Well-Posedness And Data Dependence Of Strict Fixed Point For -Hardy Roger Type Contraction And Applications

Well-Posedness And Data Dependence Of Strict Fixed Point For -Hardy Roger Type Contraction And Applications

Deepak Kumar

, Anita Tomar

, Ritu Sharma

Received 15 February 2019

Abstract

We …rst familiarise with -Hardy Roger type contraction in the frame work of metric space. Then, well-posedness, data dependence, existence and uniqueness results of strict …xed point for -Hardy Roger type contraction are presented. The obtained results generalize the existing results in the literature.

Applications to an integral inclusion equation and Fractals concludes the paper.

1 Introduction

Banach contraction principle [1] has been generalised in numerous directions and one such generalisation is due to Nadler [14], who generalised it considering set-valued contraction. There after many results are established for set-valued mappings (see for instance [3]-[8], [16]-[21]). In this paper, considering the fact that Hardy-Rogers type operator is a ´Ciri´c type operator (however the reverse need not be true), we introduce - Hardy Roger type contraction and establish strict …xed point for it using iterations of a delta distance which is not even a metric. Also, we present well-posedness and data dependence of strict …xed point problem and utilise it to solve an integral inclusion equation and in presenting a novel iterated function framework via

-Hardy-Roger type operator to obtain attractor of multifunction system.

Let(X; d)be a metric space,

P(X) =fY X :Y 6=;g; Pb(X) =fY 2P(X) :Y is boundedg; P_cl(X) =fY 2P(X) :Y is closedg and P_cp(X) =fY 2P(X) :Y is compactg:

De…ne a set-valued operator as T : X ! P(X) and T(Y) = [x2YT(x) for Y 2 P(X). Also, F_T = fx 2 X : x 2 T(x)g is a set of …xed points and (SF)_T = fx 2 X : Tx = fxgg is a set of strict …xed points of the set-valued operator T: Chifu and Petrusel [10] introduced the generalized functional as:

:P(X) P(X)!R⁺[ f1g;

(A; B) = supfd(a; b) :a2A; b2Bg:

2 Main Results

Firstly, we de…ne -Hardy Roger type contraction and establish strict …xed point making use of iterations of a delta distance which is not even a metric.

De…nition 1 If T :X !Pb(X)is a set-valued operator of a metric space(X; d) satisfying (T(x);T(y)) a₁d(x; y) +a₂ (x;T(x)) +a₃ (y;T(y)) +a₄ (x;T(y)) +a₅ (y;T(x));

wherea⁰_is2R⁺,P5

i=1a_i<1,x; y2X, then T is called a -Hardy Roger type contraction.

Mathematics Sub ject Classi…cations: 47H10; 54H25; 46J10; 46J15.

yDepartment of Mathematics, Lovely Professional University, Phagwara, Punjab-144411, India

zDepartment of Mathematics, Government Degree College Thatyur, Tehri Garhwal, (Uttrakhand), India

xDepartment of Mathematics, V. S. K. C. Government P. G. College Dakpathar Dehradun (Uttrakhand), India

46

Theorem 1 IfT is a -Hardy Roger type contraction of a complete metric space(X; d), then(SF)_T =fu g: Proof. Letu02X. Then there existsu12 T(u0)and

(u0;T(u0)) qd(u0; u1); q >1 is arbitrary.

Now,

(u1;T(u1)) (T(u0);T(u1))

a1d(u0; u1) +a2 (u0;T(u0)) +a3 (u1;T(u1)) +a₄ (u₀; T(u₁)) +a₅ (u₁;T(u₀))

a1d(u0; u1) +a2qd(u0; u1) +a3 (u1;T(u1)) +a₄fd(u₀; u₁) + (u₁;T(u₁))g+a₅ (u₁; u₁) (a1+a2q+a4)d(u0; u1) + (a3+a4) (u1;T(u1)):

This implies

(1 a₃ a₄) (u₁;T(u₁)) (a₁+a₂q+a₄)d(u₀; u₁);

or

(u1;T(u1)) a₁+a₂q+a₄

1 (a3+a4) d(u0; u1):

Also,u₁2 T(u₀); 9 u₂2 T(u₁)and

(u₁;T(u₁)) qd(u₁; u₂):

Therefore,

(u2;T(u2)) (T(u1);T(u2))

a₁d(u₁; u₂) +a₂ (u₁;T(u₁)) +a₃ (u₂;T(u₂)) +a₄ (u₁;T(u₂)) +a5 (u2;T(u1))

a₁d(u₁; u₂) +a₂qd(u₁; u₂) +a₃ (u₂;T(u₂)) +a₄fd(u₁; u₂) + (u2;T(u2))g+a5 (u2; u2)

= (a₁+a₂q+a₄)d(u₁; u₂) + (a₃+a₄) (u₂;T(u₂)):

This implies

(1 (a3+a4)) (u2;T(u2)) (a1+a4+a2q)d(u1; u2);

or

(u2;T(u2)) a1+a4+a2q

1 (a₃+a₄) d(u1; u2) a1+a4+a2q

1 (a3+a4) (u1;T(u1)) a1+a4+a2q

1 (a3+a4)

2

d(u0; u1):

Following the similar pattern, we construct a sequence(un)n2Nsatisfying the following properties:

(i) u_n2 T(u_{n 1});

(ii) d(un; un+1) (un;T(un))

ha1+a4+a2q 1 a3 a4

in

d(u0; u1):

Next, consider

d(u_n; u_n+p) d(u_n; u_n+1) +d(u_n+1; u_n+2) +:::+d(u_{n+p 1}; u_n+p) a₁+a₄+a₂q

1 a3 a4 n

+ a₁+a₄+a₂q 1 a3 a4

n+1

+:::+ a1+a4+a2q 1 a₃ a₄

n+p 1

d(u₀; u₁):

Assume that = ^a₁^1+a_a^4+a2q

3 a4 . Then

d(un; un+p) ⁿ[1 + +:::+ ^{p 1}]d(u0; u1) = ⁿ

p 1

1 d(u0; u1):

Chooseq < ^{1 a1}_a^{a3 2a4}

2 then <1:Lettingn! 1, we get d(u_n; u_n+p)!0;

i.e., (u_n)_n2N is a Cauchy sequence. Since(X; d) is a complete metric space, there existsu 2X such that u_n!u as n! 1:Now, we shall demonstrate thatu 2(SF)_T. Consider,

(u ;T(u )) d(u ; un) + (un;T(un)) + (T(un);T(u ))

d(u ; un) + (un;T(un)) +a1d(un; u ) +a2 (un;T(un)) +a₃ (u ;T(u )) +a₄ (u_n;T(u )) +a₅ (u ;T(u_n)) d(u ; un) + (un;T(un)) +a1d(un; u ) +a2 (un;T(un)) +a₃ (u ;T(u )) +a₄(d(u_n; u ) + (u ;T(u )))

+a5(d(u ; un) + (un;T(un)))

= (1 +a1+a4+a5)d(u ; un) + (1 +a2+a5) (un;T(un)) +(a₃+a₄) (u ;T(u )):

This implies

(1 a3 a4) (u ;Tu ) (1 +a1+a4+a5)d(u ; un) + (1 +a2+a5) (un;T(un));

or

(u ;Tu ) 1 +a₁+a₄+a₅ 1 a3 a4

d(u ; u_n) + 1 +a₂+a₅ 1 a3 a4

(u_n;T(u_n)):

Since (un;T(un)) ⁿd(u0; u1);we see that (u ;Tu ) = 0. It implies thatT(u ) =fu g;i.e.,u 2(SF)_T: For uniqueness, assume that there exists two distinct pointsu ; v 2(SF)_T: So

d(u ; v ) = (T(u );T(v ))

a₁d(u ; v ) +a₂ (u ;T(u )) +a₃ (v ;T(v )) +a₄ (u ;T(v )) +a₅ (v ;T(u ))

a₁d(u ; v ) +a₂ (u ;T(u )) +a₃ (v ;T(v )) +a₄(d(u ; v ) + (v ; T(v ))) +a5(d(v ; u ) + (u ;T(u )))

(a1+a4+a5)d(u ; v ) +a2 (u ;Tu ) +a3 (v ;Tv ) +a4 (v ;Tv ) + a₅ (u ;Tu ):

This implies

(1 a1 a4 a5)d(u ; v ) a2 (u ;Tu ) +a3 (v ;Tv ) +a4 (v ;Tv ) +a5 (u ;Tu );

or

(1 a1 a4 a5)d(u ; v ) 0;

or1 a1 a4 a5 0ora1+a4+a5 1, a contradiction to the fact thata1+a2+a3+a4+a5<1:Hence u =v .

Example 1 Let X = [0;3]be a complete metric space and the mappingT :X !Pb(X)be de…ned as

T(x) =

( [0;1]; 0 x <2;

f2g; 2 x 3:

Taking a₁= ₁₀³; a₂= 0; a₃ = ₂₀¹; a₄ = ¹₄; a₅= ₂₀⁷; a₁+a₂+a₃+a₄+a₅ = ¹⁹₂₀ <1: Now we have following cases:

Case I: Whenx; y2[0;2);

(T(x);T(y)) = 1 a1:2 +a2:2 +a3:2 +a4:2 +a5 2:19 20: Case II: Whenx2[0;2) andy2[2;3];

(T(x);T(y)) = 2 a1:3 +a2:2 +a3:1 +a4:2 +a5:3 45 20: Case III: Whenx2[2;3]andy2[0;2);

(T(x);T(y)) = 2 a1:2 +a2:1 +a3:2 +a4:3 +a5:2 43 20: Case IV: Whenx; y2[2;3],

(T(x);T(y)) = 0 a₁+a₂+a₃+a₄+a₅ 19 20:

Subsequently, all the hypotheses of Theorem 1 are veri…ed and x= 2is the only strict …xed point of a discontinuous set-valued operatorT.

Next, we try to establish su¢ cient conditions for the well-posedness of a strict …xed point problem for the set-valued operator.

Theorem 2 IfT is a -Hardy Roger type contraction of a complete metric space(X; d), then the strict …xed point is well-posed forT with respect to _d:

Proof. Using Theorem1,(SF)_T =fu g:Supposeun2X; n2N satisfying

d(un;T(un))!0 as n! 1: We next prove thatu_n!u as n! 1. Now,

d(un; u ) d(un;T(un)) + d(T(un);T(u ))

d(u_n;T(u_n)) +a₁d(u_n; u ) +a_{2 d}(u_n;T(u_n)) +a_{3 d}(u ;T(u )) +a₄

d(un;T(u )) +a5 d(u ;T(un))

d(u_n;T(u_n)) +a₁d(u_n; u ) +a_{2 d}(u_n;T(u_n)) +a_{3 d}(u ;T(u )) +a₄ (d(un; u ) + d(u ;T(u )) +a5(d(u ; un) + d(un;T(un)):

This implies

(1 a1 a4 a5)d(un; u ) (1 +a2+a5) d(un;T(un));

or

d(u_n; u ) 1 +a₂+a₅

1 a1 a4 a5 d(u_n;T(u_n))!0 as n! 1; i.e.,un!u asn! 1:

Now, we establish a data dependence result.

Theorem 3 IfT1is a -Hardy Roger type contraction of a complete metric space(X; d)andT2:X !P_b(X) is a set-valued operator such that

(i) (SF)_T26=;;

(ii) 9 >0 satisfying (T¹(x);T²(x)) ,x2X;

then

(u₁;(SF)_T2) 1 +a3+a4

1 a1 a4 a5

:

Proof. Letu₂2(SF)_(T2). So, (u₂;T2(u₂)) = 0:Now, d(u₁; u₂) = (T¹(u₁);T²(u₂))

(T1(u₁);T1(u₂)) + (T1(u₂);T2(u₂))

a1d(u₁; u₂) +a2 (u₁;T¹(u₁)) +a3 (u₂;T¹(u₂)) +a4 (u₁;T¹(u₂)) + a₅ (u₂;T1(u₁)) + (T1(u₂);T2(u₂))

a1d(u₁; u₂) +a3 (u₂;T¹(u₂)) +a4(d(u₁; u₂) + (u₂;T¹(u₂))) + a5(d(u₂; u₁) + (u₁;T¹(u₁))) + (T¹(u₂);T²(u₂)):

This implies

(1 a₁ a₄ a₅)d(u₁; u₂) a₃ (u₂;T1(u₂)) +a₄ (u₂;T1(u₂)) + (T1(u₂);T2(u₂)) a3 +a4 +

(1 +a3+a4) ; or

d(u₁; u₂) 1 +a3+a4

1 a₁ a₄ a₅ : By taking supremum, it follows that

(u₁;(SF)_T2) 1 +a₃+a₄ 1 a1 a4 a5

:

3 Applications

3.1 Application to Volterra Integral Inclusion.

Now, we utilise Theorem1 to solve the following Volterra integral inclusion

x(t)2q(t) + Z(t)

0

k(t; s)F(s; x(s))ds; (1)

for t 2 J, : J ! J; k : J J ! R, q : J ! E are continuous, F : J E ! C(E); E is a real Banach space with normk:k^E, C(E) is the class of all non-empty closed subsets ofE and J = [0;1]in R is a closed and bounded interval. Let C(J; E) is the space of all continuous E-valued functions onJ and kxk=supt2Jjx(t)j^E:We use the following de…nitions.

De…nition 2 A set-valued function :J E!2^E is Caratheodory if (i) t! (t; x)is measurable, x2E and

(ii) x! (t; x)is upper semi-continuous a. e. for t2J.

De…nition 3 A Caratheodory multifunctionF(t; X)isL¹-Caratheodory if for every real numberr >0 9a function hr2L¹(J; R) satisfyingkF(t; x)k hrt for almost every t2J and x2E andkxk^E r. Denote kF(t; x(t))k= supfkuk^E :u2F(t; x(t))g; TF¹=fv2B(J; E) :v(t)2F(t; x(t))a.e. t2Jg;where B(J; E) is the space of all E-valued Bochner-integrable functions onJ.

Lemma 1 ([13]) If diam(E)<1andF :J E!2^E is L¹- Caratheodory, thenTF¹6= ,x2E:

Lemma 2 ([13]) Let E be a Banach space, L : L¹(J; E)! C(J; E) a continuous linear mapping and F a Caratheodory set-valued mapping such that TF¹ 6= . Then LoTF¹ : C(J; E) ! 2^C(J;E) is a closed graph operator onC(J; E) C(J; E).

Theorem 4 Suppose that the following set of hypotheses hold.

(i) The functionk(t; s)is non-negative onJ J andM = sup_t;s2J[k(t; s)];

(ii) the set-valued function F(t; x)is Caratheodory;

(iii) the set-valued functionF(t; x)is nondecreasing inxa.e. fort2J;

(iv) jF(s; x(s)) F(s; y(s))j _M¹( (x; y)); s2J; x2E;where

(x; y) =a₁d(x; y) +a₂ (x;T(x)) +a₃ (y;T(y)) +a₄ (x;T(y)) +a₅ (y;T(x));

a⁰_is2R⁺,P5

i=1ai <1;

(v) TF¹6= ,x2C(J; E):

Then the integral inclusion (1) has a solution inJ.

Proof. A continuous functionx:J!Eis a solution of the integral inclusion (1), if

x(t) =q(t) + Z(t)

0

k(t; s)v(s)ds;

wherev2B(J; E)such thatv(t)2F(t; x(t)):De…ne the set-valued mappingT : [0;1]!2^X as T(x) =q(t) +

Z (t) 0

k(t; s)v(s)ds;

wherev 2 TF¹(x)for everyt2[0;1]. Clearly T is well-de…ned, since, from (v),TF¹6= . For all t2[0;1]by (ii) and (iv), we get

jT(x) T(y)j= Z (t)

0

k(t; s)v1(s) k(t; s)v2(s) ds

E

; v1; v22T_F¹(x):

Taking supremum on both sides

(T(x);T(y)) MjF(s; x(s)) F(s; y(s))j (x; y);

i.e., the operator T verify the hypotheses of the Theorem1 on [0;1] and consequently, the given integral inclusion has a unique solution.

3.2 Application to Fractals.

Fixed point theory performs a signi…cant role in fractals that are the self-similar sets. Iterated function systems de…ne fractals as attractors in discrete dynamical frameworks and can be applied to wavelet analy- sis, quantum physics, computer graphics and di¤erent applied sciences. This concept was …rst introduced by Hutchinson [11] and popularized by Barnsley [2] as a natural generalization of the celebrated Banach contraction principle. Now, we present novel iterated function framework utilizing the -Hardy Rogers type operators which covers a large range of operators. The operator

T :Pcp(X)!Pcp(X);T(Y) =[^mi=1Tⁱ(Y); Y 2P(X)

is the multifractal operator generated by T = (T¹; :::;T^m); such that Tⁱ : X ! Pcp(X): A …xed point V 2Pcp(X)ofT is an attractor of the iterated multifunction systemT. Next, we establish existence of an attractor.

Theorem 5 Let Tⁱ : X ! Pcp(X); i 2 f1; :::; mg be a …nite family of set-valued operator of a complete metric space(X; d)such that

(Tⁱ(x);Tⁱ(y)) A1d(x; y) +A2 (x;Tⁱ(x)) +A3 (y;Tⁱ(y)) +A4 (x;Tⁱ(y)) +A₅ (y;Ti(x));

A⁰_js2R⁺,P5

j=1A_j<1,x; y2X. Then the operatorT :P_cp(X)!P_cp(X)de…ned byT(B) =[^mi=1Ti(B) for allB2P_cp(X)satis…es:

(T(B);T(C)) A₁d(B; C) +A₂ (B;T(B)) +A₃ (C;T(C)) +A₄ (B;T(C)) +A5 (C;T(B));

where B; C 2 Pcp(X) and has attractor A in (Pcp(X); (d)) such that A = T(A) = [^mi=1Tⁱ(A); A = limn!1T^on(B)andB2Pcp(X).

Proof. Let F 2 Pcp(X), then F is a non-empty and compact in X. Clearly T(F) is non-empty. Now we establish that T(F) is compact in X. If fyng T(F), then there is a sequence fung F satisfying yn = Tun(n = 1;2; :::). Compactness of F implies that there is a subsequence fxnkg fung such that fxnkg ! x 2 F. Since T is continuous, fynkg = Txnk ! Tx 2 T(F). Hence, T(F) is compact in X. De…nition of -Hardy Roger type contraction shows thatT satis…es

(T(B);T(C)) A11d(B; C) +A21 (B;T(B)) +A31 (C;T(C)) +A41 (B;T(C)) +A₅₁ (C;T(B)); 8 B; C 2P_cp(X):

Now, we shall use the principle of mathematical induction. The statement is clearly true form= 1. Now, form= 2;

(T(B);T(C)) = (T¹(B)[ T²(B);T¹(C)[ T²(C))

maxfA₁₂g (B; C) + maxfA₂₂g (B;T1(B)) + maxfA32g (C;T¹(C)) + maxfA42g (B;T¹(C)) + maxfA52g (C;T¹(B)):

Hence by induction inequality is true for alli21; :::; m;i.e.,

(T(B);T(C)) A₁d(B; C) +A₂ (B;Tn(B)) +A₃ (C;Tn(C)) +A₄ (B;Tn(C)) +A5 (C;Tⁿ(B));

where A₁= maxfA_1jg; A₂= maxfA_2jg; :::; A₅= maxfA_5jg, i.e., the operatorT satisfy all the hypotheses of Theorem1and consequentlyT has a attractorA=T(A) =[^mi=1Ti(A)andA= limT^on(B); B2Pcp(X).

4 Remarks

Remark 1 Choosing suitably the values of constants in Theorems 1,2 and3, similar results can be estab- lished for Kannan [12], Chatterjee [9], Reich [15] and Banach [1] type set-valued contractions.

Remark 2 In Theorem 5, we established the attractors of -Hardy Roger type set-valued iterated function systems, which generalizes the celebrated Hutchinson iterated function systems.

Acknowledgment. The authors are thankful to the learned referees for the very careful reading of the manuscript and valuable suggestions.

References

[1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux l’équations intégrales, Fund. Math., 3(1922), 133–181.

[2] M. F. Barnsley, “Lecture Notes on Iterated Function Systems," Chaos and Fractals (Providence, RI, 1988), 39 Proceedings of Symposia in Applied Mathematics, 127?44, American Mathematical Society, Providence, RI, USA, 1989.

[3] S. Beloul and A. Tomar, A coincidence and common …xed point theorem for subsequentially continuous hybrid pairs of maps satisfying an implicit relation, Math. Morav., 21(2017), 15–25.

[4] S. Chandok, Some common …xed point results for rational type contraction mappings in partially ordered metric spaces, Math. Bohemica, 138(2013), 403–413.

[5] S. Chandok, B. S. Choudhury and N. Metiya, Some …xed point results in ordered metric spaces for rational type expressions with auxiliary functions, J. Egypt. Math Soc., 23(2015), 95–101.

[6] S. Chandok and D. Kumar, Some common …xed point results for rational type contraction mappings in complex valued metric spaces, J. Operator, (2013), Article ID 813707, 6 pages.

[7] S. Chandok, T. D. Narang and M. A. Taoudi, Some common …xed point results in partially ordered metric spaces for generalized rational type contraction mappings, Veitnam J. Math., 41(2013), 323–331.

[8] S. Chandok, M. S. Khan and M. Abbas, Common …xed point theorems for nonlinear weakly contractive mappings, Ukrainian Math. J., 66(2014), 531–537.

[9] S. K. Chatterjee, Fixed point theorems, Cr. Acad. Bulg. Sc., 25(1972), 727–730.

[10] C. Chifu and G. Petrusel, Well-posedness and fractals via …xed point theory, Fixed Point Theory and A., (2008), Article ID 645419, 9 pages.

[11] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30(1981), 713–747.

[12] R. Kannan, Some results on …xed points, Bull. Cal. Math. Soc., 60(1968), 71–76.

[13] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary di¤erential equations, Bull. Acad. Pol. Sci. Ser Sci. Math. Astronom. Phy., 13 (1965), 781–786.

[14] S. B. J. Nadler, Multivalued contraction mappings, Paci…c J. Math., 20(1969), 457–488.

[15] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14(1971), 121–124.

[16] A. Tomar, R. Sharma and A. H. Ansari, Strict coincidence and common strict …xed point of a faintly compatible hybrid pair of maps via C-class function and applications, Palestine J. Math., 9(2020), 274–288.

[17] A. Tomar, R. Sharma and S. Upadhyay, Some Applications of Existence of Common Fixed and Common Stationary Point of a Hybrid Pair, Bull. Internat. Math. Virtual Inst., 9(2019), 73–84.

[18] A. Tomar, S. Beloul, S. Upadhyay and R. Sharma, Strict coincidence and strict common …xed point via strongly tangential property with an application, Electron. J. Math. Anal. Appl., 7(2019), 82–94.

[19] A. Tomar, S. Upadhyay and R. Sharma, On existence of strict coincidence and common strict …xed point of a faintly compatible hybrid pair of maps, Electron. J. Math. Anal. Appl., 5(2017), 298-305.

[20] A. Tomar, S. Upadhyay and R. Sharma, Strict coincidence and common strict …xed point of hybrid pairs of self-map with an application, Math. Sci. Appl. E-notes 5(2017), 51–59.

[21] A. Tomar, S. Upadhyay and R. Sharma, Common …xed point theorems with an application, Recent Advances in Fixed Point Theory and Applications, Nova Science Publishers, USA, (2017), 157–169.