Dhananjay GOPALDepartment of Applied Mathematics&Humanities，S.V.National Institute of Technology，Surat-395007，Gujarat，IndiaE-mail:dg@ashd.svnit.ac.in;gopaldhananjay@yahoo.in

Mujahid ABBASDepartment of Mathematics and Applied Mathematics，University of Pretoria，Lynnwood Road，Pretoria 0002，South Africa;Department of Mathematics，King AbdulAziz University，P.O.Box 80203 Jeddah 21589，Saudi ArabiaE-mail:Mujahid.Abbas@up.ac.za;abbas.muajahid@gmail.com

Deepesh Kumar PATELDepartment of Mathematics Visvesvaraya National Institute of Technology，Nagpur-440010，Maharashtra，IndiaE-mail:deepesh456@gmail.com

Calogero VETROUniversit`a degli Studi di Palermo，Dipartimento di Matematica e Informatica，Via Archirafi，34-90123 Palermo，ItalyE-mail:calogero.vetro@unipa.it

FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION?

Dhananjay GOPAL?
Department of Applied Mathematics&Humanities，S.V.National Institute of Technology，Surat-395007，Gujarat，India
E-mail:dg@ashd.svnit.ac.in;gopaldhananjay@yahoo.in

Mujahid ABBAS
Department of Mathematics and Applied Mathematics，University of Pretoria，Lynnwood Road，Pretoria 0002，South Africa;Department of Mathematics，King AbdulAziz University，P.O.Box 80203 Jeddah 21589，Saudi Arabia
E-mail:Mujahid.Abbas@up.ac.za;abbas.muajahid@gmail.com

Deepesh Kumar PATEL
Department of Mathematics Visvesvaraya National Institute of Technology，Nagpur-440010，Maharashtra，India
E-mail:deepesh456@gmail.com

Calogero VETRO
Universit`a degli Studi di Palermo，Dipartimento di Matematica e Informatica，Via Archirafi，34-90123 Palermo，Italy
E-mail:calogero.vetro@unipa.it

In this paper，we introduce new concepts of α-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in［21，22］and different from α-GF-contractions given in［8］.Then，sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings，in the setting of complete metric space.Consequently，the obtained results encompass various generalizations of the Banach contraction principle.Moreover，some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

fixed points;nonlinear fractional differential equations;periodic points

2010 MR Subject Classification37C25;34A08

1　Introduction

The contraction mapping principle appeared in explicit form in Banach’s thesis in 1922［3］，where it was used to establish the existence of a solution for an integral equation.Since then，because of its simplicity and usefulness，this fixed point theorem became a very popular tool in solving existence problems in many branches of mathematical analysis.Consequently，it was largely studied and generalized;see［4-7，11，14，17，20］and others.

Recently，Wardowski［21，22］introduced the concepts of F-contraction and F-weak contraction to generalize the Banach’s contraction in many ways，see also［19］.On the other hand，Hussain et al.［8］introduced the concept of α-GF-contraction as a generalization of F-contraction and obtained some interesting fixed point results.

Following this direction of research，we introduce new concepts of α-type F-contractive mappings and prove some fixed point and periodic point theorems concerning such contractions. Moreover，some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

2　Preliminaries

The aim of this section is to present some notions and results used in the paper.Throughout the article N，R+and R will denote the set of natural numbers，non-negative real numbers and real numbers respectively.

Definition 2.1（see［21］）Let Fbe a mapping satisfying:

（F1）F is strictly increasing，that is

We denote with F the family of all functions F that satisfy conditions（F1）-（F3）.

Example 2.2The following functionbelongs to F:

Definition 2.3（see［21］）Let （X，d）be a metric space.A mapping f:X→X is called an F-contraction on X if there exist F∈F and τ＞0 such that for all x，y∈X with d（fx，fy）＞0，we have

Definition 2.4（see［22］）Let（X，d）be a metric space.A mapping f:X→X is called an F-weak contraction on X if there exist F∈F and τ＞0 such that，for all x，y∈X with d（fx，fy）＞0，we have

Remark 2.5（see［22］）Every F-contraction is an F-weak contraction but converse is not necessarily true.

Definition 2.6Let ΔGdenote the set of all functions G:（R+）4→R+satisfying the condition:

Example 2.7The following functionbelongs to

Definition 2.8（see［8］）Let（X，d）be a metric space and f:X→X.Also suppose thatare two functions.We say that f is an α-GF-contraction if for all x，with，we have

Definition 2.9（see［18］）A mappingis α-admissible if there exists a functionsuch that

3　Fixed Point Results for α-type F-contractions

In this section，we first introduce the concepts of α-type F-contractions and then we prove some fixed point theorems for these contractions in a complete metric space.For convenience，we assume that an expression-∞·0 has value-∞.

We begin with the following definitions:

Definition 3.1Let（X，d）be a metric space.A mapping f:X→X is said to be an α-type F-contraction on X if there exist τ＞0 and two functionssuch that for all，the following inequality holds

Definition 3.2Let（X，d）be a metric space.A mapping f:X → X is said to be an α-type F-weak contraction on X if there exists τ＞0 and two functions F ∈F andsuch that，for all xsatisfying，the following inequality holds

Remark 3.3Every α-type F-contraction is an α-type F-weak contraction，but converse is not necessarily true.

Then，for x=0 and y=1，by puttingwe get

Further，since

therefore，inequality（2.1）reduces to

a contradiction and hence f is not an F-weak contraction.

However，since

then f is an α-type F-weak contraction for the choice

Remark 3.5Definition 3.1（respectively，Definition 3.2）reduces to F-contraction（respectively，F(xiàn)-weak contraction）for α（x，y）=1.

The next two examples demonstrate that α-type F-contractions（defined above）and α-GF-contractions［8］are independent.

Example 3.6Let X=［0，1］and d be the usual metric on X.Define f:X→X by

Then f is an α-type F-weak contraction with α（x，y）=1 for all xsuch thatBut f is not an α-GF-contraction［8］.To see this，considersuch that

and hence f is not an α-GF-contraction.

Then，one can easily verify by simple calculations that f is an α-GF-contraction;but it is not an α-type F-weak contraction.To see this consider x=0 and y=2，then we get

and so the inequality 6≤4e-τdoes not hold for any τ＞0.Hence f is not an α-type F-weak contraction.

Now，we prove our first result.

Theorem 3.8Let（X，d）be a complete metric space and f:X→X be an α-type-F-weak contraction satisfying the following conditions:

（i）f is α-admissible，

（ii）there exists x0∈X such that α（x0，fx0）≥1，

（iii）f is continuous.

By induction we get

Since f is an α-type F-weak contraction，then，for every n∈N，we write

Consequently，we have

If there exists n∈N such thatthen（3.3）becomes

This implies that

Taking limit as n→+∞in（3.4），we get

that together with（F2）gives us

From（3.4），for all n∈N，we deduce that

Next using（3.5），（3.6）and taking limit as n→+∞in（3.7），we get

Since X is complete，there existsFinally，the continuity of f yields

i.e.，x?is a fixed point of f.

In the next theorem we omit the continuity hypothesis of f.

Theorem 3.9Let（X，d）be a metric space and f:X → X be an α-type F-weak contraction satisfying the following conditions:

（ii）f is α-admissible，

（iv）F is continuous.

that is x?is a fixed point of f.

By（3.9），we get

Example 3.4 above satisfies all the hypothesis of Theorem 3.9，consequently f has at least a fixed point.Hereare two fixed point of f.

To ensure the uniqueness of the fixed point，we will consider the following hypothesis:

Theorem 3.10Adding condition（H）to the hypotheses of Theorem 3.8（respectively，Theorem 3.9）the uniqueness of the fixed point is obtained.

ProofSuppose that y?is an another fixed point of f，soThen，we get easily

a contradiction，which implies that

Example 3.6 above satisfies all the hypothesis of Theorem 3.10，hence f has unique fixed point

From Remark 3.3，we deduce the following corollary.

Corollary 3.11Let（X，d）be a complete metric space and f:X → X be an α-type F-contraction satisfying the hypotheses of Theorem 3.10，then f has unique fixed point.

Finally，we conclude that many existing results in the literature can be deduced easily from our Theorem 3.10.In fact，taking in Theorem 3.10，we obtain the following fixed point result.

Corollary 3.12（see［22］）Let（X，d）be a complete metric space and f:X→X be an F-weak contraction.If F is continuous，then f has a unique fixed point x?in X.

Since the above Corollary 3.12 implies the corresponding theorems in Wardowski［21］，′Ciri′c［6］，Hardy and Rogers［7］，thus these results are consequences of our Theorem 3.10.

4　Periodic Point Results

It is an obvious fact that，if f is a mapping which has a fixed point x，then x is also a fixed point of fnfor every n∈N.However，the converse is false.Indeed，let X=［0，1］andThen f has a unique fixed point atbut fn=I（identity map on X）for each even n＞1，has every point of X as a fixed point. On the other hand，ifgiven by fx=cosx for all，is nonexpansive and every iterative of f has the same fixed point as f.

In this section we prove some periodic point results for self-mappings on a complete metric space.In the sequel，we need the following definition.

Definition 4.1A mapping f:X→X is said to have property（P）iffor every

For further details on these property，we refer to［10］.

Theorem 4.2Let（X，d）be a complete metric space and f:X → X be a mapping satisfying the following conditions

（i）there exists τ＞0 and two functionsthat

holds for all x∈X with d（fx，f2x）＞0，

（iii）f is α-admissible，

（v）if w∈Fix（fn）and w/∈Fix（f），then α（fn-1w，fnw）≥1. Then f has property（P）.

and by induction we write

If there exists n0∈N such thatis a fixed point of f and the proof is finished.Hence，we assume

From（4.1）and（i），we have

By using a similar reasoning as in the proof of Theorem 3.8，we get that the sequenceis a Cauchy sequence and hence the completeness of（X，d）ensures that there existssuch that

Corollary 4.3Let（X，d）be a complete metric space andbe a continuous mapping satisfying

5　Application

In this section，we present an application of Theorem 3.9 to establishing the existence of solutions for a nonlinear fractional differential equation considered in［2］.

We will study the existence of solutions for the nonlinear fractional differential problem

via the integral boundary conditions

Note that，for a continuous function g:R+→R，the Caputo derivative of fractional order β is defined as

where［β］denotes the integer part of the real number β.Also，the Riemann-Liouville fractional derivatives of order β for a continuous function g:R+→R is defined by

provided the right-hand side is point-wise defined on（0，+∞），see for instance［15］.

Now，we prove the following existence theorem.

Theorem 5.1Suppose that

（i）there exist a function ξ:R×R→R and τ＞0 such that

Then，problem（5.1）has at least one solution.

Then，problem（5.1）is equivalent to findwhich is a fixed point of T.

By passing to logarithm，we write

Therefore

This implies that T is an α-type F-contraction.Next，by using condition（iii），we get

is an easy example of function suitable for Theorem 5.1，whereis given by

6　Conclusion

Taking into account its interesting applications，searching for fixed point and periodic point theorems involving contractive type conditions received considerable attention through the last few decades.In this connection，the main aim of our paper is to present new concepts of α-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in［21，22］and different from α-GF-contractions given in［8］.For these type of contractions，the existence and uniqueness of fixed point in complete metric space are established.An application to nonlinear fractional differential equation illustrates the usability of the obtained results for studying problems arising in pure and applied sciences.The new concepts lead to further investigations and applications.For instance，using the recent ideas in the literature［12，13，16］，it is possible to extend our results to the case of coupled and cyclical fixed points in partially ordered metric spaces.It will be also interesting to apply these concepts in a metric space having graphical structure on it，see［9］.

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?August 26，2015;revised December 6，2015.

?Dhananjay GOPAL.

AcknowledgementsThe thanks for the support of CSIR，Govt.of India，Grant No.-25（0215）/13/EMR-II and C.Vetro is member of the Gruppo Nazionale per l’Analisi Matematica，la Probabilit`a e le loro Applicazioni（GNAMPA）of the Istituto Nazionale di Alta Matematica（INdAM）.