Co-Variational Inequality Problem involving Two Generalized Yosida Approximation Operators

In this paper, we study a co-variational inequality problem involving Yosida approximation operators in real uniformly smooth Banach space and define an iterative method to obtain its solution. We prove some properties of Yosida approximation operator for our purpose, that is strongly accretivity and Lipschitz continuity. An existence as well as convergence result is proved for co-variational inequality problem involving Yosida approximation operators using the concept of nonexpansive sunny retraction.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

MSC: 65J15; 47J25; 65K15

1. Introduction

Variational inequality theory is an influential unifying methodology for solving many obstacles of pure as well as applied sciences. Hartman and Stampacchia [1] in 1966 initiated the study of variational inequalities while dealing with some problems of mechanics.

The concept of variational inequalities furnish us various devices for modelling many problems existing in variational analysis related to applicable sciences. One can ensure the existence of solution and convergence of iterative sequences using these devices. For applications, see [2,3,4,5,6,7,8,9,10] and references therein.

Alber and Yao [11] first considered and studied co-variational inequalities using nonexpansive sunny retraction concept. They obtained solution of co-variational inequality problem and discussed the convergence criteria. Their work is extended by Ahmad and Irfan [12].

Yosida approximation operators are useful for obtaining solution of various types of differential equations. Petterson [13] first solved the stochastic differential equation by using Yosida approximation operator approach. For the study of heat equations, problem of couple sound and heat flow in compressible fluids, wave equations, etc., the concept of Yosida approximation operator is applicable. For more details, we refer to [14,15,16,17,18].

After above important discussion, the aim of this work is to introduce a different version of co-variational inequality which involve two generalized Yosida approximation operators. We obtain the solution of our problem and we also discuss convergence criteria for the sequences achieved by iterative method.

2. Preface

Throughout this document, we denote real Banach space by

\tilde{E}

and its dual space by

{\tilde{E}}^{*}

. Let

〈 \dot{a}, \dot{b} 〉

be the duality pairing between

\dot{a} \in \tilde{E}

and

\dot{b} \in {\tilde{E}}^{*}

. The usual norm on

\tilde{E}

is denoted by

∥ \cdot ∥

, the class of nonempty subsets of

\tilde{E}

2^{\tilde{E}}

and the class of nonempty compact subsets of

\tilde{E}

\hat{C} (\tilde{E})

The Normalized duality operator

J : \tilde{E} \to {\tilde{E}}^{*}

is defined by

J (\dot{a}) = \{\dot{b} \in {\tilde{E}}^{*} : 〈 \dot{a}, \dot{b} 〉 = ∥ \dot{a} ∥^{2} = {∥ \dot{b} ∥}^{2}\}, \forall \dot{a} \in \tilde{E} .

Some characteristics of normalized duality operator can be discovered in [19].

For the space

\tilde{E}

, modulus of smoothness is given by the function

ρ_{\tilde{E}} (t) = sup \{\frac{∥ \dot{c} + \dot{d} ∥ + ∥ \dot{c} - \dot{d} ∥}{2} - 1 : ∥ \dot{c} ∥ = 1, ∥ \dot{d} ∥ = t\} .

In fact

\tilde{E}

is uniformly smooth if and only if

lim_{t \to 0} t^{- 1} ρ_{\tilde{E}} (t) = 0 .

The following result is instrumental for our main result.

Proposition 1.

[11] Let

\tilde{E}

be a uniformly smooth Banach space and J be the normalized duality operator. Then, for any

\dot{a}, \dot{b} \in \tilde{E}

, we have

(i): $∥ \dot{a} + \dot{b} ∥^{2} \leq {∥ \dot{a} ∥}^{2} + 2 〈 \dot{b}, J (\dot{a} + \dot{b}) 〉$ ,
(ii): $〈 \dot{a} - \dot{b}, J (\dot{a}) - J (\dot{b}) 〉 \leq 2 d^{2} ρ_{\tilde{E}} (4 ∥ \dot{a} - \dot{b} ∥ / d)$ , where $d = \sqrt{∥ \dot{a} ∥^{2} + {∥ \dot{b} ∥}^{2} / 2}$

Definition 1.

The operator

\hat{h_{1}} : \tilde{E} \to \tilde{E}

is called

(i): accretive, if

$〈 \hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}), J (\dot{a} - \dot{b}) 〉 \geq 0, \forall \dot{a}, \dot{b} \in \tilde{E} .$
(ii): Strongly accretive, if

$〈 \hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}), J (\dot{a} - \dot{b}) 〉 \geq r_{1} {∥ \dot{a} - \dot{b} ∥}^{2}, \forall \dot{a}, \dot{b} \in \tilde{E},$

where $r_{1} > 0$ is a constant.
(iii): Lipschitz continuous, if

$∥ \hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}) ∥ \leq λ_{\hat{h_{1}}} ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E},$

where $λ_{\hat{h_{1}}} > 0$ is a constant.
(iv): expansive, if

$∥ \hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}) ∥ \geq β_{\hat{h_{1}}} ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E}$

where $β_{\hat{h_{1}}} > 0$ is a constant.

Definition 2.

Let

\tilde{A} : \tilde{E} \to \tilde{E}

be an operator. The operator

S : \tilde{E} \times \tilde{E} \times \tilde{E} \to \tilde{E}

is said to be

(i): Lipschitz continuous in the first slot, if

$∥ S ({\hat{u}}_{1}, \cdot, \cdot) - S ({\hat{u}}_{2}, \cdot, \cdot) ∥ \leq δ_{S_{1}} ∥ {\hat{u}}_{1} - {\hat{u}}_{2} ∥, \forall \dot{a}, \dot{b} \in \tilde{E} and for some {\hat{u}}_{1} \in \tilde{A} (\dot{a}), {\hat{u}}_{2} \in \tilde{A} (\dot{b}),$

where $δ_{S_{1}} > 0$ is a constant.

Similarly, we can obtain Lipschitz continuity of S in other slots.
(ii): Strongly accretive in the first slot with respect to $\tilde{A}$ , if

$〈 S ({\hat{u}}_{1}, \cdot, \cdot) - S ({\hat{u}}_{2}, \cdot, \cdot), J (\dot{a} - \dot{b}) 〉 \geq λ_{S_{1}} {∥ \dot{a} - \dot{b} ∥}^{2}, \forall \dot{a}, \dot{b} \in \tilde{E} and for some {\hat{u}}_{1} \in \tilde{A} (\dot{a}), {\hat{u}}_{2} \in \tilde{A} (\dot{b}),$

where $λ_{S_{1}} > 0$ is a constant.

Similarly strong accretivity of S in other slots and with respect to other operators can be obtained.

Definition 3.

The operator

\tilde{A} : \tilde{E} \to \hat{C} (\tilde{E})

is called D-Lipschitz continuous if

D (\tilde{A} (\dot{a}), \tilde{A} (\dot{b})) \leq α_{\tilde{A}} ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E},

where

α_{\tilde{A}} > 0

is a constant and

D (\cdot, \cdot)

denotes the Housdörff metric.

Definition 4.

[11] Suppose

\tilde{Ω}

is the nonempty closed convex subset of

\tilde{E}

. Then an operator

Q_{\tilde{Ω}} : \tilde{E} \to \tilde{Ω}

is called:

(i): retraction on $\tilde{Ω}$ , if $Q_{\tilde{Ω}}^{2} = Q_{\tilde{Ω}}$ ,
(ii): nonexpansive retraction on $\tilde{Ω}$ , if it satisfies the inequality:

$∥ Q_{\tilde{Ω}} (\dot{a}) - Q_{\tilde{Ω}} (\dot{b}) ∥ \leq ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E},$
(iii): nonexpansive sunny retraction on $\tilde{Ω}$ , if

$Q_{\tilde{Ω}} (Q_{\tilde{Ω}} (\dot{a}) + \hat{t} (\dot{a} - Q_{\tilde{Ω}} (\dot{a}))) = Q_{\tilde{Ω}} (\dot{a}),$

for all $\dot{a} \in \tilde{E}$ and for $0 \leq \hat{t} < + \infty$ .

Some characteristics of nonexpansive sunny retraction operator are mentioned below, which can be found in [20,21,22].

Proposition 2.

The operator

Q_{\tilde{Ω}}

is a nonexpansive sunny retraction, if and only if

〈 \dot{a} - Q_{\tilde{Ω}} (\dot{a}), J (Q_{\tilde{Ω}} (\dot{a}) - \dot{b}) 〉 \geq 0,

for all

\dot{a} \in \tilde{E}

and

\dot{b} \in \tilde{Ω}

Proposition 3.

Suppose

\tilde{m} = \tilde{m} (\dot{a}) : \tilde{E} \to \tilde{E}

and

Q_{\tilde{Ω}} : \tilde{E} \to \tilde{Ω}

is a nonexpansive sunny retraction. Then for all

\dot{a} \in \tilde{E}

, we have

Q_{\tilde{Ω} + \tilde{m} (\dot{a})} (\dot{a}) = \tilde{m} (\dot{a}) + Q_{\tilde{Ω}} (\dot{a} - \tilde{m} (\dot{a})) .

Definition 5.

The multi-valued operator

\hat{M} : \tilde{E} \to 2^{\tilde{E}}

is called accretive, if

〈 \hat{u} - \hat{v}, J (\dot{a} - \dot{b}) 〉 \geq 0, \forall \dot{a}, \dot{b} \in \tilde{E} and for some \hat{u} \in \hat{M} (\dot{a}), \hat{v} \in \hat{M} (\dot{b}) .

Definition 6.

Let

\hat{h_{1}} : \tilde{E} \to \tilde{E}

be an operator. The multi-valued operator

\hat{M} : \tilde{E} \to 2^{\tilde{E}}

is said to be

\hat{h_{1}}

-accretive if

\hat{M}

is accretive and

[\hat{h_{1}} + λ \hat{M}] (\tilde{E}) = \tilde{E}, w h e r e λ > 0 i s a c o n s t a n t .

Definition 7.

We define

R_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

such that

R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) = {[\hat{h_{1}} + λ \hat{M}]}^{- 1} (\dot{a}), \forall \dot{a} \in \tilde{E}, w h e r e λ > 0 i s a c o n s t a n t .

We call it generalized resolvent operator.

Definition 8.

We define

Y_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

such that

Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) = \frac{1}{λ} [\hat{h_{1}} - R_{\hat{h_{1}}, λ}^{\hat{M}}] (\dot{a}), \forall \dot{a} \in \tilde{E}, w h e r e λ > 0 i s a c o n s t a n t .

We call it generalized Yosida approximation operator.

Proposition 4.

[23] Let

\hat{h_{1}} : \tilde{E} \to \tilde{E}

r_{1}

-strongly accretive and

\hat{M} : \tilde{E} \to 2^{\tilde{E}}

\hat{h_{1}}

-accretive multi-valued operator. Then, the operator

R_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

satisfies the following condition.

∥R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})∥ \leq \frac{1}{r_{1}} ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E} .

That is,

R_{\hat{h_{1}}, λ}^{\hat{M}}

\frac{1}{r_{1}}

-Lipschitz continuous.

Proposition 5.

\hat{h_{1}} : \tilde{E} \to \tilde{E}

r_{1}

-strongly accretive,

β_{\hat{h_{1}}}

-expansive,

λ_{\hat{h_{1}}}

-Lipschitz continuous operator and

R_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

\frac{1}{r_{1}}

-Lipschitz continuous, then the operator

Y_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

satisfy the following condition.

〈Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b}), J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 \geq δ_{Y_{\hat{h_{1}}}} {∥ \dot{a} - \dot{b} ∥}^{2}, \forall \dot{a}, \dot{b} \in \tilde{E},

where

δ_{Y_{\hat{h_{1}}}} = \frac{β_{\hat{h_{1}}}^{2} r_{1} - λ_{\hat{h_{1}}}}{λ r_{1}},

β_{\hat{h_{1}}}^{2} r_{1} > λ_{\hat{h_{1}}}, λ r_{1} \neq 0

and all the constants involved are positive. That is,

Y_{\hat{h_{1}}, λ}^{\hat{M}}

δ_{Y_{\hat{h_{1}}}}

-strongly accretive with respect to the operator

\hat{h_{1}}

Proof.

Since

Y_{\hat{h_{1}}, λ}^{\hat{M}} = \frac{1}{λ} [\hat{h_{1}} - R_{\hat{h_{1}}, λ}^{\hat{M}}]

, we evaluate

\begin{matrix} 〈Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b}), J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 \\ = & \frac{1}{λ} 〈\hat{h_{1}} (\dot{a}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - [\hat{h_{1}} (\dot{b}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})], J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 \\ = & \frac{1}{λ} 〈\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}), J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 \\ - \frac{1}{λ} 〈R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b}), J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 . \end{matrix}

\hat{h_{1}}

is expansive and Lipschitz continuous and the generalized resolvent operator

R_{\hat{h_{1}}, λ}^{\hat{M}}

is Lipschitz continuous, we obtain

\begin{matrix} 〈Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b}), J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 \\ \geq & \frac{1}{λ} {∥\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b})∥}^{2} - \frac{1}{λ} ∥R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})∥ ∥\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b})∥ \\ \geq & \frac{1}{λ} {∥\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b})∥}^{2} - \frac{1}{λ} \frac{1}{r_{1}} ∥\dot{a} - \dot{b}∥ ∥\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b})∥ \\ \geq & \frac{1}{λ} β_{\hat{h_{1}}}^{2} {∥\dot{a} - \dot{b}∥}^{2} - \frac{1}{λ} \frac{1}{r_{1}} λ_{\hat{h_{1}}} ∥\dot{a} - \dot{b}∥ ∥\dot{a} - \dot{b}∥ \\ \geq & \frac{β_{\hat{h_{1}}}^{2}}{λ} {∥\dot{a} - \dot{b}∥}^{2} - \frac{λ_{\hat{h_{1}}}}{λ r_{1}} {∥\dot{a} - \dot{b}∥}^{2} \\ \geq & \frac{β_{\hat{h_{1}}}^{2} r_{1} - λ_{\hat{h_{1}}}}{λ r_{1}} {∥\dot{a} - \dot{b}∥}^{2} \\ = & δ_{Y_{\hat{h_{1}}}} {∥\dot{a} - \dot{b}∥}^{2} . \end{matrix}

That is,

〈Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b}), J (\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b}))〉 \geq δ_{Y_{\hat{h_{1}}}} {∥\dot{a} - \dot{b}∥}^{2} .

That is,

Y_{\hat{h_{1}}, λ}^{\hat{M}}

δ_{Y_{\hat{h_{1}}}}

-strongly accretive with respect to

\hat{h_{1}}

. □

Proposition 6.

Let

\hat{h_{1}} : \tilde{E} \to \tilde{E}

λ_{\hat{h_{1}}}

-Lipschitz continuous,

r_{1}

-strongly accretive operator and

R_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

\frac{1}{r_{1}}

-Lipschitz continuous, then the operator

Y_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

satisfy the following condition.

∥Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})∥ \leq λ_{Y_{\hat{h_{1}}}} ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E},

where

λ_{Y_{\hat{h_{1}}}} = \frac{λ_{\hat{h_{1}}} r_{1} + 1}{λ r_{1}}, λ r_{1} \neq 0

. That is,

Y_{\hat{h_{1}}, λ}^{\hat{M}}

λ_{Y_{\hat{h_{1}}}}

-Lipschitz continuous.

Proof.

Since

\hat{h_{1}}

and the generalized resolvent operator

R_{\hat{h_{1}}, λ}^{\hat{M}}

are Lipschitz continuous, we obtain

\begin{matrix} ∥Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})∥ & = & ∥\frac{1}{λ} [\hat{h_{1}} (\dot{a}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a})] - \frac{1}{λ} [\hat{h_{1}} (\dot{b}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})]∥ \\ = & \frac{1}{λ} ∥\hat{h_{1}} (\dot{a}) - \hat{h_{1}} (\dot{b})∥ + \frac{1}{λ} ∥R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - R_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})∥ \\ \leq & \frac{1}{λ} λ_{\hat{h_{1}}} ∥\dot{a} - \dot{b}∥ + \frac{1}{λ} \frac{1}{r_{1}} ∥\dot{a} - \dot{b}∥ \\ = & (\frac{λ_{\hat{h_{1}}} r_{1} + 1}{λ r_{1}}) ∥\dot{a} - \dot{b}∥ \\ = & λ_{Y_{\hat{h_{1}}}} ∥\dot{a} - \dot{b}∥ . \end{matrix}

That is

∥Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{b})∥ \leq λ_{Y_{\hat{h_{1}}}} ∥ \dot{a} - \dot{b} ∥, \forall \dot{a}, \dot{b} \in \tilde{E} .

Thus, the operator

Y_{\hat{h_{1}}, λ}^{\hat{M}}

λ_{Y_{\hat{h_{1}}}}

-Lipschitz continuous. □

3. Problem Formation and Iterative Method

Suppose

S : \tilde{E} \times \tilde{E} \times \tilde{E} \to \tilde{E}

is a nonlinear operator,

\tilde{A}, \tilde{B}, \tilde{C} : \tilde{E} \to \hat{C} (\tilde{E})

be multi-valued operators and

\tilde{K} : \tilde{E} \to 2^{\tilde{E}}

be a multi-valued operator such that

\tilde{K} (\dot{a})

is a nonempty, closed and convex set for all

\dot{a} \in \tilde{E}

. Let

\hat{h_{1}}, \hat{h_{2}} : \tilde{E} \to \tilde{E}

be the single-valued operators,

\hat{M} : \tilde{E} \to 2^{\tilde{E}}

\hat{h_{1}}

-accretive multi-valued operator and

\hat{N} : \tilde{E} \to 2^{\tilde{E}}

\hat{h_{2}}

-accretive multi-valued operator,

Y_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

and

Y_{\hat{h_{2}}, λ}^{\hat{N}} : \tilde{E} \to \tilde{E}

be the generalized Yosida approximation operators.

We consider the problem of finding

\dot{a} \in \tilde{E}

\hat{u} \in \tilde{A} (\dot{a}), \hat{v} \in \tilde{B} (\dot{a}), \hat{w} \in \tilde{C} (\dot{a})

such that

〈Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} (\dot{a}), J (S (\hat{u}, \hat{v}, \hat{w}))〉 \geq 0, \forall S (\hat{u}, \hat{v}, \hat{w}) \in \tilde{K} (\dot{a}), a n d λ > 0 i s a c o n s t a n t .

(1)

We call problem (1) as co-variational inequality problem involving generalized Yosida approximation operators..

Clearly for problem (1), it is easily accessible to obtain co-variational inequalities studied by Alber and Yao [11] and Ahmad and Irfan [12].

We provide few characterizations of solution of problem (1).

Theorem 1.

Let

\tilde{A}, \tilde{B}, \tilde{C} : \tilde{E} \to \hat{C} (\tilde{E})

be the multi-valued operators,

S : \tilde{E} \times \tilde{E} \times \tilde{E} \to \tilde{E}

be the nonlinear operator and

\tilde{K} : \tilde{E} \to 2^{\tilde{E}}

be a multi-valued operator such that

\tilde{K} (\dot{a})

is a nonempty, closed and convex set for all

\dot{a} \in \tilde{E}

. Let

\hat{h_{1}}, \hat{h_{2}} : \tilde{E} \to \tilde{E}

be the single-valued operators,

\hat{M} : \tilde{E} \to 2^{\tilde{E}}

\hat{h_{1}}

-accretive multi-valued operator and

\hat{N} : \tilde{E} \to 2^{\tilde{E}}

\hat{h_{2}}

-accretive multi-valued operator,

Y_{\hat{h_{1}}, λ}^{\hat{M}} : \tilde{E} \to \tilde{E}

and

Y_{\hat{h_{2}}, λ}^{\hat{N}} : \tilde{E} \to \tilde{E}

be the generalized Yosida approximation operators, where

λ > 0

is a constant. Then the following assertions are similar:

(i): $\dot{a} \in \tilde{E}, \hat{u} \in \tilde{A} (\dot{a}), \hat{v} \in \tilde{B} (\dot{a}), \hat{w} \in \tilde{C} (\dot{a})$ constitute the solution of problem (1).
(ii): $\dot{a} \in \tilde{E}, \hat{u} \in \tilde{A} (\dot{a}), \hat{v} \in \tilde{B} (\dot{a}), \hat{w} \in \tilde{C} (\dot{a})$ such that

$S (\hat{u}, \hat{v}, \hat{w}) = Q_{\tilde{K} (\dot{a})} [S (\hat{u}, \hat{v}, \hat{w}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} (\dot{a}))] .$

Proof.

For proof, see [6,19]. □

Combining Proposition 3 and Theorem 1, we obtain the theorem mentioned below.

Theorem 2.

Suppose all the conditions of Theorem 1 are fulfilled and additionally

\tilde{K} (\dot{a}) = \tilde{m} (\dot{a}) + F

, for all

\dot{a} \in \tilde{E}

, where F is nonempty closed convex subset of

\tilde{E}

and

Q_{F} : \tilde{E} \to F

be a nonexpansive sunny retraction. Then

\dot{a} \in \tilde{E}, \hat{u} \in \tilde{A} (\dot{a}), \hat{v} \in \tilde{B} (\dot{a})

and

\hat{w} \in \tilde{C} (\dot{a})

form the solution of problem (1), if and only if

\dot{a} = \dot{a} + \tilde{m} (\dot{a}) - S (\hat{u}, \hat{v}, \hat{w}) + Q_{F} [S (\hat{u}, \hat{v}, \hat{w}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} (\dot{a})) - \tilde{m} (\dot{a})],

(2)

where

λ > 0

is a constant.

Using Theorem 2, we construct the following iterative method.

Iterative Method 3.1. For initial

{\dot{a}}_{0} \in \tilde{E}, {\hat{u}}_{0} \in \tilde{A} ({\dot{a}}_{0}), {\hat{v}}_{0} \in \tilde{B} ({\dot{a}}_{0}), {\hat{w}}_{0} \in \tilde{C} ({\dot{a}}_{0})

, let

{\dot{a}}_{1} = {\dot{a}}_{0} + \tilde{m} ({\dot{a}}_{0}) - S ({\hat{u}}_{0}, {\hat{v}}_{0}, {\hat{w}}_{0}) + Q_{F} [S ({\hat{u}}_{0}, {\hat{v}}_{0}, {\hat{w}}_{0}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{0}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{0})) - \tilde{m} ({\dot{a}}_{0})] .

Since

\tilde{A} ({\dot{a}}_{0}), \tilde{B} ({\dot{a}}_{0})

and

\tilde{C} ({\dot{a}}_{0})

are nonempty convex sets, by Nadler[24], there exists

{\hat{u}}_{1} \in \tilde{A} ({\dot{a}}_{1}), {\hat{v}}_{1} \in \tilde{B} ({\dot{a}}_{1})

and

{\hat{w}}_{1} \in \tilde{C} ({\dot{a}}_{1})

such that

\begin{matrix} ∥ {\hat{u}}_{1} - {\hat{u}}_{0} ∥ & \leq & D (\tilde{A} ({\dot{a}}_{1}), \tilde{A} ({\dot{a}}_{0})), \\ ∥ {\hat{v}}_{1} - {\hat{v}}_{0} ∥ & \leq & D (\tilde{B} ({\dot{a}}_{1}), \tilde{B} ({\dot{a}}_{0})), \\ and ∥ {\hat{w}}_{1} - {\hat{w}}_{0} ∥ & \leq & D (\tilde{C} ({\dot{a}}_{1}), \tilde{C} ({\dot{a}}_{0})), \end{matrix}

where

D (\cdot, \cdot)

denotes the Hausdorff metric.

Proceeding in a similar manner, we can find the sequences

{{\dot{a}}_{n}}, {{\hat{u}}_{n}}, {{\hat{v}}_{n}}

and

{{\hat{w}}_{n}}

by the following method:

{\dot{a}}_{n + 1} = {\dot{a}}_{n} + \tilde{m} ({\dot{a}}_{n}) - S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) + Q_{F} [S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n})) - \tilde{m} ({\dot{a}}_{n})],

(3)

\begin{matrix} {\hat{u}}_{n} \in \tilde{A} ({\dot{a}}_{n}), ∥ {\hat{u}}_{n + 1} - {\hat{u}}_{n} ∥ \leq D (\tilde{A} ({\dot{a}}_{n + 1}), \tilde{A} ({\dot{a}}_{n})), \end{matrix}

(4)

\begin{matrix} {\hat{v}}_{n} \in \tilde{B} ({\dot{a}}_{n}), ∥ {\hat{v}}_{n + 1} - {\hat{v}}_{n} ∥ \leq D (\tilde{B} ({\dot{a}}_{n + 1}), \tilde{B} ({\dot{a}}_{n})), \end{matrix}

(5)

\begin{matrix} {\hat{w}}_{n} \in \tilde{C} ({\dot{a}}_{n}), ∥ {\hat{w}}_{n + 1} - {\hat{w}}_{n} ∥ \leq D (\tilde{C} ({\dot{a}}_{n + 1}), \tilde{C} ({\dot{a}}_{n})), \end{matrix}

(6)

for

n = 0, 1, 2, 3, \dots \dots

, where

λ > 0

is a constant.

4. Convergence Result

Theorem 3.

Suppose

\tilde{E}

be real uniformly smooth Banach space and

ρ_{\tilde{E}} (t) \leq C t^{2}

, for some

C > 0

, is the modulus of smoothness. Suppose F be a closed convex subset of

\tilde{E}, S (\cdot, \cdot, \cdot) : \tilde{E} \times \tilde{E} \times \tilde{E} \to \tilde{E}

be an operator,

\tilde{A}, \tilde{B}, \tilde{C} : \tilde{E} \to \hat{C} (\tilde{E})

be the multi-valued operators,

\tilde{m} : \tilde{E} \to \tilde{E}

be an operator. Let

Q_{F} : \tilde{E} \to F

be a nonexpansive sunny retraction and

\tilde{K} : \tilde{E} \to 2^{\tilde{E}}

be a multi-valued operator such that

\tilde{K} (\dot{a}) = \tilde{m} (\dot{a}) + F

, for all

\dot{a} \in \tilde{E}

. Let

\hat{M}, \hat{N} : \tilde{E} \to 2^{\tilde{E}}

be the multi-valued operators,

\hat{h_{1}}, \hat{h_{2}} : \tilde{E} \to \tilde{E}

be operators. Let

Y_{\hat{h_{1}}, λ}^{\hat{M}}

be the generalized Yosida approximation operator associated with the generalized resolvent operator

R_{\hat{h_{1}}, λ}^{\hat{M}}

and

Y_{\hat{h_{2}}, λ}^{\hat{N}}

be the generalized Yosida approximation operators associated with the generalized resolvent operator

R_{\hat{h_{2}}, λ}^{\hat{N}}

. Suppose that the following assertions are satisfied:

(i): $S (\cdot, \cdot, \cdot)$ is $λ_{S_{1}}$ -strongly accretive with respect to $\tilde{A}$ in the first slot, $λ_{S_{2}}$ -strongly accretive with respect to $\tilde{B}$ in the second slot, $λ_{S_{3}}$ -strongly accretive with respect to $\tilde{C}$ in the third slot and $δ_{S_{1}}$ -Lipschitz continuous in first slot, $δ_{S_{2}}$ -Lipschitz continuous in second slot, $δ_{S_{3}}$ -Lipschitz continuous in third slot.
(ii): $\tilde{A}$ is $α_{\tilde{A}}$ -D-Lipschitz continuous, $\tilde{B}$ is $α_{\tilde{B}}$ -D-Lipschitz continuous and $\tilde{C}$ is $α_{\tilde{C}}$ -D-Lipschitz continuous.
(iii): $\tilde{m}$ is $λ_{m}$ -Lipschitz continuous.
(iv): $\hat{h_{1}}$ is $r_{1}$ -strongly accretive, $β_{\hat{h_{1}}}$ -expansive and $λ_{\hat{h_{1}}}$ -Lipschitz continuous and $\hat{h_{2}}$ is $r_{2}$ -strongly accretive, $β_{\hat{h_{2}}}$ -expansive and $λ_{\hat{h_{2}}}$ -Lipschitz continuous.
(v): $R_{\hat{h_{1}}, λ}^{\hat{M}}$ is $\frac{1}{r_{1}}$ -Lipschitz continuous and $R_{\hat{h_{2}}, λ}^{\hat{N}}$ is $\frac{1}{r_{2}}$ -Lipschitz continuous.
(vi): $Y_{\hat{h_{1}}, λ}^{\hat{M}}$ is $δ_{Y_{\hat{h_{1}}}}$ -strongly accretive, $λ_{Y_{\hat{h_{1}}}}$ -Lipschitz continuous and $Y_{\hat{h_{2}}, λ}^{\hat{N}}$ is $δ_{Y_{\hat{h_{2}}}}$ -strongly accretive, $λ_{Y_{\hat{h_{2}}}}$ -Lipschitz continuous.
(vii): Suppose that

$\begin{matrix} 0 < [\sqrt{[(1 - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}})) + 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2})]} \\ + 2 λ_{m} + (δ_{S_{1}} α_{\tilde{A}} + δ_{S_{2}} α_{\tilde{B}} + δ_{S_{3}} α_{\tilde{C}}) + \sqrt{1 - 2 λ δ_{Y_{\hat{h_{1}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{1}}}}^{2}} \\ + \sqrt{1 - 2 λ δ_{Y_{\hat{h_{2}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{2}}}}^{2}}] < 1, \end{matrix}$

where

$\begin{matrix} δ_{Y_{\hat{h_{1}}}} = \frac{β_{\hat{h_{1}}}^{2} r_{1} - λ_{\hat{h_{1}}}}{λ r_{1}}, & δ_{Y_{\hat{h_{2}}}} = \frac{β_{\hat{h_{2}}}^{2} r_{2} - λ_{\hat{h_{2}}}}{λ r_{2}}, \\ λ_{Y_{\hat{h_{1}}}} = \frac{λ_{\hat{h_{1}}} r_{1} + 1}{λ r_{1}}, & λ_{Y_{\hat{h_{2}}}} = \frac{λ_{\hat{h_{2}}} r_{2} + 1}{λ r_{2}}, \\ β_{\hat{h_{1}}}^{2} r_{1} > λ_{\hat{h_{1}}} & a n d & β_{\hat{h_{2}}}^{2} r_{2} > λ_{\hat{h_{2}}} . \end{matrix}$

Then, there exist

\dot{a} \in \tilde{E}, \hat{u} \in \tilde{A} (\dot{a}), \hat{v} \in \tilde{B} (\dot{a})

and

\hat{w} \in \tilde{C} (\dot{a})

, the solution of problem (1). Also sequences

{{\dot{a}}_{n}}, {{\hat{u}}_{n}}, {{\hat{v}}_{n}}

and

{{\hat{w}}_{n}}

converge strongly to

\dot{a}, \hat{u}, \hat{v}

and

\hat{w}

, respectively.

Proof.

Using (3) of iterative method Section 3 and nonexpansive retraction property of

Q_{F}

, we estimate

\begin{matrix} ∥ {\dot{a}}_{n + 1} - {\dot{a}}_{n} ∥ & = & ∥ [{\dot{a}}_{n} + \tilde{m} ({\dot{a}}_{n}) - S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) + Q_{F} [S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) \\ - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n})) - \tilde{m} ({\dot{a}}_{n})]] \\ - [{\dot{a}}_{n - 1} + \tilde{m} ({\dot{a}}_{n - 1}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) + Q_{F} [S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) \\ - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n - 1})) - \tilde{m} ({\dot{a}}_{n - 1})]] ∥ \\ \leq & ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) ∥ + ∥ \tilde{m} ({\dot{a}}_{n}) - \tilde{m} ({\dot{a}}_{n - 1}) ∥ \\ + ∥ Q_{F} [S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n})) - \tilde{m} ({\dot{a}}_{n})] \\ - Q_{F} [S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n - 1})) - \tilde{m} ({\dot{a}}_{n - 1})] ∥ \\ \leq & ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) ∥ \\ + 2 ∥ \tilde{m} ({\dot{a}}_{n}) - \tilde{m} ({\dot{a}}_{n - 1}) ∥ + ∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥ \\ + ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1})) ∥ \\ + ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} - λ (Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n - 1})) ∥ . \end{matrix}

(7)

Applying Proposition 1, we evaluate

\begin{matrix} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) ∥^{2} \\ \leq ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} \\ - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}))) 〉 \\ = ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), \\ J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}))) - J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ = ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) \\ + S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) + S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} \\ - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) - J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ = ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ - 2 〈 S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ - 2 〈 S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), \\ J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) - J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 . \end{matrix}

(8)

Since

S (\cdot, \cdot, \cdot)

λ_{S_{1}}

-strongly accretive with respect to

\tilde{A}

in the first slot,

λ_{S_{2}}

-strongly accretive with respect to

\tilde{B}

in the second slot,

λ_{S_{3}}

-strongly accretive with respect to

\tilde{C}

in the third slot and applying (ii) of Proposition 1, (8) becomes

\begin{matrix} ∥ ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) ∥^{2} \\ \leq ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}}) {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2} \\ - 2 〈 S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}), \\ J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}))) - J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ \leq (1 - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}})) {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2} \\ + 4 d^{2} ρ_{\tilde{E}} (\frac{4 ∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥}{d}) . \end{matrix}

(9)

S (\cdot, \cdot, \cdot)

δ_{S_{1}}

-Lipschitz continuous in first slot,

δ_{S_{2}}

-Lipschitz continuous in second slot,

δ_{S_{3}}

-Lipschitz continuous in third slot and

\tilde{A}

α_{\tilde{A}}

-D-Lipschitz continuous,

\tilde{B}

α_{\tilde{B}}

-D-Lipschitz continuous,

\tilde{C}

α_{\tilde{C}}

-D-Lipschitz continuous, we have

\begin{matrix} ∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥ \\ = ∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) + S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) \\ + S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥ \\ \leq δ_{S_{1}} ∥ {\hat{u}}_{n} - {\hat{u}}_{n - 1} ∥ + δ_{S_{2}} ∥ {\hat{v}}_{n} - {\hat{v}}_{n - 1} ∥ + δ_{S_{3}} ∥ {\hat{w}}_{n} - {\hat{w}}_{n - 1} ∥ \\ \leq δ_{S_{1}} D (\tilde{A} ({\dot{a}}_{n}), \tilde{A} ({\dot{a}}_{n - 1})) + δ_{S_{2}} D (\tilde{B} ({\dot{a}}_{n}), \tilde{B} ({\dot{a}}_{n - 1})) + δ_{S_{3}} D (\tilde{C} ({\dot{a}}_{n}), \tilde{C} ({\dot{a}}_{n - 1})) \\ \leq δ_{S_{1}} α_{\tilde{A}} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ + δ_{S_{2}} α_{\tilde{B}} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ + δ_{S_{3}} α_{\tilde{C}} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ \\ \leq (δ_{S_{1}} α_{\tilde{A}} + δ_{S_{2}} α_{\tilde{B}} + δ_{S_{3}} α_{\tilde{C}}) ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ . \end{matrix}

(10)

Using equation (10) and

(i i)

of Proposition 1, we evaluate

\begin{matrix} 4 d^{2} ρ_{\tilde{E}} & (\frac{4 ∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥}{d}) \\ = 4 d^{2} ρ_{\tilde{E}} (\frac{4}{d} (∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) + S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) \\ - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) + S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥)) \\ \leq 64 C (∥ S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) ∥^{2} + ∥ S ({\hat{u}}_{n - 1}, {\hat{v}}_{n}, {\hat{w}}_{n}) \\ - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) ∥^{2} + {∥ S (n_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1}) ∥}^{2}) \\ \leq 64 C (δ_{S_{1}}^{2} ∥ {\hat{u}}_{n} - {\hat{u}}_{n - 1} ∥^{2} + δ_{S_{2}}^{2} ∥ {\hat{v}}_{n} - {\hat{v}}_{n - 1} ∥^{2} + δ_{S_{3}}^{2} {∥ {\hat{w}}_{n} - {\hat{w}}_{n - 1} ∥}^{2}) \\ \leq 64 C (δ_{S_{1}}^{2} D^{2} (\tilde{A} ({\dot{a}}_{n}), \tilde{A} ({\dot{a}}_{n - 1})) + δ_{S_{2}}^{2} D^{2} (\tilde{B} ({\dot{a}}_{n}), \tilde{B} ({\dot{a}}_{n - 1})) + δ_{S_{3}}^{2} D^{2} (\tilde{C} ({\dot{a}}_{n}), \tilde{C} ({\dot{a}}_{n - 1}))) \\ \leq 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2} {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2}) \\ = 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2}) {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2} . \end{matrix}

(11)

Combining (9) and (11), we have

\begin{matrix} ∥ ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) - (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) ∥^{2} & \leq [(1 - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}})) \\ + 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2})] {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2}, \end{matrix}

which implies that

\begin{matrix} ∥ ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) - & (S ({\hat{u}}_{n}, {\hat{v}}_{n}, {\hat{w}}_{n}) - S ({\hat{u}}_{n - 1}, {\hat{v}}_{n - 1}, {\hat{w}}_{n - 1})) ∥ \\ \leq \sqrt{[(1 - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}})) + 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2})]} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ . \end{matrix}

(12)

Since

\tilde{m}

λ_{m}

-Lipschitz continuous, we have

∥ \tilde{m} ({\dot{a}}_{n}) - \tilde{m} ({\dot{a}}_{n - 1}) ∥ \leq λ_{m} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ .

(13)

As Yosida approximation operator

Y_{\hat{h_{1}}, λ}^{\hat{M}}

δ_{Y_{\hat{h_{1}}}}

-strongly accretive,

λ_{Y_{\hat{h_{1}}}}

-Lipschitz continuous and applying Proposition 1, we evaluate

\begin{matrix} {∥({\dot{a}}_{n} - {\dot{a}}_{n - 1}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}))∥}^{2} \\ \leq ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 λ 〈Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1})))〉 \\ = ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 λ 〈 Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ - 2 λ 〈 Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}), J ({\dot{a}}_{n} - {\dot{a}}_{n - 1} - (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1}))) - J ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) 〉 \\ \leq ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 λ δ_{Y_{\hat{h_{1}}}} {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2} + 4 d^{2} ρ_{\tilde{E}} (\frac{4 λ^{2} ∥Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1})∥}{d}) \\ = ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 λ δ_{Y_{\hat{h_{1}}}} {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2} + 64 C λ^{4} {∥Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1})∥}^{2} \\ = ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥^{2} - 2 λ δ_{Y_{\hat{h_{1}}}} {∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥}^{2} + 64 C λ^{4} λ_{Y_{\hat{h_{1}}}}^{2} {∥{\dot{a}}_{n} - {\dot{a}}_{n - 1}∥}^{2} \\ = (1 - 2 λ δ_{Y_{\hat{h_{1}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{1}}}}^{2}) {∥{\dot{a}}_{n} - {\dot{a}}_{n - 1}∥}^{2}, \end{matrix}

that is

∥ ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n}) - Y_{\hat{h_{1}}, λ}^{\hat{M}} ({\dot{a}}_{n - 1})) ∥ \leq \sqrt{1 - 2 λ δ_{Y_{\hat{h_{1}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{1}}}}^{2}} ∥{\dot{a}}_{n} - {\dot{a}}_{n - 1}∥ .

(14)

Using the same arguments as for (14), we have

∥ ({\dot{a}}_{n} - {\dot{a}}_{n - 1}) - λ (Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} ({\dot{a}}_{n - 1})) ∥ \leq \sqrt{1 - 2 λ δ_{Y_{\hat{h_{2}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{2}}}}^{2}} ∥{\dot{a}}_{n} - {\dot{a}}_{n - 1}∥ .

(15)

Using (10), (12), (13), (14), (15), (7) becomes

\begin{matrix} ∥ {\dot{a}}_{n + 1} - {\dot{a}}_{n} ∥ & \leq & \sqrt{[(1 - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}})) + 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2})]} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ \\ + 2 λ_{m} ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ + (δ_{S_{1}} α_{\tilde{A}} + δ_{S_{2}} α_{\tilde{B}} + δ_{S_{3}} α_{\tilde{C}}) ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥ \\ + \sqrt{1 - 2 λ δ_{Y_{\hat{h_{1}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{1}}}}^{2}} ∥{\dot{a}}_{n} - {\dot{a}}_{n - 1}∥ \\ + \sqrt{1 - 2 λ δ_{Y_{\hat{h_{2}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{2}}}}^{2}} ∥{\dot{a}}_{n} - {\dot{a}}_{n - 1}∥ \\ = & θ ∥ {\dot{a}}_{n} - {\dot{a}}_{n - 1} ∥, \end{matrix}

(16)

\begin{matrix} where θ & = & [\sqrt{[(1 - 2 (λ_{S_{1}} + λ_{S_{2}} + λ_{S_{3}})) + 64 C (δ_{S_{1}}^{2} α_{\tilde{A}}^{2} + δ_{S_{2}}^{2} α_{\tilde{B}}^{2} + δ_{S_{3}}^{2} α_{\tilde{C}}^{2})]} \\ + 2 λ_{m} + (δ_{S_{1}} α_{\tilde{A}} + δ_{S_{2}} α_{\tilde{B}} + δ_{S_{3}} α_{\tilde{C}}) + \sqrt{1 - 2 λ δ_{Y_{\hat{h_{1}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{1}}}}^{2}} \\ + \sqrt{1 - 2 λ δ_{Y_{\hat{h_{2}}}} + 64 C λ^{4} λ_{Y_{\hat{h_{2}}}}^{2}}] . \end{matrix}

(17)

In view of the assumption

(v i i)

0 < θ < 1

and clearly

{{\dot{a}}_{n}}

is a Cauchy sequence such that we have

\dot{a} \in \tilde{E}

and

{\dot{a}}_{n} \to \dot{a}

. Using (4), (5), (6) of iterative method Section 3, D-Lipschitz continuity of

\tilde{A}, \tilde{B}

\tilde{C}

and the techniques of Ahmad and Irfan [12], it can be shown easily that

{{\hat{u}}_{n}}, {{\hat{v}}_{n}}

and

{{\hat{w}}_{n}}

are all Cauchy sequences in

\tilde{E}

. Thus,

{\hat{u}}_{n} \to \hat{u} \in \tilde{E}, {\hat{v}}_{n} \to \hat{v} \in \tilde{E}

and

{\hat{w}}_{n} \to \hat{w} \in \tilde{E}

. Since

Q_{F}, S (\cdot, \cdot, \cdot), \tilde{A}, \tilde{B}, \tilde{C}, \hat{h_{1}}, \hat{h_{2}}, \hat{M}, N, Y_{\hat{h_{1}}, λ}^{\hat{M}}

and

Y_{\hat{h_{2}}, λ}^{\hat{N}}

are all continuous operators in

\tilde{E}

, we have

\dot{a} = \dot{a} + \tilde{m} (\dot{a}) - S (\hat{u}, \hat{v}, \hat{w}) + Q_{F} [S (\hat{u}, \hat{v}, \hat{w}) - λ (Y_{\hat{h_{1}}, λ}^{\hat{M}} (\dot{a}) - Y_{\hat{h_{2}}, λ}^{\hat{N}} (\dot{a}))] .

It is remaining to show that

\hat{u} \in \tilde{A} (\dot{a}), \hat{v} \in \tilde{B} (\dot{a})

and

{\hat{w}}_{n} \to \hat{w} \in \tilde{C} (\dot{a})

. In fact

\begin{matrix} d (\hat{u}, \tilde{A} (\dot{a})) & = & inf {∥ \hat{u} - h ∥ : h \in \tilde{A} (\dot{a})} \\ \leq & ∥ \hat{u} - {\hat{u}}_{n} ∥ + d ({\hat{u}}_{n}, \tilde{A} (\dot{a})) \\ \leq & ∥ \hat{u} - {\hat{u}}_{n} ∥ + D (\tilde{A} ({\dot{a}}_{n}), \tilde{A} (\dot{a})) \\ \leq & ∥ \hat{u} - {\hat{u}}_{n} ∥ + α_{\tilde{A}} ∥ {\dot{a}}_{n} - \dot{a} ∥ \to 0 . \end{matrix}

Hence,

d (\hat{u}, \tilde{A} (\dot{a})) = 0

and thus

\hat{u} \in \tilde{A} (\dot{a})

. Similarly, it can be shown that

\hat{v} \in \tilde{B} (\dot{a})

and

\hat{w} \in \tilde{C} (\dot{a})

. From Theorem 2, the result follows. □

5. Conclusions

In this work, we consider a different version of co-variational inequalities existing in available literature. We call it co-variational inequality problem which involves two generalized Yosida approximation operators depending on different generalized resolvent operators. Some properties of generalized Yosida approximation operators are proved. Using the concept of nonexpansive sunny retraction, we prove an existence and convergence result for problem (1).

Our results may be used for further generalization and experimental purposes.

Author Contributions

Conceptualization, R.A.; methodology, R.A. and A.K.R.; software, M.I.; validation, R.A., M.I. and H.A.R.; formal analysis, Y.W.; resources, Y.W. and H.A.R.; writing—original draft preparation, M.I.; writing—review and editing, A.K.R.; visualization, A.K.R.; supervision, R.A.; project administration, R.A. and Y.W.; funding acquisition, Y.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (Grant no. 12171435).

Conflicts of Interest

All authors declare that they do not have conflict of interest.

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Co-Variational Inequality Problem involving Two Generalized Yosida Approximation Operators

Abstract

1. Introduction

2. Preface

3. Problem Formation and Iterative Method

4. Convergence Result

5. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

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