Thyrotoxic periodic paralysis together with thyrotoxic heart disease in a Ghanaian man: case report and literature review

Gordon Manu Amponsah; Yaw Adu-Boakye; Maureen Nyarko; Henry Kofi Andoh; Kwaku Gyasi Danso; Manolo Agbenoku; Isaac Kofi Owusu; Gordon Manu Amponsah; Yaw Adu-Boakye; Maureen Nyarko; Henry Kofi Andoh; Kwaku Gyasi Danso; Manolo Agbenoku; Isaac Kofi Owusu

doi:10.3934/medsci.2023005

AIMS Medical Science

2023, Volume 10, Issue 1: 46-54. doi: 10.3934/medsci.2023005

Previous Article Next Article

Case report Topical Sections

Thyrotoxic periodic paralysis together with thyrotoxic heart disease in a Ghanaian man: case report and literature review

1.
Department of Physiology, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
2.
Directorate of Medicine, Komfo Anokye Teaching Hospital, Kumai, Ghana
3.
Department of Medicine, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Received: 22 January 2023 Revised: 28 February 2023 Accepted: 06 March 2023 Published: 15 March 2023

Thyrotoxic periodic paralysis (TPP) is an uncommon symmetrical paralysis usually affecting proximal muscles, which occurs in the hyperthyroid state with associated hypokalemia. It is more prevalent in East Asian males and extremely rare in blacks. Data on TPP is scarce in Africa and no report has been made in Ghana. We report a case of a middle-aged Ghanaian man who had three episodes of paralysis in all four limbs occurring at night with the second and third episodes requiring hospital visit. He had no clinical signs of hyperthyroidism during his first hospital visit but had developed clinical and biochemical evidence of hyperthyroidism on the second visit with serum potassium levels of 1.9 mmol/l; and he was eventually diagnosed with TPP. His paralysis resolved with correction of the hypokalemia. It is important to evaluate patients presenting with paralysis comprehensively. Less common differential diagnosis such as TPP may also be considered in such patients to ensure early diagnosis and treatment which can prevent complications.

Keywords:

Citation: Gordon Manu Amponsah, Yaw Adu-Boakye, Maureen Nyarko, Henry Kofi Andoh, Kwaku Gyasi Danso, Manolo Agbenoku, Isaac Kofi Owusu. Thyrotoxic periodic paralysis together with thyrotoxic heart disease in a Ghanaian man: case report and literature review[J]. AIMS Medical Science, 2023, 10(1): 46-54. doi: 10.3934/medsci.2023005

Related Papers:

[1]	Yong-Ki Ma, N. Valliammal, K. Jothimani, V. Vijayakumar . Solvability and controllability of second-order non-autonomous impulsive neutral evolution hemivariational inequalities. AIMS Mathematics, 2024, 9(10): 26462-26482. doi: 10.3934/math.20241288
[2]	Ahmed Salem, Kholoud N. Alharbi . Fractional infinite time-delay evolution equations with non-instantaneous impulsive. AIMS Mathematics, 2023, 8(6): 12943-12963. doi: 10.3934/math.2023652
[3]	Zainab Alsheekhhussain, Ahmed Gamal Ibrahim, Rabie A. Ramadan . Existence of $S$ -asymptotically $\omega$ -periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $1 < \alpha < 2$ . AIMS Mathematics, 2023, 8(1): 76-101. doi: 10.3934/math.2023004
[4]	Misbah Iram Bloach, Muhammad Aslam Noor . Perturbed mixed variational-like inequalities. AIMS Mathematics, 2020, 5(3): 2153-2162. doi: 10.3934/math.2020143
[5]	Lu-Chuan Ceng, Li-Jun Zhu, Tzu-Chien Yin . Modified subgradient extragradient algorithms for systems of generalized equilibria with constraints. AIMS Mathematics, 2023, 8(2): 2961-2994. doi: 10.3934/math.2023154
[6]	Ebrahem A. Algehyne, Abdur Raheem, Mohd Adnan, Asma Afreen, Ahmed Alamer . A study of nonlocal fractional delay differential equations with hemivariational inequality. AIMS Mathematics, 2023, 8(6): 13073-13087. doi: 10.3934/math.2023659
[7]	Qiang Li, Jina Zhao . Extremal solutions for fractional evolution equations of order $1 < \gamma < 2$ . AIMS Mathematics, 2023, 8(11): 25487-25510. doi: 10.3934/math.20231301
[8]	Ramkumar Kasinathan, Ravikumar Kasinathan, Dumitru Baleanu, Anguraj Annamalai . Well posedness of second-order impulsive fractional neutral stochastic differential equations. AIMS Mathematics, 2021, 6(9): 9222-9235. doi: 10.3934/math.2021536
[9]	Chunli You, Linxin Shu, Xiao-bao Shu . Approximate controllability of second-order neutral stochastic differential evolution systems with random impulsive effect and state-dependent delay. AIMS Mathematics, 2024, 9(10): 28906-28930. doi: 10.3934/math.20241403
[10]	Zhi Guang Li . Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems. AIMS Mathematics, 2023, 8(8): 18174-18184. doi: 10.3934/math.2023923

Abstract

1. Introduction

Differential variational inequality is a dynamical system that includes variational inequalities and ordinary differential equations. Differential variational inequalities plays an important role for formulating models involving both dynamics and inequality constraints. Aubin and Cellina ^[3] introduced the concept of differential variational inequality and after that it was studied by Pang and Stewart ^[25]. The partial differential variational inequalities was studied by Liu, Zeng and Motreanu ^[15] and shown that the solution set is compact and continuous. There are some obstacles in their work that constraint set necessarily be compact and only local boundary conditions are satisfied. Liu, Migórskii and Zeng ^[14] relaxed the conditions of ^[15] and proved the existence of partial differential variational inequality in non-compact setting. Properties of solution set like strong-weak upper semicontinuity and measurability was proved by them.

Differential variational inequalities are application oriented and have several applications in engineering and physical sciences, operation research, etc. In particular, they are applicable in electrical circuits with ideal diodes, economical dynamics, dynamic traffic network, functional problems, differential Nash games, control systems, etc., see for example ^{[1,2,16,17,18,19,20,23,26,27,31]}.

Evolution equation can be explained as the differential law of the development (evolution) in time of a system. The evolution character of the equation make easier its numerical solution. Variational-like inequality is a generalized form of a variational inequalities and has many applications in operations research, optimization, convex mathematical programming, etc. On the other hand, many problems of engineering and applied sciences can be solved by using second order evolution equation, see for example ^{[5,6,9,10,12,13,22,28,30,32,33]}.

2. Preliminaries and formulation of the problem

Throughout the paper, we assume $\widetilde{B_1}$ and $\widetilde{B_2}$ denote separable reflexive Banach spaces and $\widehat{K}(\neq \phi)$ be convex and closed subset of $\widetilde{B_1}$ . We define some mapping below, that is,

$\begin{align*} &\widetilde{{\mathcal{F}}} : [0, \mathcal{T}]\times\widetilde{B_2}\times\widetilde{B_2}\longrightarrow\mathcal{L}\left(\widetilde{B_1}, \widetilde{B_2}\right), \\ &\widetilde{\mathfrak{f}} : [0, \mathcal{T}]\times\widetilde{B_2}\times\widetilde{B_2}\longrightarrow\widetilde{B_2}, \\ &\widetilde{\mathfrak{g}} : [0, \mathcal{T}]\times\widetilde{B_2}\times\widetilde{B_2}\longrightarrow\widetilde{B_2}, \\ &\widetilde{A} : \widehat{K}\longrightarrow\widetilde{B_1^*}, \\ &\eta : \widehat{K}\times\widehat{K}\longrightarrow\widetilde{B_1}, \\ &\psi : \widetilde{K}\longrightarrow\mathbb{R}\cup\{+\infty\}, \ {\rm{ where }}\ \mathcal{T} > 0. \end{align*}$

Inspired by the above discussed work, in this paper, we introduce and study a second order evolutionary partial differential variational-like inequality in Banach spaces. We mention our problem below:

$\begin{align} \begin{cases} \mathfrak{y}''(x) = \widetilde{A}\mathfrak{y}(x)+\widetilde{\mathcal{F}}(x, \mathfrak{y}(x), \mathfrak{y}'(x))\widehat{\mathfrak{u}}(x)+\widetilde{\mathfrak{f}}(x, \mathfrak{y}(x), \mathfrak{y}'(x)), \ {\rm{ a.e. }}\ x\in[0, \mathcal{T}], \\ \widehat{\mathfrak{u}}(x)\in Sol(\widetilde{K}, \widetilde{\mathfrak{g}}(x, \mathfrak{y}(x), \mathfrak{y}'(x))+\widetilde{\mathcal{A}}(\cdot), \psi), \ {\rm{ a.e. }}\ x\in[0, \mathcal{T}], \\ \mathfrak{y}(0) = \mathfrak{y}_0, \mathfrak{y}'(0) = y_0. \end{cases} \end{align}$

(2.1)

We also consider a variational-like inequality problem of finding $\widehat{\mathfrak{u}}:[0, \mathcal{T}]\rightarrow \widehat{K}$ such that

$\begin{align} \langle\widetilde{\mathfrak{g}}(x, \mathfrak{y}(x), \mathfrak{y}'(x))+\widetilde{\mathcal{A}}(\widehat{\mathfrak{u}}(x)), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}(x))\rangle+\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}(x))\geq0, \forall\; \widehat{\mathfrak{v}}\in\widehat{K}, \ {\rm{ a.e. }}\ x\in[0, \mathcal{T}]. \end{align}$

(2.2)

The solution set of problem (2.2) is denoted by $Sol[(2.1)].$

The mild solution of problem (2.1) is described by the following definition.

Definition 2.1. A pair of function $(\mathfrak{y}, \widehat{\mathfrak{u}})$ such that $\mathfrak{y}\in C^1([0, \mathcal{T}], \widetilde{B_2})$ and $\widehat{\mathfrak{u}}:[0, \mathcal{T}]\to\widehat{K}(\subset\widetilde{B_1})$ measurable, called mild solution of problem (2.1) if

$\mathfrak{y}(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_0^x R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\widehat{\mathfrak{u}}(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))]dp,$

where $x\in[0, \mathcal{T}]$ and $\widehat{\mathfrak{u}}(p)\in Sol(\widehat{K}, \widetilde{\mathfrak{g}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))+\mathcal{A}(\cdot, \psi)$ . $R(x)$ will be defined in continuation. Here, $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)$ denotes the solution set of mixed variational-like inequality (3.1). If $(\mathfrak{y}, \widehat{\mathfrak{u}})$ is a mild solution of above assumed problem, then $\mathfrak{y}$ is said to be the mild trajectory and $\widehat{\mathfrak{u}}$ is called the variational control trajectory. Here $C^1([0, \mathcal{T}], \widetilde{B_2})$ denotes the Banach space of all continuous differentiable mappings $\mathfrak{y}:[0, \mathcal{T}]\to\widetilde{B_2}$ with norm

$\|\mathfrak{y}\|_{C^1} = \max\left\{\max\limits_{x\in[0, \mathcal{T}]}\|\mathfrak{y}(x)\|, \max\limits_{x\in[0, \mathcal{T}]}\|\mathfrak{y}'(x)\|\right\},$

and $L(\widetilde{B_2})$ denotes the Banach space of bounded linear operators from $\widetilde{B_2}$ into $\widetilde{B_2}$ .

The subsequent part of this paper is organised in this way. In the next section, some definitions and results are defined, which will be used to achieve our goal. In Section 3, an existence result for variational-like inequalities is proved. Also, we have proved that $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)$ is nonempty, closed and convex. The upper semicontinuity of the multi-valued mapping $\mathfrak{F}:[0, \mathcal{T}]\times\widetilde{B_2}\times\widetilde{B_2}\to\Pi_{bv}(\widehat{B_1})$ is discussed. In the last section, we have proved that the existence result for the mild solution of second order evolutionary partial differerntial variational-like inequalities under some appropriate conditions.

Let $\widehat{X_1}$ and $\widehat{X_2}$ are topological spaces. We shall use $\Pi(\widehat{X_2})$ to denote the family of all nonempty subsets of $X_2$ , and

$\Pi_c(\widehat{X_2}) : = \{\widehat{D}\in\Pi(\widehat{X_2}):\widehat{D}$ is closed $\}$ ;

$\Pi_b(\widehat{X_2}) : = \{\widehat{D}\in\Pi(\widehat{X_2}):\widehat{D}$ is bounded $\}$ ;

$\Pi_{bc}(\widehat{X_2}) : = \{\widehat{D}\in\Pi(\widehat{X_2}):\widehat{D}$ is bounded and closed $\}$ ;

$\Pi_{cv}(\widehat{X_2}) : = \{\widehat{D}\in\Pi(\widehat{X_2}):\widehat{D}$ is closed and convex $\}$ ;

$\Pi_{bv}(\widehat{X_2}) : = \{\widehat{D}\in\Pi(\widehat{X_2}):\widehat{D}$ is bounded and convex $\}$ ;

$\Pi_{kv}(\widehat{X_2}) : = \{\widehat{D}\in\Pi(\widehat{X_2}):\widehat{D}$ is compact and convex $\}$ .

One parameter family $Q(x)$ , where $x$ is real number, of bounded linear operators from a Banach space $\widehat{B_2}$ into itself is called a strongly continuous cosine family if and only if

(1) $Q(x+p)+Q(x-p) = 2C(x)C(p), \forall x, p\in \mathbb{R}$ ,

(2) $Q(0) = I, (I\ {\rm{ is\ the\ identity\ operator\ in }}\ \widehat{B_2})$ ,

(3) $Q(x)w$ is continuous in $x$ on $\mathbb{R}$ for every fixed $w\in \widehat{B_2}$ .

We associate with the strongly continuous cosine family $Q(x)$ in $\widehat{B_2}$ the strongly continuous sine family $R(x)$ , such that

$R(x)W = \int_{0}^{x}Q(p)wdp, \; \; w\in \widehat{B_2}, \; x\in \mathbb{R},$

and the two sets

$\begin{eqnarray*} E_1& = & \{w\in \widehat{B_2}: Q(x)u \; \; \mbox{is one time continuously differentiable in }\ x\ \mbox{ on}\ \mathbb{R} \}, \\ E_2& = & \{w\in \widehat{B_2}: Q(x)w \; \; \mbox{is two times continuously differentiable in }\ x \ \mbox{ on} \mathbb{R} \}. \end{eqnarray*}$

The operator $A:D(A)\subset \widehat{B_2}\to \widehat{B_2}$ is the infinitesimal generator of a strongly continuous cosine family $Q(x)$ , $x\in \mathbb{R}$ defined by $A(\mathfrak{y}) = d^2/dx^2Q(0)\mathfrak{y}\; \mbox{with $D(A) = E_2$.}$

Proposition 2.1. ^[29] Let $Q(x), \; x\in \mathbb{R}$ be a strongly continuous cosine family in $\widehat{B_1}$ . Then the following hold:

(i) $Q(x) = Q(-x), \; \forall$ $x\in \mathbb{R}$ ,

(ii) $Q(p), R(p), Q(x), \; \mathit{\mbox{and}}\; R(x)\; {commute } \ \forall \; x, \; p\in \mathbb{R}$ ,

(iii) $R(x+p)+Q(x-p) = 2R(x)Q(p), \; \forall$ $x, p\in \mathbb{R}$ ,

(iv) $R(x+p) = R(x)Q(p)+R(p)C(x), \; \forall$ $x, \; p\in \mathbb{R}$ ,

(v) $R(x) = -R(-x), \; \forall$ $x\in \mathbb{R}$ .

For furthure information related to the properties of the sine and cosine families, see ^[12,23,27] and references therein.

Definition 2.2. ^[21] Let $\widehat{X_1}, \; \widehat{X_2}$ are topological spaces. Then the multi-valued mapping $\widehat{F}: \widehat{X_1}\to \Pi(\widehat{X_2})$ is said to be:

(i) Upper semicontinuous (u.s.c., in short) at $x\in \widehat{X_1}$ , if for each open set $\mathfrak{U}\subset \widehat{X_2}$ with $\widehat{F}(x)\subset \mathfrak{U}$ , $\exists$ a neighbourhood $\mathcal{N}(x)$ of $x$ such that

$\widehat{F}(\mathcal{N}(x)): = \underset{y\in \mathcal{N}(x)}{\widehat{F}(y)}\subset\mathfrak{U}.$

If $\widehat{F}$ is u.s.c. $\forall$ $x\in \widehat{X_1}$ , then $\widehat{F}$ is said to be upper semicontinuous on $\widehat{X_1}.$

(ii) Lower semicontinuous (l.s.c., in short) at $x\in \widehat{X_1}$ if, for each open set $\mathfrak{U}\subset \widehat{X_2}$ satisfying $\widehat{F}\cap \mathfrak{U}\neq \phi,$ $\exists$ a neighbourhood $\mathcal{N}(x)$ of $x$ such that $\widehat{F}\cap\mathfrak{U}\neq \phi$ $\forall$ $y\in \mathcal{N}(x).$ If $\widehat{F}$ is l.s.c. $\forall$ $x\in \widehat{X_1}$ , then $\widehat{F}$ is called lower semicontinuous on $\widehat{X_1}$ .

Proposition 2.2. ^[21] Let $\widehat{F}:\widehat{X_1}\to \Pi(\widehat{X_2})$ be a multi-valued mapping, where $\widehat{X_1}, \; \mathit{\mbox{and}}\; \widehat{X_2}$ denote topological vector spaces. Then the following are equivalent:

(i) $\widehat{F}$ is upper semicontinuous,

(ii) the set

$\widehat{F}^{-}(\mathcal{C}) = \{x\in \widehat{X_1}: \widehat{F}(x)\cap \mathcal{C}\neq \phi\},$

is closed in $\widehat{X_1}$ , for each closed set $\mathcal{C}\subset \widehat{X_2}$ ,

(iii) the set

$\widehat{F}^{+}(\mathcal{C}) = \{x\in \widehat{X_1}: \widehat{F}(x)\subset \mathfrak{U}\},$

is open in $\widehat{X_1}$ , for each open set $\mathfrak{U}\subset \widehat{X_2}.$

Proposition 2.3. ^[4] Let $\Omega(\neq \phi)$ subset of Banach space $\widehat{X}$ . Assume that the multi-valued mapping $\widehat{F}: \Omega\to \Pi(\widehat{X})$ is weakly compact and convex. Then, $\widehat{F}$ is strongly-weakly u.s.c. if and only if $\{x_n\}\subset \Omega$ with $x_n\to x_0\in \Omega$ and $y_n\in \widehat{F}(x_n)$ implies $y_n\rightharpoonup y_0\in \widehat{F}(x_0)$ up to a subsequence.

Lemma 2.1. ^[7] Let $\{x_n\}$ be a sequence such that $x_n\rightharpoonup \bar{x}$ in a normed space $V$ . Then there is a sequence of combinations $\{y_n\}$ such that

$y_n = \sum\limits_{i = n}^{\infty}\lambda_ix_i, \; \; \sum\limits_{i = n}^{\infty}\lambda_i = 1\; \mathit{\mbox{and}}\; \lambda_i\geq 0, \; \; \; 1\leq i\leq \infty,$

which converges to $\bar{x}$ in norm.

Now we define the measurability of a multi-valued mapping, which is needed in the proof of existence of solution of second order evolutionary partial differential variational-like inequality problem (2.1).

Definition 2.3. ^[11,21]

(i) A multi-valued mapping $\widehat{F}:I \to \Pi(\widehat{X})$ is called measurable if for each open subset $\mathfrak{U}\subset \widehat{X}$ the set $\widehat{F}^{+}(\mathfrak{U})$ is measurable in $\mathbb{R}$ .

And

(ii) the multi-valued mapping $\widehat{F}:I\to \Pi_{bc}(\widehat{X})$ is called strongly measurable if $\exists$ a sequence $\{\widehat{F}_n\}_{n = 1}^{\infty}$ of step set-valued mappings such that

$\widehat{\mathcal{H}}(\widehat{F}(t), \widehat{F}_n(t))\to 0, \; \; {as}\; n\to \infty, t\in I\ { a.e.},$

here $\widehat{X}$ denotes Banach space, $I$ be an interval of real numbers and $\widehat{\mathcal{H}}(., .)$ denotes the Hausdorff metric on $\Pi_{bc}(\widehat{X}).$

Definition 2.4. ^[11,34] Let $\widehat{X}$ be Banach space and $(\mathbb{F}, \preceq)$ be a partial ordered set. A function $\mathcal{\beta} :\Pi_b(\widehat{X})\to \mathbb{F}$ is called a measure of non compactness (MNC, for short) in $\widehat{X}$ if

$\mathcal{\beta} (\overline{conv}\mathcal{O}) = \mathcal{\beta}(\mathcal{O})\; \; \; {for\; every\; }\mathcal{O}\in \Pi_b(\widehat{X}),$

here $\overline{conv}\mathcal{O}$ showing the closure of convex hull of $\mathcal{O}.$

Definition 2.5. ^[34] A measure of non compactness $\mathcal{\beta}$ is called

(i) monotone, if $\mathcal{O}_0, \mathcal{O}_1\in \Pi_b(\widehat{X})$ and $\mathcal{O}_0\subseteq \mathcal{O}_1$ implies $\mathcal{\beta}(\mathcal{O}_0)\preceq \mathcal{\beta} (\mathcal{O}_1)$ ,

(ii) nonsingular, if $\mathcal{\beta}({a}\cup \mathcal{O}) = \mathcal{\beta} (\mathcal{O})$ $\forall$ $a\in \widehat{X}$ and $\mathcal{O} \in \Pi_b(\widehat{X})$ ,

(iii) invariant with respect to union of compact set, if $\mathcal{\beta}(K\cup \mathcal{O}) = \mathcal{\beta}(\mathcal{O})$ for each relatively compact set $K\subset \widehat{X}$ and $\mathcal{O} \in \Pi_b(\widehat{X})$ ,

(iv) algebraically semiadditive, if $\mathcal{\beta} (\mathcal{O}_0+\mathcal{O}_1)\preceq \mathcal{\beta}(\mathcal{O}_0)+\mathcal{\beta}(\mathcal{O}_1)$ for every $\mathcal{O}_0, \; \mathcal{O}_1\in \Pi_b(\widehat{X})$ ,

(v) regular, if $\mathcal{\beta}(\mathcal{O}) = 0$ is equivalent to the relative compactness of $\mathcal{O}.$

A very famous example of measure of non compactness is the following Hausdorff measure of non compactness on $C([0, \mathcal{T}], \widehat{X})$ with $0 < \mathcal{T} < \infty$ calculated by the following formula:

$\begin{equation} \chi_T(\mathcal{O}) = \frac{1}{2}\underset{\delta\to 0}{lim}\; \underset{x\in \mathcal{O}}{sup}\underset{|t_1-t_2|\leq \delta }{max}\|x(t_1)-x(t_2)\|_{\widehat{X}}. \end{equation}$

(2.3)

Here, $\chi_T(\mathcal{O})$ is said to be the modulus of equicontinuity of $\mathcal{O}\subset C([0, \mathcal{T}], \widehat{X})$ . Definition (2.4) is applicable on (2.3).

Definition 2.6. ^[11] A multi-valued mapping $\widehat{F}:\widehat{K}\subset \widehat{X}\to \Pi(\widehat{X})$ is said to be condensing relative to measure of non compactness $\mathcal{\beta}$ (or $\mathcal{\beta}$ -condensing) if for each $\mathcal{O}\subset \widehat{K}$ , we have

$\mathcal{\beta}(\widehat{F}(\mathcal{O}))\nsucceq \mathcal{\beta}(\mathcal{O}).$

That is not relatively compact.

Definition 2.7. ^[8] A single valued mapping $T:\widehat{K}\to \widehat{X}^*$ is called relaxed $\eta$ - $\alpha$ monotone if $\exists$ a mapping $\eta: \widehat{K}\times \widehat{K}\to \widehat{X}$ and a real-valued mapping $\alpha: \widehat{X} \to \mathbb{R}$ , with $\alpha(tz) = t^p\alpha(z)$ , ${ \forall } t > 0,$ and $z\in \widehat{X}$ , such that

$\begin{equation} \langle Tx-Ty, \eta(x, y)\rangle\geq- \alpha(x-y), {\rm{ \forall }}x, y\in \widehat{K}, \end{equation}$

(2.4)

where $p > 1$ is a constant.

Definition 2.8. ^[8] A mapping $T:\widehat{K}\to \widehat{X}^*$ is called $\eta$ -coercive with respect to $\psi$ if $\exists$ $x_0\in \widehat{K}$ such that

$\begin{equation} \underset{x\in K, \; \|x\|\to \infty}{lim \; inf}\frac{\langle T(x)-T(y), \eta(x, x_0)\rangle+\psi(x)-\psi(x_0)}{\|\eta(x, x_0)\|}\to +\infty. \end{equation}$

(2.5)

Where $\eta: \widehat{K}\times \widehat{K}\to \widehat{X}$ be a mapping and $\psi:\widehat{X}\to \mathbb{R}\cup\{+\infty\}$ is proper convex lower semicontinuous function.

Theorem 2.1. ^[11] Let $\widehat{X}$ be a Banach space and $\mathcal{M}$ its closed convex subset, then the fixed point set of $\mathcal{\beta}$ -condensing multi-valued mapping $\widehat{F}:\mathcal{M}\to \Pi_{kv}(\mathcal{M})$ is nonempty. That is $\mathit{\mbox{Fix}}\widehat{F}: = \{x\in \mathcal{M}:x\in \widehat{F}(x)\}\neq \phi.$ Where $\mathcal{\beta}$ is a nonsingular measure of non compactness defined on subsets of $\mathcal{M}$ .

3. Some properties of $Sol(K, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)$

Let $\widehat{B_1}$ and $\widehat{B_2}$ are real reflexive Banach spaces and ${\widehat{B_1}^*}$ be the dual of $\widehat{B_1}$ and $\widehat{K}$ be a nonempty closed, convex subset of $\widehat{B_1}$ .

We consider the following problem of finding $\widehat{\mathfrak{u}}\in \widehat{K}$ such that

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\geq 0, {\rm{ \forall }}\widehat{\mathfrak{v}}\in \widehat{K}, \end{equation}$

(3.1)

where $\widehat{\mathfrak{w}}\in \widehat{B_1}^*$ , $\mathcal{A}:\widehat{K}\to \widehat{B_1}^*$ and $\eta:\widehat{K}\times \widehat{K}\to \widehat{B_1}.$ Problem (3.1) is called generalized mixed variational-like inequality. We prove the following lemma.

Lemma 3.1. Suppose that the following conditions are satisfied:

$(I_1)$ $\mathcal{A}: \widehat{B_1}\to \widehat{B_1}^*$ is an $\eta$ -hemicontinuous and $\eta$ - $\alpha$ monotone mapping;

$(I_2)$ $\psi: \widehat{B_1}\to \mathbb{R}\cup\{+\infty\}$ be a proper convex and lower semicontinuous;

$(I_3)$ the mapping $\widehat{\mathfrak{u}}\to\langle \mathcal{A}\widehat{\mathfrak{z}}, \eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}})\rangle$ is convex, lower semicontinuous for fixed $\widehat{\mathfrak{v}}, \widehat{\mathfrak{z}}\in \widehat{K}$ and $\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{u}}) = 0, \mathit{\rm{\forall }\; }\widehat{\mathfrak{u}}\in \widehat{K}.$

Then $\widehat{\mathfrak{u}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)$ , if and only if $\widehat{\mathfrak{u}}$ is the solution of following inequality:

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\geq \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}), \mathit{\rm{ \forall }}\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.2)

Proof. Let $\widehat{\mathfrak{u}}$ is a solution of problem (3.1), then

$\begin{equation*} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\geq 0. \end{equation*}$

Since $\mathcal{A}$ is relaxed $\eta$ - $\alpha$ monotone, we have

$\begin{eqnarray*} && \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\\ & = &\langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+\langle \mathcal{A}(\widehat{\mathfrak{v}})-\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\\ &\geq& \langle \mathcal{A}(\widehat{\mathfrak{v}})-\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle\geq\alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}), {\rm{ \forall }}\widehat{\mathfrak{v}}\in \widehat{K}. \end{eqnarray*}$

Hence, $\widehat{\mathfrak{u}}$ is the solution of inequality (3.2).

Conversely, let $\widehat{\mathfrak{u}}\in \widehat{K}$ be a solution of problem (3.2) and let $\widehat{\mathfrak{v}}\in \widehat{K}$ be any point $\psi(\widehat{\mathfrak{v}}) < \infty$ . We define $\widehat{\mathfrak{v}}_s = (1-s)\widehat{\mathfrak{u}}+s\widehat{\mathfrak{v}}, \; \; s\in (0, 1),$ then due to convexity of $\widehat{K}$ $\widehat{\mathfrak{v}}_s\in K$ . Since $\widehat{\mathfrak{v}}_s\in \widehat{K}$ is the solution of inequality (3.2), it follows from $(I_1)$ – $(I_3)$

$\begin{eqnarray*} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta(\widehat{\mathfrak{v}}_s, \widehat{\mathfrak{u}})\rangle +\psi(\widehat{\mathfrak{v}}_s)-\psi(\widehat{\mathfrak{u}})&\geq& \alpha(\widehat{\mathfrak{v}}_s-\widehat{\mathfrak{u}})\\ \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta((1-s)\widehat{\mathfrak{u}}+s\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +\psi((1-s)\widehat{\mathfrak{u}}+s\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})&\geq& \alpha((1-s)\widehat{\mathfrak{u}}\\ &&+s\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}})\\ \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), (1-s)\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{u}})+s\eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +(1-s)\psi(\widehat{\mathfrak{u}})+s\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})&\geq& \alpha(s(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}})). \end{eqnarray*}$

Using $(I_3)$ , we have

$\begin{eqnarray*} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), s\eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +s(\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}))&\geq&s^p\alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}})\\ \langle \widehat{\mathfrak{w}}+\mathcal{A}((1-s)\widehat{\mathfrak{u}}+s\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +(\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}))&\geq&s^{p-1}\alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}), \end{eqnarray*}$

letting $s\to 0^{+}$ , we get

$\begin{equation*} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\geq 0, {\rm{ \forall }}\widehat{\mathfrak{v}}\in K. \end{equation*}$

Theorem 3.1. Suppose that the conditions $(I_1)$ – $(I_3)$ are satisfied. Additionally, if the following conditions hold.

$(I_4)$ $\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}})+\eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}) = 0,$

$(I_5)$ $\exists$ $\; \; \widehat{\mathfrak{v}}_0\in \widehat{K}\cap D(\psi)$ such that

$\begin{equation} \underset{\widehat{\mathfrak{u}}\in \widehat{K}, \; \|\widehat{\mathfrak{u}}\|\to \infty}{lim \; inf}\frac{\langle \mathcal{A}(\widehat{\mathfrak{u}})-\mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\rangle+\psi(\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}}_0)}{\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|}\xrightarrow{} +\infty. \end{equation}$

(3.3)

Then, $Sol(K, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi) = \{\widehat{\mathfrak{u}}\in \widehat{K}: \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})\geq0, \mathit{\rm{\forall }\; }\widehat{\mathfrak{v}}\in K\}\neq \phi$ , bounded, closed and convex, for $\widehat{\mathfrak{w}}\in \widehat{B_1}^*.$

Proof. Clearly, $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)\neq \phi$ , as $\widehat{\mathfrak{v}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ , for each $\widehat{\mathfrak{v}}\in \widehat{K}.$

Now, we have to show that $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is bounded. Suppose to contrary that $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is not bounded, then there exists a sequence $\{\widehat{\mathfrak{u}}_n\}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ such that $\|\widehat{\mathfrak{u}}_n\|_{\widehat{B_1}}\to \infty$ as $n\to \infty$ . We can consider, $\forall$ $n\in \mathbb{N}$ , $\|\widehat{\mathfrak{u}}_n\| > n.$ By $\eta$ -coercive condition (3.3), $\exists$ a constant $M > 0$ and a mapping $\kappa:[0, \infty)\to [0, \infty)$ with $\kappa(k)\to \infty$ such that for every $\|\widehat{\mathfrak{u}}\|_{\widehat{B_1}}\geq M,$

$\begin{equation*} \langle \mathcal{A}(\widehat{\mathfrak{u}})-\mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\rangle+\psi(\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}}_0)\geq \kappa(\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}})\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}}. \end{equation*}$

Thus, if $n$ is sufficiently large as $\kappa(n) > (\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|+\|\widehat{\mathfrak{w}}\|),$

$\begin{eqnarray*} 0&\leq& \langle \mathcal{A}(\widehat{\mathfrak{u}}_n)+\widehat{\mathfrak{w}}, \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_n)\rangle+\psi(\widehat{\mathfrak{v}}_0)-\psi(\widehat{\mathfrak{u}}_n)\\ & = &\langle \mathcal{A}(\widehat{\mathfrak{u}}_n), \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_n)\rangle+\langle \widehat{\mathfrak{w}}, \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_n)\rangle+\psi(\widehat{\mathfrak{v}}_0)-\psi(\widehat{\mathfrak{u}}_n)\\ & = &-\langle \mathcal{A}(\widehat{\mathfrak{u}}_n)-\mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}_n, \widehat{\mathfrak{v}}_0)\rangle+\psi(\widehat{\mathfrak{v}}_0)-\psi(\widehat{\mathfrak{u}}_n)+\langle \mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_n)\rangle\\ &&+\langle \widehat{\mathfrak{w}}, \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_n)\rangle\\ &\leq& -\kappa(\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|)\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}}+\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|.\|\eta(\widehat{\mathfrak{u}}_n, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}}+\|\widehat{\mathfrak{w}}\|.\|\eta(\widehat{\mathfrak{u}}_n, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}}\\ & = &\|\eta(\widehat{\mathfrak{u}}_n, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}}\Big[-\kappa(\|\eta(\widehat{\mathfrak{u}}_n, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}})+\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|+\|\widehat{\mathfrak{w}}\|\Big]\\ & < &0. \end{eqnarray*}$

Which is not possible. Thus, $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is bounded.

Now it remains to prove that $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is closed.

Let $\{\widehat{\mathfrak{u}}_n\}$ be a sequence in $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ such that $\widehat{\mathfrak{u}}_n\to \widehat{\mathfrak{u}}\in \widehat{K}.$ Then, $\forall$ $n\in \mathbb{N}$

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{u}}_n), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}_n)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}_n)\geq 0, \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.4)

From Lemmas (3.1) and (3.4) same as

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{u}}_n, \widehat{\mathfrak{v}})\rangle +\psi(\widehat{\mathfrak{u}}_n)-\psi(\widehat{\mathfrak{v}})\geq \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}_n), \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.5)

By using $(I_4)$ , we have

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}_n)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}_n)+ \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}_n)\leq 0, \ {\rm{ \forall }}\ \widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.6)

Which implies that

$\begin{equation} \underset{n\to 0^+}{lim\; sup}\{\langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}_n)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}_n)+ \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}_n)\}\leq 0, {\rm{ \forall }}\; \widehat{\mathfrak{v}}\in \widehat{K}, \end{equation}$

(3.7)

as $\widehat{\mathfrak{u}}\to\langle \mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle$ , $\psi$ and $\alpha$ are lower semicontinuous functions. From (3.7), we have

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}})+ \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}})\leq0, \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}, \end{equation}$

(3.8)

that is,

$\begin{equation} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}})\rangle +\psi(\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}})\geq \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}), \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.9)

By Lemma 3.1, we get $\widehat{\mathfrak{u}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ , that is $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is closed.

Lastly, we show that $Sol(\widehat{K}, \widehat{\mathfrak{w}}+G(.), \; \psi)$ is convex. For any $\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ and $s\in [0, 1],$ let $\widehat{\mathfrak{v}}_s = (1-s)\widehat{\mathfrak{v}}+s\widehat{\mathfrak{u}}$ . Since $\widehat{K}$ is convex, so that $\widehat{\mathfrak{u}}_s\in \widehat{K}$ . Using $(I_3)$ and letting $s\to 0^+$ , we obtain

$\begin{align*} \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta(\widehat{\mathfrak{v}}_s, \widehat{\mathfrak{v}})\rangle+\psi(\widehat{\mathfrak{v}}_s)&-\psi(\widehat{\mathfrak{v}})\rangle\\ & = \langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta((1-s)\widehat{\mathfrak{v}}+s\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}})\rangle +\psi((1-s)\widehat{\mathfrak{v}}+s\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}})\\ &\leq (1-s)\langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{v}})\rangle +s\langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle\\ &+s(\psi(\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}}))\\ &\leq s[\langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}})\rangle+(\psi(\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}}))]\\ &\leq0, \end{align*}$

that is,

$\langle \widehat{\mathfrak{w}}+\mathcal{A}(\widehat{\mathfrak{v}}_s), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{v}}_s)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{v}}_s)\geq 0.$

Hence, $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is convex.

Boundedness of $\widehat{\mathfrak{w}}$ implies that $Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ is bounded.

Theorem 3.2. Suppose that all the conditions and mappings are same as considered in Theorem 3.1. Additionally, $\forall$ $\widehat{\mathfrak{w}}\in \overline{B}(n, \widehat{B_1}^*),$ $\exists$ a constant $M_n > 0,$ depending on $n$ , such that

$\begin{equation} \|\widehat{\mathfrak{u}}\|_{\widehat{B_1}}\leq M_n, \; \mathit{\rm{ \forall }\;}\widehat{\mathfrak{u}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi), \end{equation}$

(3.10)

where $\overline{B}(n, \widehat{B_1}^*) = \{\widehat{\mathfrak{w}}\in \widehat{B_1}^*:\|\widehat{\mathfrak{w}}\|_{\widehat{B_1}^*}\leq n\}.$

Proof. On contrary let us suppose that $\exists$ $N_0 > 0$ and

$\begin{equation*} \underset{\widehat{\mathfrak{w}}\in\overline{B}(N_0, \widehat{B_1}^*)}{Sup}\Big\{\|\widehat{\mathfrak{u}}\|_{\widehat{B_1}}:\widehat{\mathfrak{u}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)\Big\} = +\infty. \end{equation*}$

Therefore, $\exists$ $\widehat{\mathfrak{w}}_{\hat{k}}\in \overline{B}(N_0, \widehat{B_1}^*)$ and $\widehat{\mathfrak{u}}_{\hat{k}}\in Sol(\widehat{K}, \widehat{\mathfrak{w}}+\mathcal{A}(.), \; \psi)$ with $\|\eta(\widehat{\mathfrak{u}}_{\hat{k}}, \widehat{\mathfrak{v}}_0)\| > \hat{k}$ $(\hat{k} = 1, 2, 3, \cdots).$ By $\eta$ -coercivity assumption, $\exists$ a constant $M > 0$ such that $\forall$ $\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|\geq M$ and a function $\kappa:[0, \infty)\to [0, \infty)$ with $\kappa(\hat{k})\to \infty$ as $\hat{k}\to \infty$ , we have

$\begin{equation*} \langle \mathcal{A}(\widehat{\mathfrak{u}}), \eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\rangle+\psi(\widehat{\mathfrak{u}})-\psi(\widehat{\mathfrak{v}}_0)\geq \kappa(\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|)\|\eta(\widehat{\mathfrak{u}}, \widehat{\mathfrak{v}}_0)\|_{\widehat{B_1}}. \end{equation*}$

Thus, for $\hat{k} > M$ sufficiently large such that $\kappa(\hat{k}) > \frac{N_0+\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|}{\hat{k}}$ , one has

$\begin{eqnarray*} 0&\leq& \langle \widehat{\mathfrak{w}}_{\hat{k}}+\mathcal{A}(\widehat{\mathfrak{u}}_{\hat{k}}), \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\rangle+\psi(\widehat{\mathfrak{v}}_0)-\psi(\widehat{\mathfrak{u}}_{\hat{k}})\\ & = & \langle \widehat{\mathfrak{w}}_{\hat{k}}, \eta(\widehat{\mathfrak{v}}_0, u_{\hat{k}})\rangle- \langle \mathcal{A}(\widehat{\mathfrak{u}}_{\hat{k}})-\mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}_{\hat{k}}, \widehat{\mathfrak{v}}_0)\rangle+\langle \mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}_{\hat{k}}, \widehat{\mathfrak{v}}_0)\rangle\\ &&+\psi(\widehat{\mathfrak{v}}_0)-\psi(\widehat{\mathfrak{u}}_{\hat{k}})\\ & = &\langle \widehat{\mathfrak{w}}_{\hat{k}}, \eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\rangle- [\langle \mathcal{A}(\widehat{\mathfrak{u}}_{\hat{k}})-\mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}_{\hat{k}}, \widehat{\mathfrak{v}}_0)\rangle+\psi(\widehat{\mathfrak{u}}_{\hat{k}})-\psi(\widehat{\mathfrak{v}}_0)]\\ &&+\langle \mathcal{A}(\widehat{\mathfrak{v}}_0), \eta(\widehat{\mathfrak{u}}_{\hat{k}}, \widehat{\mathfrak{v}}_0)\rangle\\ & = &\|\widehat{\mathfrak{w}}_{\hat{k}}\|_{\widehat{B_1}^*}\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|-r(\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|)\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|+\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|\\ & = &N_0\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|-r(\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|)\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|+\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|\|\eta(\widehat{\mathfrak{v}}_0, \widehat{\mathfrak{u}}_{\hat{k}})\|\\ &\leq& (N_0+\|\mathcal{A}(\widehat{\mathfrak{v}}_0)\|)\hat{k}-r(\hat{k}) < 0, \end{eqnarray*}$

which is a contradiction. Hence our supposition is wrong.

Let $\widetilde{\mathfrak{g}}:[0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}\rightarrow \widehat{B_1}^*$ be the single valued mapping and a multi-valued mapping $\mathfrak{F}:[0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}\to \Pi(\widehat{K})$ is defined as follows:

$\begin{equation*} \mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x)): = \Big\{\widehat{\mathfrak{u}}\in \widehat{K}: \widehat{\mathfrak{u}}\in Sol(\widehat{K}, \widetilde{\mathfrak{g}}(x, \mathfrak{y}(x), \mathfrak{y}'(x))+\mathcal{A}(.), \; \psi)\Big\}. \end{equation*}$

It follows from Theorem 3.1 that $\mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))$ is nonempty, bounded, closed and convex that is, $\mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))\in \Pi_{bcv}(\widehat{B_1})$ $\forall$ $(x, \mathfrak{y}(x), \mathfrak{y}'(x))\in [0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}.$

Theorem 3.3. Suppose that all the conditions and mappings are same as considered in Theorem 3.1 and the mapping $\widetilde{g}:[0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}\to \widehat{B_1}^*$ is bounded and continuous, then the following assertions hold:

(i) $\mathfrak{F}$ is strongly weakly u.s.c.;

(ii) $x\to \mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))$ is measurable $\forall$ $\mathfrak{y}, \mathfrak{y}'\in \widehat{B_2}$ ;

(iii) for every bounded subset $\Omega^* = \Omega_1\times \Omega_2$ of $C^1\Big([0, \mathcal{T}], \widehat{B_2}\times \widehat{B_2}\Big)$ , $\exists$ a constant $M_{\Omega^*}$ such that

$\begin{equation} \|\mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))\|: = sup\{\|\widehat{\mathfrak{u}}\|_{\widehat{B_1}}: \widehat{\mathfrak{u}}\in \mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))\}\leq M_{\Omega^*}, \mathit{\rm{ \forall }\;}x\in [0, \mathcal{T}]\; \end{equation}$

(3.11)

$\mathit{\mbox{and}}\; (\mathfrak{y}, \mathfrak{y}')\in \Omega^*.$

Proof. (ⅰ) Let $\mathcal{C}\subset \widehat{B_1}$ be any weakly closed subset of $\widehat{B_1}$ , suppose that $\{(x_n, \mathfrak{y}_n, \mathfrak{y'}_n)\} \subset [0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}$ such that $(x_n, \mathfrak{y}_n, {\mathfrak{y}'}_n)\to (x^*, \mathfrak{y}^{*}, \mathfrak{y}^*{'})$ in $[0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}$ with $(x_n, \mathfrak{y}_n, \mathfrak{y}_n{'})\in \mathfrak{F}^{-1}(\mathcal{C}): = \{(x, \mathfrak{y}, \mathfrak{y'})\; |\; \mathcal{C}\cap \mathfrak{F}(x, \mathfrak{y}, \mathfrak{y}')\neq \phi\}.$ Therefore, for any $n\in\mathbb{N}$ , there exists $\widehat{\mathfrak{u}}_n\in \mathcal{C}\cap \mathfrak{F}(x_n, \mathfrak{y}_n, \mathfrak{y}_n{'})$ such that

$\begin{equation} \langle \widetilde{\mathfrak{g}}(x_n, \mathfrak{y}_n, \mathfrak{y}_n{'})+\mathcal{A}(\widehat{\mathfrak{u}}_n), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}_n)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}_n)\geq 0, {\rm{ \forall }}\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.12)

By Lemma 3.1, (3.12) is equivalent to

$\begin{equation} \langle \widetilde{\mathfrak{g}}(x_n, \mathfrak{y}_n, \mathfrak{y}_n{'})+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}_n)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{v}}_n)\geq \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}_n), \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.13)

Which implies that,

$\begin{eqnarray} \underset{n\to 0^+}{lim\; sup}\{\langle \widetilde{\mathfrak{g}}(x_n, \mathfrak{y}_n, \mathfrak{y}_n{'})+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}_n)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{v}}_n)\}\geq \underset{n\to 0^+}{lim\; sup}\{ \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}_n)\}, \\ \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}.\; \; \; \; \; \; \; \end{eqnarray}$

(3.14)

Since $\widetilde{\mathfrak{g}}$ is continuous. Therefore, by Theorem 3.3, it implies that $\{\widehat{\mathfrak{u}}_n\}$ is bounded. Hence, by reflexivity of $\widehat{B_1},$ we can suppose that $\widehat{\mathfrak{u}}_n\to \widehat{\mathfrak{u}}^*\in \mathcal{C}$ in $\widehat{B_1}.$

From (3.14), we get

$\begin{equation} \langle \widetilde{\mathfrak{g}}(x^*, \mathfrak{y}^{*}, \mathfrak{y}^*{'})+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}^*)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}^*)\geq \alpha(\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}^*), \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation}$

(3.15)

Using Lemma 3.1, we have

$\begin{equation*} \langle \widetilde{\mathfrak{g}}(x^*, \mathfrak{y}^{*}, \mathfrak{y}^*{'})+\mathcal{A}(\widehat{\mathfrak{v}}), \eta(\widehat{\mathfrak{v}}, \widehat{\mathfrak{u}}^*)\rangle +\psi(\widehat{\mathfrak{v}})-\psi(\widehat{\mathfrak{u}}^*)\geq 0, \; \ {\rm{ \forall }}\;\widehat{\mathfrak{v}}\in \widehat{K}. \end{equation*}$

It follows from weakly closeness of $\mathcal{C}$ that

$\begin{equation*} (x^*, \mathfrak{y}^{*}, \mathfrak{y}^*{'})\in \mathfrak{F}^{-1}(\mathcal{C}): = \{(x, \mathfrak{y}, \mathfrak{y}'):\mathcal{C}\cap \mathfrak{F}(x, \mathfrak{y}, \mathfrak{y}')\neq \phi\}. \end{equation*}$

Hence, $\mathfrak{F}$ is strongly weakly u.s.c..

(ⅱ) Define a set

$L_\lambda: = \{x\in [0, \mathcal{T}]; d(v, \mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))) > \lambda\}, \; { \forall }\; (\mathfrak{y}, \mathfrak{y}')\in \widehat{B_2}\times \widehat{B_2}, \; \widehat{\mathfrak{v}}\in \widehat{B_1}.$

Now we will show that $L_\lambda$ is an open set for all $\lambda\geq 0.$ For this let $\{x_n\} \subset (L_\lambda)^c = [0, \mathcal{T}]\setminus L_\lambda$ be a sequence with $x_n\to x.$ Then $\forall$ $n\in \mathbb{N}$ , we have $d(v, \mathfrak{F}(x_n, \mathfrak{y}, \mathfrak{y}'))\leq \lambda$ . As for every $(x, \mathfrak{y}, \mathfrak{y}')\in [0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2},$ the multi-valued mapping $\mathfrak{F}(x, \mathfrak{y}, \mathfrak{y}')$ is bounded, closed and convex by Theorem 3.1, we get $\forall$ $n\in \mathbb{N},$ $\widehat{\mathfrak{u}}_n\in \mathfrak{F}(x_n, \mathfrak{y}, \mathfrak{y}')$ such that $\|\widehat{\mathfrak{v}}-\widehat{\mathfrak{u}}_n\|\leq \lambda.$ By Theorem 3.3, $\{\widehat{\mathfrak{u}}_n\}$ is bounded, so we may assume that $\widehat{\mathfrak{u}}_n\rightharpoonup \widehat{\mathfrak{u}}\in \widehat{K}.$ By $(i)$ , $\widehat{\mathfrak{u}}\in \mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))$ . Hence, we obtain

$\begin{equation*} d(v, \mathfrak{F}(x, \mathfrak{y}, \mathfrak{y}'))\leq \|\widehat{\mathfrak{u}}-\widehat{\mathfrak{v}}\|_{\widehat{B_1}} = \underset{n\to \infty}{\liminf}\|\widehat{\mathfrak{u}}_n-\widehat{\mathfrak{v}}\|_{\widehat{B_1}}\leq \lambda, \end{equation*}$

that is $x\in (L_\lambda)^c$ , thus $[0, \mathfrak{F}]\setminus L_\lambda$ is closed. Hence, $L_\lambda$ is open, consequently $L_\lambda$ is measurable. By [,Proposition 6.2.4], the mapping $x\mapsto \mathfrak{F}(x, \mathfrak{y}, \mathfrak{y}')$ is measurable $\forall$ $(\mathfrak{y}, \mathfrak{y}')\in \widehat{B_2}\times \widehat{B_2}$ .

(ⅲ) As $\widetilde{\mathfrak{g}}$ is bounded. Therefore

$\widetilde{\mathfrak{g}}_{\Omega^*}: = \{\widetilde{\mathfrak{g}}(x, \mathfrak{y}(x), \mathfrak{y}'(x)):\; x\in [0, \mathcal{T}] \mbox{ and } (\mathfrak{y}, \mathfrak{y}')\in \Omega^*\},$

is also bounded in $\widehat{B_1}$ for every bounded subset $\Omega^*$ of $C^1\Big([0, \mathcal{T}], \widehat{B_2}\times \widehat{B_2}\Big)$ . Then, by Theorem 3.3, $\mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))$ is bounded, $\forall\; x\in [0, \mathcal{T}]$ and $(\mathfrak{y}, \mathfrak{y}')\in\Omega^*$ . Hence, $\exists$ a constant $M_{\Omega^*} > 0$ such that 3.11 holds.

4. Main results

Before proving our main result, we mention that by Theorem 3.3, $\mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))$ is measurable and $\widehat{B_1}$ is a separable Banach space. Hence, by [,Theorem 3.17] $\mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x))$ possess a measurable selection $\xi$ such that $\xi\in L^\infty \Big([0, \mathcal{T}]; \widehat{B_1}\Big)\subset L^2\Big([0, \mathcal{T}], \widehat{B_1}\Big)$ $\forall$ $(\mathfrak{y}, \mathfrak{y}')\in C^1\Big([0, \mathcal{T}], \widehat{B_2}\times \widehat{B_2}\Big).$ So

$\begin{equation} P_\mathfrak{F}(\mathfrak{y}, \mathfrak{y}'): = \Big\{\xi\in L^2([0, \mathcal{T}], \widehat{B_1})\; |\; \xi(t)\in \mathfrak{F}(x, \mathfrak{y}(x), \mathfrak{y}'(x)), \; \; a.e., \; x\in [0, \mathcal{T}]\Big\}, \end{equation}$

(4.1)

is well defined $\forall$ $(\mathfrak{y}, \mathfrak{y}')\in C^1([0, \mathcal{T}], \widehat{B_2}\times \widehat{B_2}).$

Lemma 4.1. Suppose that $(I_1)-(I_4)$ hold and $\widetilde{g}:[0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}\to \widehat{B_1}^*$ is bounded and continuous. Then, multi-valued mapping $P_\mathfrak{F}$ is strongly upper semicontinuous.

Proof. Let $\{\mathfrak{y}_n, \mathfrak{y}_n'\}\subset C^1([0, \mathcal{T}], \widehat{B_2}\times \widehat{B_2})$ with $(\mathfrak{y}_n, \; \mathfrak{y}_n')\to (\mathfrak{y}_0, \mathfrak{y}_0')$ in $C^1([0, \mathcal{T}], \widehat{B_2}\times \widehat{B_2})$ and $\xi_n\in P_\mathfrak{F}(\mathfrak{y}_n, \mathfrak{y}_n')$ for $n\in \mathbb{N}$ . Now, we need to prove that $\exists$ a subsequence of $\{\xi_n\}$ , such that $\xi_n\to\xi_0\in P_\mathfrak{F}(\mathfrak{y}_0, \mathfrak{y}_0')$ .

Indeed, $(I_5)$ confirms that the sequence $\{\xi_n\}$ is bounded in $L^2([0, \mathcal{T}], \widehat{B_1})$ . Therefore, we can suppose $\xi_n\to \xi_0$ weakly in $L^2([0, T], \widehat{B_1})$ . By Lemma 2.1, there is $\{\xi\}$ , a finite combination of the $\{\xi_i: i\geq n\}$ with $\bar{\xi_n}\to \xi_0$ converges strongly in $L^2([0, \mathcal{T}], \widehat{B_1}).$

Since $\mathfrak{F}$ is strongly weakly upper semicontinuous and $(\mathfrak{y}_n, \mathfrak{y}_n')\to (\mathfrak{y}_0, \mathfrak{y}_0')\in C^1([0, \mathcal{T}], \widehat{B_2}),$ therefore for every weak neighborhood $\mathcal{Y}_x$ of $\mathfrak{F}(x, \mathfrak{y}_0(x), \mathfrak{y}_0'(x))$ there exists a strong neighborhood

$\mathfrak{F}(x, \mathfrak{y}, \mathfrak{y}')\subset \mathcal{Y}_x, \; \; \; { \forall }\; (\mathfrak{y}, \mathfrak{y}')\in \mathcal{U}_x.$

Which shows that $\xi \in P_\mathfrak{F}(\mathfrak{y}_0, \mathfrak{y}_0').$ Thus, by Proposition 2.3, $P_\mathfrak{F}$ is strongly upper semi continuous.

We also need the following assumptions for achieving the goal.

$(I_6)$ $\widetilde{\mathfrak{g}}: [0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}\to \widehat{B_1}^*$ is continuous and bounded;

$(I_7)$ $\widetilde{\mathcal{F}}(., \mathfrak{y}, .):[0, \mathcal{T}]\to \mathcal{L}(\widehat{B_1}, \widehat{B_2})$ , $\widetilde{\mathcal{F}}(., ., \mathfrak{y}'):[0, \mathcal{T}]\to \mathcal{L}(\widehat{B_1}, \widehat{B_2})$ are measurable for all $\mathfrak{y}, \mathfrak{y}'\in \widehat{B_2}$ and $\widetilde{\mathcal{F}}(x, ., .):\widehat{B_2}\to \mathcal{L}(\widehat{B_2}, \widehat{B_1})$ is continuous for a.e. $x\in [0, \mathcal{T}],$ where $\mathcal{L}(\widehat{B_1}, \widehat{B_2})$ denotes the class of bounded linear operators from $\widehat{B_1}$ to $\widehat{B_2}$ , and there exists $\rho_{\widetilde{\mathcal{F}}}\in L^2([0, \mathcal{T}], \mathbb{R_+})$ and a non-decreasing continuous mapping $\gamma_{\widetilde{\mathcal{F}}}:[0, \infty)\to [0, \infty)$ such that

$\|\widetilde{\mathcal{F}}(x, \mathfrak{y}(x), \mathfrak{y}'(x))\|\leq \rho_{\widetilde{\mathcal{F}}}(x)\gamma_{\widetilde{\mathcal{F}}}\Big(\|\mathfrak{y}(x)\|_{\widehat{B_2}}+\|\mathfrak{y}'(x)\|_{\widehat{B_2}}\Big), \ {\rm{ \forall }}\;(x, \mathfrak{y}, \mathfrak{y}')\in [0, \mathcal{T}]\times \widehat{B_2}\times \widehat{B_2}.$

$(I_8)$ $\widetilde{\mathfrak{f}}(., \mathfrak{y}, .), \; \widetilde{\mathfrak{f}}(., ., \mathfrak{y}'):[0, \mathcal{T}]\to \widehat{B_2}$ are measurable for all $\mathfrak{y}, \mathfrak{y}'\in \widehat{B_2}$ and there exists $\rho_{\widetilde{\mathfrak{f}}}\in L^2\Big([0, \mathcal{T}], \mathbb{R_+}\Big)$ such that for $x\in [0, \mathcal{T}]$ $\widetilde{\mathfrak{f}}(x, ., .):\widehat{B_2}\to \widehat{B_2}$ satisfies

$\begin{equation} \begin{cases} \|\widetilde{\mathfrak{f}}(x, \mathfrak{y}, \mathcal{y})-\widetilde{\mathfrak{f}}(x, \mathfrak{y}', \mathcal{y})\|\leq \rho_{\widetilde{\mathfrak{f}}}(x)\|\mathfrak{y}-\mathfrak{y}'\|_{\widehat{B_2}}, \cr \|\widetilde{\mathfrak{f}}(x, \mathfrak{y}, \mathcal{y})-\widetilde{\mathfrak{f}}(x, \mathfrak{y}, \mathcal{y}')\|\leq \rho_{\widetilde{\mathfrak{f}}}(x)\|\mathcal{y}-\mathcal{y}'\|_{\widehat{B_2}}, \cr \|\widetilde{\mathfrak{f}}(x, 0, 0)\|\leq \rho_{\widetilde{\mathfrak{f}}}(x). \end{cases} \end{equation}$

(4.2)

The following result ensures the existence of solution of problem (2.1).

Theorem 4.1. Under the assumptions $(I_1)$ – $(I_8)$ , if the following inequalities hold

$\begin{equation} \underset{\hat{k}\to \infty}{lim\; inf}\Big[\frac{\gamma_{\widetilde{\mathcal{F}}}(\hat{k})}{\hat{k}}\|\rho_{\widetilde{\mathcal{F}}}(x)\|M_{\|\widetilde{\mathfrak{g}}\|}+\|\rho_{\widetilde{\mathfrak{f}}}(x)\|_{L^2}+\frac{\|\mathfrak{y}_0\|+\|y_0\|}{\hat{k}\mathcal{T}^{1/2}}\Big] < \frac{1}{\delta \mathcal{T}^{1/2}}, \end{equation}$

(4.3)

$\begin{equation} \|Q(x_1)-Q(x_2)\|\leq \|x_1-x_2\|\; \mathit{\mbox{and}}\; \|R(x_1)-R(x_2)\|\leq \|x_1-x_2\|, \end{equation}$

(4.4)

where

$\begin{equation*} \delta = max\Big\{\underset{x\in J}{sup}\|Q(x)\|_{L(\widehat{B_2})}, \; \; \underset{x\in J}{sup}\|R(x)\|_{L(\widehat{B_2})}\Big\}, \end{equation*}$

and $M_{\|\widetilde{\mathfrak{g}}\|} > 0$ is a constant stated in Theorem 3.2, then, the problem (2.1) has at least one mild solution $(\mathfrak{y}, \widehat{\mathfrak{u}}).$

Proof. We define the multi-valued mapping $\Gamma: C^1([0, \mathcal{T}], \widehat{B_2})\to \Pi(C^1([0, \mathcal{T}], \widehat{B_2}))$ such that

$\begin{eqnarray} \Gamma(\mathfrak{y}): = \Big\{y\in C^1([0, \mathcal{T}], \widehat{B_2})\Big{|}\; y(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)\Big[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\\ +\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\Big]dp, \; x\in [0, \mathcal{T}], \; \xi\in P_\mathfrak{F}(x)\Big\}, \\ \end{eqnarray}$

(4.5)

where $P_\mathfrak{F}$ is defined in (4.1). Our aim is to show that $Fix(\Gamma)\neq \phi$ .

Step-Ⅰ. $\Gamma(\mathfrak{y})\in \Pi_{cv}\Big(C^1([0, \mathcal{T}], \widehat{B_2})\Big)$ for each $\mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}).$

Clearly, $\Gamma(\mathfrak{y})$ is convex for every $\mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2})$ due to the convexity of $P_\mathfrak{F}(\mathfrak{y}).$

Since for each $y\in \Gamma(\mathfrak{y}),$ we can choose $\xi\in P_\mathfrak{F}(\mathfrak{y})$ such that

$\begin{eqnarray*} y(x)& = &Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\\ &&+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp, \end{eqnarray*}$

which implies that,

$\begin{eqnarray*} \|y(x)\|&\leq& \|\mathfrak{y}_0Q(x)\|+\|y_0R(x)\|+\Big\|\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\\ &&+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\Big\|\\ &\leq& \delta \|\mathfrak{y}_0\|+\delta \|y_0\|+\delta \Big[\int_{0}^{x}\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\xi(p)\|dp\\ &&+\int_{0}^{x}\|\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|dp\Big]. \end{eqnarray*}$

Using $(I_7)$ and $(I_8)$ and applying Hölder's inequality,

$\begin{eqnarray*} \|y(x)\|&\leq& \delta \|\mathfrak{y}_0\|+\delta \|y_0\|+\delta \Big[\int_{0}^{x}\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\xi(p)\|dp\\ &&+\int_{0}^{x}\|\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|dt\Big], \\ & = &\delta(\|\mathfrak{y}_0\|+\|y_0\|)+\delta \Big[\int_{0}^{x}\rho_{\widetilde{\mathcal{F}}}(p)\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}\|+\|\mathfrak{y}'\|)M_{\|\widetilde{\mathfrak{g}}\|}dp\\ &&+\int_{0}^{x}\rho_{\widetilde{\mathfrak{f}}}(p)(1+\|\mathfrak{y}\|+\|\mathfrak{y}'\|)dp\Big]\\ & = &\delta(\|\mathfrak{y}_0\|+\|y_0\|)+\delta \gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}\|+\|\mathfrak{y}'\|)M_{\|\widetilde{\mathfrak{g}}\|}\int_{0}^{x}\rho_{\widetilde{\mathcal{F}}}(p)dp\\ &&+\delta(1+\|\mathfrak{y}\|+\|\mathfrak{y}'\|)\int_{0}^{x}\rho_{\widetilde{\mathfrak{f}}}(p)dp, \\ & = &\delta\Big(\|\mathfrak{y}_0\|+\|y_0\|+\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}\|+\|\mathfrak{y}'\|)M_{\|\widetilde{\mathfrak{g}}\|}\|\rho_{\widetilde{\mathcal{F}}}\|\mathcal{T}^{1/2}\\ &&+(1+\|\mathfrak{y}\|+\|\mathfrak{y}'\|)\|\rho_{\widetilde{\mathfrak{f}}}\|\mathcal{T}^{1/2}\Big)\\ & = &\delta \mathcal{T}^{1/2}\Big[\frac{\|\mathfrak{y}_0\|+\|y_0\|}{\mathcal{T}^{1/2}}+\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}\|+\|\mathfrak{y}'\|)M_{\|\widetilde{\mathfrak{g}}\|}\|\rho_{\widetilde{\mathcal{F}}}\|\\ &&+(1+\|\mathfrak{y}\|+\|\mathfrak{y}'\|)\|\rho_{\widetilde{\mathfrak{f}}}\|\Big]. \end{eqnarray*}$

Hence, $\Gamma(\mathfrak{y})$ is bounded in $C^1([0, \mathcal{T}], \widehat{B_2})$ for each $\mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}).$

Next we shall prove that $\Gamma(\mathfrak{y})$ is a collection of equicontinuous mappings $\forall\; \mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}).$

$\begin{eqnarray} \|y(x_2)-y(x_1)\|_{\widehat{B_2}}& = &\Big\|\mathfrak{y}_0Q(x_2)+y_0R(x_2)+\int_{0}^{x_2}R(x_2-p)\Big[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\nonumber\\ &&+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\Big]dp-\mathfrak{y}_0Q(x_1)-y_0R(x_1)\nonumber\\ &&-\int_{0}^{x_1}R(x_1-t)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\Big\|_{\widehat{B_2}} \\ &\leq&\|\mathfrak{y}_0\|\|Q(x_2)-Q(x_1)\|+\|y_0\|\|R(x_2)-R(x_1)\|\\ &&+\Big\|\int_{0}^{x_2}R(x_2-t)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\\ &&-\int_{0}^{x_1}R(x_1-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\\ &&+\int_{0}^{x_1}R(x_2-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\\ &&-\int_{0}^{x_1}R(x_2-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\Big\|\\ &\leq& \|\mathfrak{y}_0\|\|x_2-x_1\|+\|y_0\|\|x_2-x_1\|+\int_{x_1}^{x_2}\Big\|R(x_2-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \\ &&\mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\Big\|+\int_{0}^{x_1}\Big\|(R(x_2-p) \\ &&-R(x_1-p))[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]\Big\|dp\\ & = &(\|\mathfrak{y}_0\|+\|y_0\|)\|x_2-x_1\|+I_1+I_2, \end{eqnarray}$

(4.6)

$\begin{eqnarray} \label{eq4.6} \text{where} \quad I_1& = &\int_{x_1}^{x_2}\Big\|R(x_2-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp\Big\|, \\ \text{and}\ \ \ \ \quad I_2& = &\int_{0}^{x_1}\Big\|(R(x_2-p)-R(x_1-p))[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\\ &&+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]\Big\|dp. \end{eqnarray}$

Applying Hölder's inequality, we have

$\begin{eqnarray} I_1&\leq& \int_{x_1}^{x_2}\|R(x_2-p)\|\|[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\|dp\\ &&+\int_{x_1}^{x_2}\|R(x_2-p)\|\|\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]\|dp\\ &\leq& \int_{x_1}^{x_2}\delta M_{\|\widetilde{\mathfrak{g}}\|}\rho_{\widetilde{\mathcal{F}}}(p)\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)dp\\ &&+\int_{x_1}^{x_2}\delta\gamma_{\widetilde{\mathfrak{g}}}(1+\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)dp, \\ & = &\delta M_{\|\widetilde{\mathfrak{g}}\|}\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)\|\rho_{\widetilde{\mathcal{F}}}(p)\|(x_2-x_1)^{1/2}\\ &&+\delta \gamma_{\widetilde{\mathfrak{f}}}(1+\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)(x_2-x_1)^{1/2}\\ & = &\delta (x_2-x_1)^{1/2}\Big[M_{\|\widetilde{\mathfrak{g}}\|}\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p))\|\rho_{\widetilde{\mathcal{F}}}(p)\|\\ &&+\gamma_{\widetilde{\mathfrak{f}}}(1+\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)\Big]\rightarrow 0\text{ as } x_1\to x_2. \end{eqnarray}$

(4.7)

Further by Proposition 2.2 and (4.4) and Hölder's inequality, we have

$\begin{eqnarray} I_2& = &\int_{0}^{x_1}\Big\|(R(x_2-p)-R(x_1-p))[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)\\ &&+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]\Big\|dp\\ & = &\Big\|\int_{0}^{x_1}[R(p)(Q(x_2)-Q(x_1))+Q(p)(R(x_1)-R(x_2))]\\ &&\times[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]\Big\|dp\\ &\leq& \int_{0}^{x_1}\|R(p)\|\|Q(x_1)-Q(x_2)\|\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\xi(p)\\ &&+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|dp+\int_{0}^{x_1}\|Q(p)\|\|R(x_1)-R(x_2)\|\\ &&\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\xi(p)+\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|dp\\ &\leq& \int_{0}^{x_1}\delta\|x_1-x_2\|\Big[\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|\|\xi(p)\|\\ &&+\|\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|\Big]dp+\int_{0}^{x_1}\delta \|x_1-x_2\|\\ &&\times[\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|+\|\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}'(p))\|]dp\\ &\leq&2\delta(\|x_1-x_2\|)\int_{0}^{x_1}[M_{\|\widetilde{\mathfrak{g}}\|}\rho_{\widetilde{\mathcal{F}}}(p)\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)\\ &&+\rho_{\widetilde{\mathfrak{f}}}(p)\gamma_{\widetilde{\mathfrak{f}}}(1+\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)]dp, \\ &\leq&2\delta\|x_1-x_2\|\Big[M_{\|\widetilde{\mathfrak{g}}\|}\|\rho_{\widetilde{\mathcal{F}}}(p)\|\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)\\ &&+\gamma_{\widetilde{\mathcal{F}}}(1+\|\mathfrak{y}(p)\|+\|\mathfrak{y}'(p)\|)\Big]x^{1/2}\to0\text{ as } x_1\to x_2. \end{eqnarray}$

(4.8)

From (4.6)–(4.8), we have

$\begin{equation*} \|y(x_2)-y(x_1)\|_{\widehat{B_2}}\longrightarrow 0, \; \mbox{as}\; \; x_1\to x_2. \end{equation*}$

Hence, $\Gamma(\mathfrak{y})$ is equicontinuous, $\forall\; \mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}).$ By Arzela-Ascoli theorem ^[34], we obtained that $\Gamma(\mathfrak{y})$ is relatively compact $\forall\; \mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}).$

Now, we have to check that $\Gamma(\mathfrak{y})$ is closed in $C^1([0, \mathcal{T}], \widehat{B_2})$ $\forall\; \mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}).$

Let $\{y_n\}\subset \Gamma(\mathfrak{y})$ is a sequence with $y_n\to y^*$ in $C^1([0, \mathcal{T}]; \widehat{B_2})$ as $n\to \infty$ . Hence, there exist a sequence $\{\xi_n\}\subset P_\mathfrak{F}(\mathfrak{y})$ such that

$\begin{equation*} y_n(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi_n(p) +\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp, \end{equation*}$

$x\in [0, \mathcal{T}].$ By $(iii)$ of Theorem 3.3, it follows that the sequence $\{\xi_n\}$ is weakly relatively compact. Since $P_\mathfrak{F}(\mathfrak{y})$ is upper semicontinuity (see Lemma 4.1), we may assume $\xi_n\to \xi^*\in P_\mathfrak{F}(\mathfrak{y})$ in $L^2([0, \mathcal{T}], \widehat{B_1})$ , where $\xi^*\in P_\mathfrak{F}(\mathfrak{y})$ . On the other hand, by strongly continuity of $Q(x)$ and $R(x)$ for $x > 0$ , we have

$\begin{equation*} y^*(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))\xi^*(p) +\widetilde{\mathfrak{f}}(p, \mathfrak{y}(p), \mathfrak{y}{'}(p))]dp, \end{equation*}$

$x\in [0, \mathcal{T}].$ Which implies that $y^*\in \Gamma(\mathfrak{y})$ , that is $\Gamma(\mathfrak{y})\in \Pi_{cv}(C^1[0, \mathcal{T}], \widehat{B_2})$ .

Step-Ⅱ. The multi-valued mapping $\Gamma$ is closed.

For this assume $\mathfrak{y}_n\to \mathfrak{y}^*$ and $y_n\to y^*$ in $C^1([0, \mathcal{T}], \widehat{B_2})$ with $y_n\in \Gamma(\mathfrak{y}_n)$ $\forall \; n\in \mathbb{N}.$ We need to prove that $y^*\in \Gamma(\mathfrak{y}^*).$ From the definition of multi-valued map $\Gamma$ , we may take $\xi_n\in P_\mathfrak{F}(\mathfrak{y}_n)$ $\forall\; n\in \mathbb{N}$ such that

$\begin{eqnarray} y_n(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}_n(p), \mathfrak{y}_n{'}(p))\xi_n(p) +\widetilde{\mathfrak{f}}(p, \mathfrak{y}_n(p), \mathfrak{y}_n{'}(p))]dp, \\ x\in [0, \mathcal{T}].\\ \end{eqnarray}$

(4.9)

With the help of Theorem 3.3 and Lemma 4.1, we may consider that $\xi_n\rightharpoonup \xi^*\in P_\mathfrak{F}(\mathfrak{y}^*)$ . By using, $I_8$ we get that $\widetilde{\mathfrak{f}}(., \mathfrak{y}_n(.), \mathfrak{y}_n'(.))\to \widetilde{\mathfrak{f}}(., \mathfrak{y}^*, \mathfrak{y}^*{'})$ in $L^2([0, \mathcal{T}], \widehat{B_2})$ .

By using the continuity of $\widetilde{\mathcal{F}}(x, ., .)$ and strongly continuity of $Q(x)$ , $R(x)$ for $x > 0$ , we obtain from (4.9) that

$\begin{eqnarray*} y^*(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}^*(p), \mathfrak{y}^*{'}(p))\xi^*(p) +\widetilde{\mathfrak{f}}(p, \mathfrak{y}^*(p), \mathfrak{y}^*{'}(p))]dp, \\ x\in [0, \mathcal{T}], \end{eqnarray*}$

and $\xi^*\in P_\mathfrak{F}(\mathfrak{y})$ . Thus $\mathfrak{y}^*\in \Gamma(\mathfrak{y}^*).$

Step-Ⅲ. $\Gamma$ is $\chi_T$ condensing.

Let $\mathcal{D}\subset \Pi_b(C^1([0, \mathcal{T}], \widehat{B_2})).$ Therefore, $\mathcal{D}$ is not relatively compact subset of $C^1([0, \mathcal{T}], \widehat{B_2}).$ For $\mathcal{D}$ , we need to prove that $\chi_\mathcal{T}(\mathcal{(D)})\nleq \chi_\mathcal{T}(\Gamma(\mathcal{(D)})).$ Since $\mathcal{D}$ is bounded subset of $C^1([0, \mathcal{T}], \widehat{B_2})$ , then by applying the same technique as in Step-I, we may prove that $\Gamma(\mathcal{(D)})$ is relatively compact, that is, $\chi_\mathcal{T}(\mathcal{D}) = 0.$ Hence, $\chi_\mathcal{T}(\mathcal{(D)})\leq \chi_\mathcal{T}(\Gamma(\mathcal{D})) = 0$ implies that $\mathcal{D}$ is relatively compact by regularity of $\chi_T$ , we conclude that $\Gamma$ is $\chi_\mathcal{T}$ -condensing.

Step-Ⅳ. $\exists$ a constant $M_\mathfrak{R} > 0$ such that

$\begin{equation} \Gamma(\bar{B}_{M_\mathfrak{R}}\subset \bar{B}{M_\mathfrak{R}}): = \{\mathfrak{y}\in C^1([0, \mathcal{T}], \widehat{B_2}):\|\mathfrak{y}\|_C\leq M_\mathfrak{R}\}\subset C^1([0, \mathcal{T}], \widehat{B_2}). \end{equation}$

(4.10)

Let us assume that $\forall$ $k > 0,$ $\exists$ two sequences $\{\mathfrak{y}_k\}$ and $\{y_k\}$ such that

$\|\mathfrak{y}_k\|_{C^1([0, \mathcal{T}], \widehat{B_2})}, \; \|\mathfrak{y}_k^{'}\|_{C^1([0, \mathcal{T}], \widehat{B_2})}\leq k/2$ and $y_k\in \Gamma(\mathfrak{y}_k)$ such that $\|y_k\| > 0$ . Hence, there is $\xi_k\in P_\mathfrak{F}(\mathfrak{y}_k)$ such that

$\begin{eqnarray*} y_k(x) = Q(x)\mathfrak{y}_0+R(x)y_0+\int_{0}^{x}R(x-p)[\widetilde{\mathcal{F}}(p, \mathfrak{y}_k(p), \mathfrak{y}_k{'}(p))\xi_k(p) +\widetilde{\mathfrak{f}}(p, \mathfrak{y}_k(p), \mathfrak{y}_k{'}(p))]dp, \\ \; \; x\in [0, \mathcal{T}]. \end{eqnarray*}$

Using Hölder's inequality, for every $x\in [0, \mathcal{T}]$ , we have

$\begin{eqnarray*} \|\mathfrak{y}_k(x)\|&\leq& \|Q(x)\|\|\mathfrak{y}_0\|+\|R(x)\|\|y_0\|\\ && +\int_{0}^{x}\|R(x-p)\|\|[\widetilde{\mathcal{F}}(p, \mathfrak{y}_k(p), \mathfrak{y}_k{'}(p))\xi_k(p) +\widetilde{\mathfrak{f}}(p, \mathfrak{y}_k(p), \mathfrak{y}_k{'}(p))]\|dp, \\ & = &\delta(\|\mathfrak{y}_0\|+\|y_0\|)+\int_{0}^{x}\delta[\gamma_{\widetilde{\mathcal{F}}}(\|\mathfrak{y}_k\|+\|\mathfrak{y}_k{'}\|)\rho_{\widetilde{\mathcal{F}}}(p)M_{\|\widetilde{\mathfrak{f}}\|}]dp\\ &&+\int_{0}^{x}\delta\gamma_{\widetilde{\mathfrak{f}}}(1+\|\mathfrak{y}_k\|+\|\mathfrak{y}_k^{'}\|)\rho_{\widetilde{\mathfrak{f}}}(p)dp\\ &\leq&\delta(\|\mathfrak{y}_0\|+\|y_0\|)+\int_{0}^{x}\delta[\gamma_{\widetilde{\mathcal{F}}}(k)\rho_{\widetilde{\mathcal{F}}}(p)M_{\|\widetilde{\mathfrak{g}}\|}]dp\\ &&+\int_{0}^{x}\delta\gamma_{\widetilde{\mathfrak{f}}}(1+k)\rho_{\widetilde{\mathfrak{f}}}(p)dp\\ &\leq&\delta(\|\mathfrak{y}_0\|+\|y_0\|)+\delta\gamma_{\widetilde{\mathcal{F}}}(k)\|\rho_{\widetilde{\mathcal{F}}}(x)\|M_{\|\widetilde{\mathfrak{g}}\|}\mathcal{T}^{1/2}\\ &&+\delta\gamma_{\widetilde{\mathfrak{f}}}(1+\kappa)\|\rho_{\widetilde{\mathfrak{f}}}(x)\|\mathcal{T}^{1/2}\\ & = &\delta \mathcal{T}^{1/2}\Big[\gamma_{\widetilde{\mathcal{F}}}(k)\|\rho_{\widetilde{\mathcal{F}}}(x)\|M_{\|\widetilde{\mathfrak{g}}\|}+\gamma_{\widetilde{\mathfrak{f}}}(1+k)\|\rho_{\widetilde{\mathfrak{g}}}(x)\|+\frac{\|\mathfrak{y}_0\|+\|y_0\|}{\mathcal{T}^{1/2}}\Big], \end{eqnarray*}$

we obtain by using (4.9),

$\begin{eqnarray*} 1&\leq& \underset{k\to \infty}{lim\; inf}\frac{\|y_k\|_{C^1([0, \mathcal{T}], \widehat{B_2})}}{k}\\ &\leq&\underset{k\to \infty}{lim\; inf}\Big[\gamma_{\widetilde{\mathcal{F}}}(k)\|\rho_{\widetilde{\mathcal{F}}}(x)\|M_{\|\widetilde{\mathfrak{g}}\|}\mathcal{T}^{1/2}+\gamma_{\widetilde{\mathfrak{f}}}(1+k)\|\rho_{\widetilde{\mathfrak{f}}}(x)\|\mathcal{T}^{1/2}+(\|\mathfrak{y}_0\|+\|y_0\|)\Big]\\ & < &1, \end{eqnarray*}$

which is a contradiction. Therefore there exists $M_\mathfrak{R}$ such that (4.10) holds.

Thus, all requirements of Theorem 2.1 are fulfilled. This implies that $Fix\Gamma\neq \phi$ in $\overline{B}_{M_\mathfrak{R}}.$ Therefore, (SOEPDVLI) has at least one mild solution $(\mathfrak{y}, \widehat{\mathfrak{u}}).$

5. Conclusions

In this paper, a second order evolutionary partial differential variational-like inequality problem is introduced and studied in a Banach space, which is much more general than the considered by Liu-Migórski-Zeng ^[14], Li-Huang-O'Regan ^[13] and Wang-Li-Li-Huang ^[33] etc. We investigate suitable conditions to prove an existence theorem for our problem by using the theory of strongly continuous cosine family of bounded linear operator, fixed point theorem for condensing set-valued mapping and the theory of measure of non-compactness.

Acknowledgments

The authors are highly thankful to anonymous referees and the editor for their valuable suggestions and comments which improve the manuscript a lot.

Conflict of interest

The authors declare that they have no conflicts of interest.

Ethical approval

Patient gave written informed consent for publication of this case report. The case report including the electrocardiogram and echocardiogram images were de-identified to protect patient's privacy and maintain confidentiality.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	Kung AWC (2006) Thyrotoxic periodic paralysis: A diagnostic challenge. J Clin Endocrinol Metab 91: 2490-2495. https://doi.org/10.1210/JC.2006-0356
[2]	Taylor PN, Albrecht D, Scholz A, et al. (2018) Global epidemiology of hyperthyroidism and hypothyroidism. Nat Rev Endocrinol 14: 301-316. https://doi.org/10.1038/nrendo.2018.18
[3]	Sarfo-Kantanka O, Sarfo FS, Ansah EO, et al. (2017) Spectrum of endocrine disorders in central Ghana. Int J Endocrinol 2017: 5470731. https://doi.org/10.1155/2017/5470731
[4]	Falhammar H, Thorén M, Calissendorff J (2013) Thyrotoxic periodic paralysis: clinical and molecular aspects. Endocrine 43: 274-284. https://doi.org/10.1007/s12020-012-9777-x
[5]	Kelley DE, Gharib H, Kennedy FP, et al. (1989) Thyrotoxic periodic paralysis. Report of 10 cases and review of electromyographic findings. Arch Intern Med 149: 2597-2600. https://doi.org/10.1001/archinte.149.11.2597
[6]	McFadzean AJ, Yeung R (1967) Periodic paralysis complicating thyrotoxicosis in Chinese. Br Med J 1: 451-455. https://doi.org/10.1136/bmj.1.5538.451
[7]	Okinaka S, Shizume K, Iino S, et al. (1957) The association of periodic paralysis and hyperthyroidism in Japan. J Clin Endocrinol Metab 17: 1454-1459. https://doi.org/10.1210/jcem-17-12-1454
[8]	Tessier JJ, Neu SK, Horning KK (2010) Thyrotoxic periodic paralysis (TPP) in a 28-year-old Sudanese man started on prednisone. J Am Board Fam Med 23: 551-554. https://doi.org/10.3122/jabfm.2010.04.090220
[9]	Manoukian MA, Foote JA, Crapo LM (1999) Clinical and metabolic features of thyrotoxic periodic paralysis in 24 episodes. Arch Intern Med 159: 601-606. https://doi.org/10.1001/archinte.159.6.601
[10]	Siddamreddy S, Dandu VH (2022) Thyrotoxic periodic paralysis. StatPearls Publishing. Available from: https://www.ncbi.nlm.nih.gov/books/NBK560670/
[11]	Glass J, Osipoff J (2020) Thyrotoxic periodic paralysis presenting in an African-American teenage male: case report. Int J Pediatr Endocrinol 2020: 7. https://doi.org/10.1186/s13633-020-00077-3
[12]	Chatot-Henry C, Smadja D, Longhi R, et al. (2000) Thyrotoxic periodic paralysis. Two news cases in black race patients. Rev Med Interne 21: 632-634. (Article in French) https://doi.org/10.1016/S0248-8663(00)80010-7
[13]	Iheonunekwu NC, Ibrahim TM, Davies D, et al. (2004) Thyrotoxic hypokalaemic paralysis in a pregnant Afro-Carribean woman: A case report and review of the literature. West Indian Med J 53: 47-49.
[14]	Ngonyani M, Manji H (2021) Thyrotoxicosis: an unusual cause of periodic paralysis (a case report from Tanzania). PAMJ Clin Med 7. https://doi.org/10.11604/pamj-cm.2021.7.24.31614
[15]	Sow M, Diagne N, Djiba B, et al. (2020) Thyrotoxic hypokalemic periodic paralysis in two African black women. Pan Afr Med J 37: 207. (Article in French) https://doi.org/10.11604/pamj.2020.37.207.24900
[16]	Schoumaker V, Bovy P (2013) Clinical case of the month. Thyrotoxic periodic paralysis. Report of a case in a Somalian male. Rev Med Liege 68: 402-407. (Article in French)
[17]	Fontaine B (2008) Periodic paralysis. Adv Genet 63: 3-23. https://doi.org/10.1016/s0065-2660(08)01001-8
[18]	Venance SL, Cannon SC, Fialho D, et al. (2006) The primary periodic paralyses: diagnosis, pathogenesis and treatment. Brain 129: 8-17. https://doi.org/10.1093/brain/awh639
[19]	Ptáček LJ, Tawil R, Griggs RC, et al. (1994) Dihydropyridine receptor mutations cause hypokalemic periodic paralysis. Cell 77: 863-868. https://doi.org/10.1016/0092-8674(94)90135-x
[20]	Fouad G, Dalakas M, Servidei S, et al. (1997) Genotype-phenotype correlations of DHP receptor α1-subunit gene mutations causing hypokalemic periodic paralysis. Neuromuscul Disord 7: 33-38. https://doi.org/10.1016/s0960-8966(96)00401-4
[21]	Jurkat-rott K, Lehmann-horn F, Elbaz A, et al. (1994) A calcium channel mutation causing hypokalemic periodic paralysis. Hum Mol Genet 3: 1415-1419. https://doi.org/10.1093/hmg/3.8.1415
[22]	Sternberg D, Maisonobe T, Jurkat-Rott K, et al. (2001) Hypokalaemic periodic paralysis type 2 caused by mutations at codon 672 in the muscle sodium channel gene SCN4A. Brain 124: 1091-1099. https://doi.org/10.1093/brain/124.6.1091
[23]	Bulman DE, Scoggan KA, Van Oene MD, et al. (1999) A novel sodium channel mutation in a family with hypokalemic periodic paralysis. Neurology 53: 1932-1936. https://doi.org/10.1212/wnl.53.9.1932
[24]	Jurkat-Rott K, Mitrovic N, Hang C, et al. (2000) Voltage-sensor sodium channel mutations cause hypokalemic periodic paralysis type 2 by enhanced inactivation and reduced current. Proc Natl Acad Sci U S A 97: 9549-9554. https://doi.org/10.1073/pnas.97.17.9549
[25]	Zhao SX, Liu W, Liang J, et al. (2019) Assessment of molecular subtypes in thyrotoxic periodic paralysis and Graves disease among Chinese Han adults: A population-based genome-wide association study. JAMA Netw open 2: e193348. https://doi.org/10.1001/jamanetworkopen.2019.3348
[26]	Li GHY, Cheung CL, Zhao SX, et al. (2020) Genome-wide meta-analysis reveals novel susceptibility loci for thyrotoxic periodic paralysis. Eur J Endocrinol 183: 607-617. https://doi.org/10.1530/EJE-20-0523
[27]	Park S, Kim TY, Sim S, et al. (2017) Association of KCNJ2 genetic variants with susceptibility to thyrotoxic periodic paralysis in patients with Graves' disease. Exp Clin Endocrinol Diabetes 125: 75-78. https://doi.org/10.1055/s-0042-119527
[28]	Cheung CL, Lau KS, Ho AYY, et al. (2012) Genome-wide association study identifies a susceptibility locus for thyrotoxic periodic paralysis at 17q24.3. Nat Genet 44: 1026-1029. https://doi.org/10.1038/ng.2367
[29]	Hsu YJ, Lin YF, Chau T, et al. (2003) Electrocardiographic manifestations in patients with thyrotoxic periodic paralysis. Am J Med Sci 326: 128-132. https://doi.org/10.1097/00000441-200309000-00004
[30]	Fisher J (1982) Thyrotoxic periodic paralysis with ventricular fibrillation. Arch Intern Med 142: 1362-1364. https://doi.org/10.1001/archinte.1982.00340200130024
[31]	Pompeo A, Nepa A, Maddestra M, et al. (2007) Thyrotoxic hypokalemic periodic paralysis: An overlooked pathology in western countries. Eur J Intern Med 18: 380-390. https://doi.org/10.1016/j.ejim.2007.03.003
[32]	Talbott JH (1941) Periodic paralysis. Medicine 20: 85-143.
[33]	Schmidt ST, Ditting T, Deutsch B, et al. (2015) Circadian rhythm and day to day variability of serum potassium concentration: a pilot study. J Nephrol 28: 165-172. https://doi.org/10.1007/s40620-014-0115-7
[34]	Gumz ML, Rabinowitz L (2013) Role of circadian rhythms in potassium homeostasis. Semin Nephrol 33: 229-236. https://doi.org/10.1016/j.semnephrol.2013.04.003
[35]	Edelman J, Stewart-Wynne EG (1981) Respiratory and bulbar paralysis with relapsing hyperthyroidism. Br Med J 283: 275-276. https://doi.org/10.1136/bmj.283.6286.275-a
[36]	Lam L, Nair RJ, Tingle L (2006) Thyrotoxic periodic paralysis. Proc 19: 126-129. https://doi.org/10.1080/08998280.2006.11928143
[37]	Tella SH, Kommalapati A (2015) Thyrotoxic periodic paralysis: An underdiagnosed and under-recognized condition. Cureus 7: e342. https://doi.org/10.7759/cureus.342

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Medical Science

0.4

Metrics

Article views(1890) PDF downloads(116) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(2) / Tables(1)

AIMS Medical Science

Thyrotoxic periodic paralysis together with thyrotoxic heart disease in a Ghanaian man: case report and literature review

Related Papers:

Abstract

1. Introduction

2. Preliminaries and formulation of the problem

3. Some properties of $Sol(K, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)$

4. Main results

5. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Medical Science

Thyrotoxic periodic paralysis together with thyrotoxic heart disease in a Ghanaian man: case report and literature review

Related Papers:

Abstract

1. Introduction

2. Preliminaries and formulation of the problem

3. Some properties of Sol(K,ˆw+A(.),ψ) Sol(K, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)

4. Main results

5. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

3. Some properties of $Sol(K, \widehat{\mathfrak{w}}+\mathcal{A}(.), \psi)$