,ψ)-contractive if there exists  ψΨ  such that (6)α*(Gx,Gy)d(y,Gy)ψ(d(x,y)) for each  xX  and  yGx, where  α*(Gx,Gy)=inf{α(a,b):aGx,bGy}.

Theorem 7 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let  (X,d)  be a complete metric space and let  G:XB(X)  be a mapping such that for each  xX  and  yGx, we have (7)α*(Gx,Gy)δ(y,Gy)ψ(d(x,y)), where  ψΨ. Assume that there exist  x0X  and  x1Gx0  such that  α(x0,x1)1. Moreover  G  is an  α*-admissible mapping. Then there exists an orbit  {xn}  of  G  at  x0  and  xX  such that  limnxn=x. Moreover,  {x}=Gx  if and only if  f(ξ)=δ(ξ,Gξ)  is lower semicontinuous at  x.

2. Main Results

In this section, we state and proof our main results. We first give the definition of the following notion.

Definition 8.

Let  (X,d)  be a metric space. We say that  G:XCL(X)  is an integral type  (α*,ψ)-contractive mapping if there exist two functions  ψΨ  and  ϕΦ  such that for each  xX  and  yGx, there exists  zGy  satisfying (8)0α*(Gx,Gy)d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt), where  α*(Gx,Gy)=inf{α(a,b):aGx,bGy}.

Example 9.

Let  X=R be endowed with the usual metric  d. Define  G:XCL(X)  by (9)Gx={[x,)if  x0(-,6x]if  x<0, and  α:X×X[0,)  by (10)α(x,y)={x+y+1if  x,y00otherwise. Take  ψ(t)=t/4  and  ϕ(t)=2t  for all  t0. Then, for each  xX  and  yGx, there exists  zGy  such that (11)0α*(Gx,Gy)d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt). Hence  G  is an integral type  (α*,ψ)-contractive mapping. Note that (4) does not hold at  x=-2.

Definition 10.

We say that  ϕΦ  is an integral subadditive if, for each  a,b>0, we have (12)0a+bϕ(t)dt0aϕ(t)dt+0bϕ(t)dt.

We denote by  Φs the class of all integral subadditive functions  ϕΦ.

Example 11.

Let  ϕ1(t)=(1/2)(t+1)-1/2  for all  t0, ϕ2(t)=(2/3)(t+1)-1/3  for all  t0,  and  ϕ3(t)=e-t  for all  t0. Then  ϕiΦs, where  i=1,2,3.

Definition 12.

Let  (X,d)  be a metric space. We say that  G:XCL(X)  is a subintegral type  (α*,ψ)-contractive if there exist two functions  ψΨ  and  ϕΦs  such that for each  xX  and  yGx, there exists  zGy  satisfying (13)0α*(Gx,Gy)d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt), where  α*(Gx,Gy)=inf{α(a,b):aGx,bGy}.

Example 13.

Let  X=R be endowed with the usual metric  d. Define  G:XCL(X)  by (14)Gx={[x4,x2]if  x0,[24x,12x]if  x<0, and  α:X×X[0,)  by (15)α(x,y)={2if  x=y=0,0otherwise. Take  ψ(t)=t/3  and  ϕ(t)=(2/3)(t+1)-1/3  for all  t0. Then, for each  xX  and  yGx, there exists  zGy  such that (16)0α*(Gx,Gy)d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt). Hence  G  is an subintegral type  (α*,ψ)-contractive mapping.

Theorem 14.

Let  (X,d)  be a complete metric space and let  G:XCL(X)  be an  α*-admissible subintegral type  (α*,ψ)-contractive mapping. Assume that there exist  x0X  and  x1Gx0  such that  α(x0,x1)1. Then there exists an orbit  {xn}  of  G  at  x0  and  xX  such that  limnxn=x. Moreover,  x  is a fixed point of  G  if and only if  f(ξ)=d(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x.

Proof.

By the hypothesis, there exist  x0X  and  x1Gx0  such that  α(x0,x1)1. Since  G  is  α*-admissible, then  α*(Gx0,Gx1)1. For  x0X  and  x1Gx0, there exists  x2Gx1  such that (17)0d(x1,x2)ϕ(t)dt0α*(Gx0,Gx1)d(x1,x2)ϕ(t)dtψ(0d(x0,x1)ϕ(t)dt).   Since  ψ  is nondecreasing, we have (18)ψ(0d(x1,x2)ϕ(t)dt)ψ2(0d(x0,x1)ϕ(t)dt). As  α(x1,x2)1  by  α*-admissibility of  G, we have  α*(Gx1,Gx2)1. For  x1X  and  x2Gx1, there exists  x3Gx2  such that (19)0d(x2,x3)ϕ(t)dt0α*(Gx1,Gx2)d(x2,x3)ϕ(t)dtψ(0d(x1,x2)ϕ(t)dt)ψ2(0d(x0,x1)ϕ(t)dt). Since  ψ  is nondecreasing, we have (20)ψ(0d(x2,x3)ϕ(t)dt)ψ3(0d(x0,x1)ϕ(t)dt). By continuing the same process, we get a sequence  {xn}  in  X  such that  xnGxn-1,  α(xn-1,xn)1, and (21)0d(xn,xn+1)ϕ(t)dtψn(0d(x0,x1)ϕ(t)dt),for  each  nN. Letting  n  in above inequality, we have (22)limn0d(xn,xn+1)ϕ(t)dt=0. Also, we have (23)limn0d(xn,Gxn)ϕ(t)dt=0, which implies that (24)limnd(xn,Gxn)=0. For any  n,pN, we have (25)d(xn,xn+p)i=nn+p-1d(xi,xi+1). Since  ϕΦs, it can be shown by induction that (26)0d(xn,xn+p)ϕ(t)dti=nn+p-10d(xi,xi+1)ϕ(t)dt. From (21) and (26), we have (27)0d(xn,xn+p)ϕ(t)dti=nn+p-1ψi(0d(x0,x1)ϕ(t)dt). Since  ψΨ  it follows that  {xn}  is Cauchy sequence in  X. As  X  is complete, there exists  x*X  such that  xnx*  as  n. Suppose  f(ξ)=d(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x*; then (28)d(x*,Gx*)liminfnf(xn)=liminfnd(xn,Gxn)=0. By closedness of  G  it follows that  x*Gx*. Conversely, suppose that  x*  is a fixed point of  G  then  f(x*)=0liminfnf(xn).

Example 15.

Let  X=R be endowed with the usual metric  d. Define  G:XCL(X)  by (29)Gx={[x,x+1]if  x0,(-,6x]if  x<0, and  α:X×X[0,)  by (30)α(x,y)={x+y+1if  x,y0,0otherwise. Take  ψ(t)=t/2  and  ϕ(t)=(1/2)(t+1)-1/2  for all  t0. Then, for each  xX  and  yGx, there exists  zGy  such that (31)0α*(Gx,Gy)d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt). Hence  G  is a subintegral type  (α*,ψ)-contractive mapping. Clearly,  G  is  α*-admissible. Also, we have  x0=1  and  x1=2Gx0  such that  α(x0,x1)=4. Therefore, all the conditions of Theorem 14 are satisfied and  G  has infinitely many fixed points. Note that Theorem 2 in Section 1 is not applicable here. For example, take  x=-1  and  y=-6.

Definition 16.

Let  (X,d)  be a metric space. We say that  G:XB(X)  is an integral type  (α*,ψ,δ)-contractive mapping if there exist two functions  ψΨ  and  ϕΦ  such that (32)0α*(Gx,Gy)δ(y,Gy)ϕ(t)dtψ(0d(x,y)ϕ(t)dt) for each  xX  and  yGx, where  α*(Gx,Gy)=inf{α(a,b):aGx,bGy}.

Definition 17.

Let  (X,d)  be a metric space. We say that  G:XB(X)  is a subintegral type  (α*,ψ,δ)-contractive mapping if there exist two functions  ψΨ  and  ϕΦs  such that (33)0α*(Gx,Gy)δ(y,Gy)ϕ(t)dtψ(0d(x,y)ϕ(t)dt) for each  xX  and  yGx, where  α*(Gx,Gy)=inf{α(a,b):aGx,bGy}.

Theorem 18.

Let  (X,d)  be a complete metric space and let  G:XB(X)  be an  α*-admissible subintegral type  (α*,ψ,δ)-contractive mapping. Assume that there exist  x0X  and  x1Gx0  such that  α(x0,x1)1. Then there exists an orbit  {xn}  of  G  at  x0  and  xX  such that  limnxn=x. Moreover,  xX  such that  {x}=Gx  if and only if  f(ξ)=δ(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x.

Proof.

By the hypothesis, there exist  x0X  and  x1Gx0  such that  α(x0,x1)1. Since  G  is  α*-admissible, then  α*(Gx0,Gx1)1. For  x0X  and  x1Gx0, we have (34)0α*(Gx0,Gx1)δ(x1,Gx1)ϕ(t)dtψ(0d(x0,x1)ϕ(t)dt). Since  Gx1, then we have  x2Gx1  such that (35)0d(x1,x2)ϕ(t)dt0α*(Gx0,Gx1)δ(x1,Gx1)ϕ(t)dtψ(0d(x0,x1)ϕ(t)dt). Since  ψ  is nondecreasing, we have (36)ψ(0d(x1,x2)ϕ(t)dt)ψ2(0d(x0,x1)ϕ(t)dt). As  α(x1,x2)1  by  α*-admissibility of  G, we have  α*(Gx1,Gx2)1. Thus, we have  x3Gx2  such that (37)0d(x2,x3)ϕ(t)dt0α*(Gx1,Gx2)δ(x2,Gx2)ϕ(t)dtψ(0d(x1,x2)ϕ(t)dt)ψ2(0d(x0,x1)ϕ(t)dt). Since  ψ  is nondecreasing, we have (38)ψ(0d(x2,x3)ϕ(t)dt)ψ3(0d(x0,x1)ϕ(t)dt). By continuing the same process, we get a sequence  {xn}  in  X  such that  xnGxn-1,  α(xn-1,xn)1, and (39)0d(xn,xn+1)ϕ(t)dt0δ(xn,Gxn)ϕ(t)dtψn(0d(x0,x1)ϕ(t)dt),hihhhhhhfor  each  nN. Letting  n  in above inequality, we have (40)limn0δ(xn,Gxn)ϕ(t)dt=0, which implies that (41)limnδ(xn,Gxn)=0. For any  n,pN, we have (42)d(xn,xn+p)i=nn+p-1d(xi,xi+1). Since  ϕΦs, it can be shown by induction that (43)0d(xn,xn+p)ϕ(t)dti=nn+p-10d(xi,xi+1)ϕ(t)dt. From (39) and (43), we have (44)0d(xn,xn+p)ϕ(t)dti=nn+p-1ψi(0d(x0,x1)ϕ(t)dt). Since  ψΨ  it follows that  {xn}  is Cauchy sequence in  X. As  X  is complete, there exists  x*X  such that  xnx*  as  n. Suppose  f(ξ)=δ(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x*; then (45)δ(x*,Gx*)liminfnf(xn)=liminfnδ(xn,Gxn)=0. Hence,  {x*}=Gx*  because  δ(A,B)=0  implies  A=B={a}. Conversely, suppose that  {x*}=Gx*. Then  f(x*)=0liminfnf(xn).

Example 19.

Let  X={1,3,5,7,9,}  be endowed with the usual metric  d. Define  G:XB(X)  by (46)Gx={{x-2,x+2}if  x1,{1}if  x=1, and  α:X×X[0,)  by (47)α(x,y)={1if  x=y=1,14otherwise. Take  ψ(t)=t/2  and  ϕ(t)=(2/3)(t+1)-1/3  for all  t0. Clearly,  G  is an  α*-admissible subintegral type  (α*,ψ,δ)-contractive mapping. Also, we have  x0=1  and  x1=1Gx0  such that  α(x0,x1)=1. Therefore, all the conditions of Theorem 18 hold and  G  has fixed points.

Example 20.

Let  X=R be endowed with the usual metric  d. Define  G:XB(X)  by (48)Gx={{x,x}if  x0,(x4,x2)if  x<0, and  α:X×X[0,)  by (49)α(x,y)={1if  x,y0,0otherwise. Take  ψ(t)=t/4  and  ϕ(t)=e-t  for all  t0. Then it is easy to check that all the conditions of Theorem 18 hold. Therefore  G  has infinitely many fixed points.

Remark 21.

Let  ϕ(t)=1  for all  t0; Theorem 18 reduces to Theorem 7 in Section 1.

Remark 22.

Note that subadditivity of the integral was needed in the proofs of Theorems 14 and 18 in order to obtain inequalities (26) and (43). It is natural to ask wether the conclusions of Theorems 14 and 18 are valid if we replace subintegral contractive conditions (13) and (33) by integral contractive conditions (8) and (32), respectively. Looking at our proofs, we can say that it will be true if the inequalities (26) and (43) hold. Here we would like to mention that many authors (see for example [14, 23]) while proving the results on integral contractions have not assumed that the integral is subadditive but indeed they used the subadditivity of the integral in the proofs of their results while obtaining the inequalities comparable to inequalities (26) and (43).

3. Application

In this section, we obtain some fixed point results for partially ordered metric spaces, as consequences of aforementioned results. Moreover, we apply our result to prove the existence of solution for an integral equation.

Let  A  and  B  be subsets of a partially ordered set. We say that  ArB, if for each  aA  and  bB, we have  ab.

Theorem 23.

Let  (X,,d)  be a complete ordered metric space and let  G:XCL(X)  be a mapping such that for each  xX  and  yGx  with  xy, there exists  zGy  satisfying (50)0d(y,z)ϕ(t)dtψ(0d(x,y)ϕ(t)dt), where  ψΨ  and  ϕΦs. Assume that there exist  x0X  and  x1Gx0  such that  x0x1. Also, assume that  xy  implies  GxrGy. Then there exists an orbit  {xn}  of  G  at  x0  and  xX  such that  limnxn=x. Moreover,  x  is a fixed point of  G  if and only if  f(ξ)=d(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x.

Proof.

Define  α:X×X[0,)  by (51)α(x,y)={1if  xy,0otherwise. By using hypothesis of corollary and definition of  α, we have  α(x0,x1)=1. As  xy  implies  GxrGy, by using the definitions of  α  and  r, we have that  α(x,y)=1  implies  α*(Gx,Gy)=1. Moreover, it is easy to check that  G  is an integral type  (α*,ψ)-contractive mapping. Therefore, by Theorem 14, there exists an orbit  {xn}  of  G  at  x0  and  xX  such that  limnxn=x. Moreover,  x  is a fixed point of  G  if and only if  f(ξ)=d(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x.

Considering  G:XX  and  ϕ(t)=1  for each  t0, Theorem 23 reduces to following result.

Corollary 24.

Let  (X,,d)  be a complete ordered metric space and let  G:XX  be a nondecreasing mapping such that, for each  xX  with  xGx, we have (52)d(Gx,G2x)ψ(d(x,Gx)), where  ψΨ. Assume that there exists  x0X  such that  x0Gx0. Then there exists an orbit  {xn}  of  G  at  x0  and  xX  such that  limnxn=x. Moreover,  x  is a fixed point of  G  if and only if  f(ξ)=d(ξ,Gξ)  is  G  orbitally lower semicontinuous at  x.

Consider an integral equation of the form (53)x(t)=abK(t,s,x(s))ds,t[a,b], where  K:[a,b]×[a,b]×RR is continuous and nondecreasing.

Theorem 25.

Assume that

for  u,vC([a,b],R), with  u(t)v(t)  for each  t[a,b], we have (54)|K(t,s,u(t))-K(t,s,v(t))|ψ(d(u,v))(b-a) for each  t,s[a,b], where  ψΨ;

for each  t,s[a,b], there exists  x0C([a,b],R) such that (55)x0(t)abK(t,s,x0(s))ds.

Then there exists an iterative sequence  {xn}, starting from  x0, and  xC([a,b],R)  such that  limnxn=x. Moreover,  x  is a solution of (53) if and only if  f(ξ)=d(ξ,y)  is lower semicontinuous at  x, where  y(t)=abK(t,s,ξ(s))ds.

Proof.

It is easy to see that  X=C([a,b],R) is complete with respect to the metric  d(x,y)=maxt[a,b]|x(t)-y(t)|. We define partial ordering on  X  as follows:  xy  if and only if  x(t)y(t)  for each  t[a,b]. Define  G:XX  by  Gx=y, where  y(t)=abK(t,s,x(s))ds, for each  t,s[a,b]. From (ii), we have  x0Gx0. For  xX, let  Gx=y  and  Gy=z; that is, y(t)=abK(t,s,x(s))ds  and  z(t)=abK(t,s,y(s))ds, for each  t,s[a,b]. Then, for each  xX  with  xGx, we have (56)d(Gx,G2x)=maxt[a,b]|y(t)-z(t)|=maxt[a,b]|abK(t,s,x(s))dshhhh-abK(t,s,y(s))ds|maxt[a,b]ab|K(t,s,x(s))-K(t,s,y(s))|ds  ψ(d(x,Gx))(b-a)(b-a). That is  d(Gx,G2x)ψ(d(x,Gx)), for each  xX  with  xGx. Clearly,  G  is nondecreasing. Therefore, all conditions of Corollary 24 hold and the conclusions follow from Corollary 24.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgment

The authors are grateful to the reviewers for their careful readings and useful comments.

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